Applications of Holomorphic Functions in Geometry 9819912989, 9789819912988

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Frontiers in Mathematics

Arif Salimov

Applications of Holomorphic Functions in Geometry

Frontiers in Mathematics Advisory Editors William Y. C. Chen, Nankai University, Tianjin, China Laurent Saloff-Coste, Cornell University, Ithaca, NY, USA Igor Shparlinski, The University of New South Wales, Sydney, NSW, Australia Wolfgang Sprößig, TU Bergakademie Freiberg, Freiberg, Germany

This series is designed to be a repository for up-to-date research results which have been prepared for a wider audience. Graduates and postgraduates as well as scientists will benefit from the latest developments at the research frontiers in mathematics and at the “frontiers” between mathematics and other fields like computer science, physics, biology, economics, finance, etc. All volumes are online available at SpringerLink.

Arif Salimov

Applications of Holomorphic Functions in Geometry

Arif Salimov Department of Algebra and Geometry Baku State University Baku, Azerbaijan

ISSN 1660-8046 ISSN 1660-8054 (electronic) Frontiers in Mathematics ISBN 978-981-99-1298-8 ISBN 978-981-99-1296-4 (eBook) https://doi.org/10.1007/978-981-99-1296-4 © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 This work is subject to copyright. All rights are solely and exclusively licensed by the Publisher, whether the whole or part of the material is concerned, specifically the rights of translation, reprinting, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other physical way, and transmission or information storage and retrieval, electronic adaptation, computer software, or by similar or dissimilar methodology now known or hereafter developed. The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use. The publisher, the authors, and the editors are safe to assume that the advice and information in this book are believed to be true and accurate at the date of publication. Neither the publisher nor the authors or the editors give a warranty, expressed or implied, with respect to the material contained herein or for any errors or omissions that may have been made. The publisher remains neutral with regard to jurisdictional claims in published maps and institutional affiliations. This book is published under the imprint Birkhäuser, www.birkhauser-science.com by the registered company Springer Nature Singapore Pte Ltd. The registered company address is: 152 Beach Road, #21-01/04 Gateway East, Singapore 189721, Singapore

Preface

This book is intended to provide a systematic introduction to the theory of holomorphic manifolds. The book furnishes detailed information on holomorphic functions in algebras and discusses some of the areas in geometry with applications. Its goal is to expound the recent developments in applications of holomorphic functions in the theory of hypercomplex and anti-Hermitian manifolds as well as in the geometry of bundles. In spite of the geometric applications of holomorphic functions that are mainstream in the investigation of differential geometry, holomorphic manifolds and their recent applications are not so well known yet. The theory of holomorphic manifolds is more than 60 years old. The initial notion of a holomorphic manifold over algebras appeared in the 1960s in a series of papers of A.P. Shirokov and culminated in the book [1]. Since then, the subject has enjoyed a rapid development. Holomorphic manifolds relate with such topics as anti-Hermitian metrics and lifting of differential-geometrical objects to vector bundle. The book is organized into three chapters. Chapter 1 presents the fundamental notions and some theorems concerning holomorphic functions and holomorphic manifolds, which are needed for the later applications. Section 1.1 gives the basic definitions and theorems on hypercomplex algebras. Section 1.2 is devoted to the study of holomorphic functions in algebra. Section 1.3 introduces the hypercomplex structures on manifolds. Section 1.4 treats manifolds with integrable regular hypercomplex structures. We show that such a manifold is a realization of a holomorphic manifold over algebra. Section 1.5 is devoted to the study of pure tensor fields. Here, we find the explicit expression of the pure tensor field with respect to the regular hypercomplex structures, which shows that the pure tensor fields on real manifolds are a realization of hypercomplex tensors. Section 1.6 discusses holomorphic hypercomplex tensor fields, and by using the Tachibana operator, we give the condition of holomorphic tensors in real coordinates. In Sect. 1.7, we consider pure connections which are realizations of the hypercomplex connections. Section 1.8 is devoted to the study of pure hypercomplex torsion tensors. In Sect. 1.9, we give a realization of holomorphic hypercomplex connections by using the pure curvature tensors. Finally, in Sect. 1.10, we consider some properties of pure curvature tensors. The main theorem of this section is that the curvature tensor of a holomorphic connection is holomorphic. v

vi

Preface

In Chap. 2, we study the pseudo-Riemannian metric on holomorphic manifolds. In Sect. 2.1, we give the condition for a hypercomplex anti-Hermitian metric to be holomorphic. In Sect. 2.2, we discuss complex Norden manifolds. We define the twin Norden metric. The main theorem of this section is that the Levi-Civita connection of KählerNorden metric coincides with the Levi-Civita connection of twin Norden metric. In Sect. 2.3, we consider Norden-Hessian structures. We give the condition for a NordenHessian manifold to be Kähler. Section 2.4 is devoted to the analysis of twin Norden metric connections with torsion. In Sects. 2.5-2.10, we focus our attention to pseudoRiemannian 4-manifolds of neutral signature. The main purpose of these sections is to study complex Norden metrics on 4-dimensional Walker manifolds. In the first part of Chap. 3, we focus on lifts from a manifold to its tensor bundle. Some introductory material concerning the tensor bundle is provided in Sect. 3.1. Section 3.2 is devoted to the study of the complete lifts of (1, 1)-tensor fields along cross-sections in the tensor bundle. In Sect. 3.3, we study holomorphic cross-sections of tensor bundles. In the second part of Chap. 3, we concentrate our attention to lifts from a manifold to its tangent bundles of orders 1 and 2 by using the realization of holomorphic manifolds. The main purpose of Sects. 3.4-3.9 is to study the differential-geometrical objects on the tangent bundle of order 1 corresponding to dual-holomorphic objects of the dual-holomorphic manifold. As a result of this approach, we find a new class of lifts, that is, deformed complete lifts of functions, vector fields, forms, tensor fields and linear connections in the tangent bundle of order 1. Section 3.10 is devoted to the study of holomorphic metrics in the tangent bundle of order 2 (that is, in the bundle of 2-jets) by using the Tachibana operator. By using the algebraic approach, the problem of deformed lifts of functions, vector fields and 1-forms is solved in Sects. 3.11-3.12. In Sect. 3.13, we investigate the complete lift of the almost complex structure to cotangent bundle and prove that it is a transfer by a symplectic isomorphism of complete lift to tangent bundle if the symplectic manifold with almost complex structure is an almost holomorphic A-manifold. Finally, in Sect. 3.14, we transfer via the differential of the musical isomorphism defined by pseudoRiemannian metrics the complete lifts of vector fields and almost complex structures from the tangent bundle to the cotangent bundle. The author believes that geometric applications of holomorphic functions are a very fruitful research domain and provides many new problems in the study of modern differential geometry. Researchers in the geometry, algebra, topology and physics communities may find the book useful as a self-study guide. I warmly thank Simona-Luiza Dru¸ta˘ -Romaniuc from the Department of Mathematics and Informatics, Gheorghe Asachi Technical University of Iasi, Romania, for reading the manuscript and making useful corrections in it. Baku, Azerbaijan

Arif Salimov

Preface

vii

Reference 1. Vishnevskii, V.V., Shirokov, A.P., Shurygin, V.V.: Spaces over algebras. Kazanskii Gosudarstvennii Universitet, Kazan (1985)

Contents

1 Holomorphic Manifolds Over Algebras . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.1 Some Basic Concepts of Algebra . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.2 Holomorphic Functions . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.3 Algebraic
-Structures on Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.4 Integrable Structures and Holomorphic Manifolds . . . . . . . . . . . . . . . . . . . 1.5 Pure Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.6 Realizations of Holomorphic Tensors . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.7 Pure Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.8 Torsion Tensors of Pure Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.9 Holomorphic Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 1.10 Holomorphic Curvature Tensor . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

1 1 4 7 10 11 15 20 23 25 26

2 Anti-Hermitian Geometry . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.1 Equivalence of Holomorphic and Anti-Kähler Conditions . . . . . . . . . . . . 2.2 Kähler-Norden Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.3 Hessian-Norden Structures . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.4 Twin Norden Metric Connections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.5 Norden-Walker Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.6 Kähler-Norden-Walker Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.7 On Curvatures of Norden-Walker Metrics . . . . . . . . . . . . . . . . . . . . . . . . . . 2.8 Isotropic Anti-Kähler Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.9 Quasi-Kähler-Norden-Walker Manifolds . . . . . . . . . . . . . . . . . . . . . . . . . . . . 2.10 The Goldberg Conjecture . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

31 31 37 40 43 49 53 54 55 57 58

3 Problems of Lifts . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.1 Tensor Bundles . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.2 Complete Lifts of (1,1)-Tensor Fields . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.3 Holomorphic Cross-Sections . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.4 Dual-Holomorphic Functions and Tangent Bundles of Order 1 . . . . . . . . 3.5 Deformed Complete Lifts of Vector Fields . . . . . . . . . . . . . . . . . . . . . . . . . 3.6 Deformed Complete Lifts of Tensor Fields of Type (1,1) . . . . . . . . . . . . .

61 62 66 70 72 75 77 ix

x

Contents

3.7 3.8 3.9 3.10 3.11 3.12

Deformed Complete Lifts of 1-Forms . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . Deformed Complete Lifts of Riemannian Metrics . . . . . . . . . . . . . . . . . . . Deformed Complete Lifts of Connections . . . . . . . . . . . . . . . . . . . . . . . . . . Holomorphic Metrics in the Tangent Bundle of Order 2 . . . . . . . . . . . . . . Deformed Lifts of Vector Fields in the Tangent Bundle of Order 2 . . . . Deformed Complete and Intermediate Lifts of 1-Forms in the Tangent Bundle of Order 2 . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . 3.13 Problems of Lifts in Symplectic Geometry . . . . . . . . . . . . . . . . . . . . . . . . . 3.14 Anti-Kähler Manifolds and Musical Isomorphisms . . . . . . . . . . . . . . . . . .

79 80 81 84 89 93 98 104

References . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

109

Index . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . . .

113

About the Author

Arif Salimov is Full Professor and Head of the Department Algebra and Geometry, Faculty of Mechanics and Mathematics, Baku State University. An Azerbaijani/Soviet mathematician, honoured scientist of Azerbaijan, he is known for his research in differential geometry. He earned his B.Sc. degree from Baku State University, Azerbaijan, in 1978, a Ph.D. and Doctor of Sciences (Habilitation) degrees in geometry from Kazan State University, Russia, in 1984 and 1998, respectively. His advisor was Vladimir Vishnevskii. He is an author/co-author of more than 100 research papers. His primary areas of research are theory of lifts in tensor bundles, geometrical applications of tensor operators, special Riemannian manifolds, indefinite metrics and general geometric structures on manifolds (almost complex, almost product, hypercomplex, Norden structures, etc.).

xi

1

Holomorphic Manifolds Over Algebras

In this chapter, we give the fundamental notions and some theorems concerning holomorphic functions and holomorphic manifolds which will be needed for the later applications. In Sect. 1.1, we give the basic definitions and theorems on hypercomplex algebras. Section 1.2 is devoted to the study of holomorphic functions in algebra. In Sect. 1.3, we introduce the hypercomplex structures on manifolds. Section 1.4 treats manifolds with integrable regular hypercomplex structures. We show that such a manifold is a realization of a holomorphic manifold over algebra. Section 1.5 is devoted to the study of pure tensor fields. We find the explicit expression of the pure tensor field with respect to the regular hypercomplex structures, and we show that the pure tensor fields on real manifolds are a realization of hypercomplex tensors. In Sect. 1.6, we discuss holomorphic hypercomplex tensor fields, and using the Tachibana operator, we give the condition of holomorphic tensors in real coordinates. In Sect. 1.7, we consider pure connections which are realizations of the hypercomplex connections. Section 1.8 is devoted to the study of pure hypercomplex torsion tensors. In Sect. 1.9, we give a realization of holomorphic hypercomplex connections by using the pure curvature tensors. In the last Sect. 1.10, we consider some properties of pure curvature tensors. The main theorem of this section is that the curvature tensor of a holomorphic connection is holomorphic. Finally, we also consider a holomorphic manifold of hypercomplex dimension 1, and we show that the hypercomplex connection on such a manifold is holomorphic if and only if the real manifold is locally flat.

1.1

Some Basic Concepts of Algebra

We consider an m-dimensional associative algebra Am over the field of real numbers R γ (hypercomplex algebra) with basis {eα }, α = 1, ..., m and structure constants Cαβ :

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Salimov, Applications of Holomorphic Functions in Geometry, Frontiers in Mathematics, https://doi.org/10.1007/978-981-99-1296-4_1

1

2

1 Holomorphic Manifolds Over Algebras γ

eα eβ = Cαβ eγ . γ

We note that Cαβ are components of the tensor of type (1,2) in the vector space of Am . In this work, we suppose Am is an algebra with the unit e1 = 1. We introduce the matrices γ α = (C γ ), α, β, γ = 1, ..., m, Cα = (Cαβ ), C βα

(1.1)

α . Then the where γ denotes rows and β denotes columns of matrices Cα and C associativity condition σ σ σ σ τ (eα eβ )eγ = eα (eβ eγ ) ⇔ Cαβ eσ eγ = eα Cβγ eσ ⇔ Cαβ Cστ γ = Cβγ Cασ

can be written in one of the following three equivalent forms: σ Cα Cβ = Cαβ Cσ ,

(1.2)

αT C βT = C γ C T C αβ γ ,

(1.3)

β = C β Cα , Cα C

(1.4)

αT is the tranpose of C α . From 1 = εσ eσ follows that where C τ eα = εσ eσ eα = εσ Cστ α eτ , eα = eα εσ eσ = εσ Cασ eτ (eα = δατ eτ )

or τ σ = I (εσ Cστ α = εσ Cασ εσ Cσ = εσ C = δατ ),

(1.5)

where δατ is the Kronecker delta. αT , respectively T (A) be the algebras of matrices of types Cα and C Let C(A) and C (see (1.2) and (1.3)). A mapping ρ1 : Am → C(A) : Am  a = a σ eσ → a σ Cσ = C(a) ∈ C(A), a 1 , ..., a m ∈ R is called a regular representation of type I. The representation ρ1 is obviously isomorphic. T (A) of type II: By similar devices, we introduce a regular representation ρ2 : Am → C σT = C T (a) ∈ C T (A), a 1 , ..., a m ∈ R Am  a = a σ eσ → a σ C which is also isomorphic. In this work, the representations appearing in the discussion will be supposed to be of type I. Notice that a regular representation of type I is usually called simply a regular representation.

1.1

Some Basic Concepts of Algebra

3

σ , a 1 , ..., a m ∈ R belongs to the commutator of the From (1.4), we see that all A = a σ C algebra C(A). Conversely, using (1.5), we easily see that if AC = CA for any C ∈ C(A), σ Aγ . α . In fact, we put C = Cα , α = 1, ..., m, then we have Aσγ C γ = Cαγ then A = a α C αβ β Contracting this equation with εβ , we find γ

γ

σ Aβ ε β , Aσγ Cαβ εβ = Cαγ σ γ a , Aσγ δαγ = Cαγ σ γ , ) = aγ C A = (Aσγ ) = a γ (Cαγ γ

where a γ = Aβ εβ . Thus, we have Theorem 1.1 Let A be a square matrix of order m. Then ACα = Cα A if and only if γ . A = aγ C By similar devices, we have αT A if and only if αT = C Theorem 1.2 Let A be a square matrix of order m. Then AC γT . A = aγ C Now, we restrict ourselves to the consideration of commutative hypercomplex algebras. In this and next sections, we always assume that the Am is a commutative hypercomplex algebra. The commutativity condition eα eβ = eβ eα can be written in the following form: γ γ α ) Cαβ = Cβα (Cα = C

(1.6)

Using (1.6), from (1.4), we have γ

γ σ σ Cασ Cβδ = Cβσ Cαδ (Cα Cβ = Cβ Cα ).

(1.7)

α , then we have C T (A) = C T (A). Therefore, C T (A) is called a transSince Cα = C pose regular representations of commutative algebras. From Theorems 1.1 and 1.2, we have. Theorem 1.3 Let Am be a commutative hypercomplex algebra. Then C(A) and C T (A) are maximal commutative subalgebras of the algebra of square matrices of order m. Among commutative algebras, a special role is played by Frobenius algebras for which there exist constants λγ such that γ

ϕαβ = Cαβ λγ

(1.8)

4

1 Holomorphic Manifolds Over Algebras

form a nonsingular symmetric matrix and can be taken as components of the metric tensor of the Frobenius metric. Then, along with the basis {eα }, we can introduce the dual basis {eα }, where eα = ϕ αβ eβ . For the Frobenius metric, we have the relations σ ϕασ Cγβ = ϕβσ Cγσ α , ϕ ασ Cγβσ = ϕ βσ Cγασ , eα eβ α γ = Cβγ e , eα eβ = ϕ ασ Cγβσ eγ ,ϕαβ εβ =λα ,

(1.9)

where εβ are components of e1 = 1.

1.2

Holomorphic Functions

Let z = x α eα be an algebraic variable in Am , where x α (α = 1, ..., m) are real variables. We introduce an algebraic function w = w(z) ∈ Am of variable z in the following form: w = y β (x)eβ , where y β (x) = y β (x 1 , ..., x m ), β = 1, ..., m are real-valued C ∞ -functions. Let dz = dx α eα and dw = dy α eα be the differentials of z and w, respectively. We shall say that the function w = w(z) is a holomorphic function if there exists a functions w (z) such that dw = w (z)dz

(1.10)

We shall call w (z) the derivative of w = w(z). Theorem 1.4 [28, 71, 78, p.87] The algebraic function w = w(z) is A-holomorphic if and only if the Scheffers conditions (the generalized Cauchy-Riemann conditions) hold: Cα D = DCα where D =



∂ yα ∂xβ



(1.11)

is the Jacobian matrix of y α (x).

Proof Let w = w(z) be a holomorphic function. We put w (z) = w˜ α eα . Then from (1.10), we have dw = dy α eα =

∂ yα β γ dx eα = w˜ α eα dx β eβ = w˜ α dx β Cαβ eγ , ∂xβ

From here, we obtain ∂ yγ γ = w˜ α Cαβ ∂xβ

(1.12)

1.2

Holomorphic Functions

5

Thus, the hypercomplex  α  function w = w(z) is a holomorphic function if and only if the Jakobian matrix ∂∂ xy β has the form (1.12). Contracting (1.12) with εβ (1 = εβ eβ ) and using (1.5), we find w˜ γ = εβ

∂ yγ , ∂xβ

i.e. w (z) = εβ

∂ yγ eγ . ∂xβ

Now applying Theorem 1.1 to (1.12), we see that the condition (1.12) is equivalent to the Scheffers conditions. Thus, the theorem is proved. The concept of the holomorphic hypercomplex functions can be immediately extended to the case of several algebraic variables. Let z v = x (v−1)m+β eβ (v = 1, ..., r ) be an r variables in Am . In fact the functions wu (z 1 , ..., z r ) = (u−1)m+α 1 r m (x , ..., x )eα , u = 1, ..., r are a holomorphic functions of the variables y z 1 , ..., z r if and only if Cα Du,v = Du,v Cα

(1.13)

for any u and v, where  Du,v =

 ∂ y (u−1)m+α , u, v = 1, ..., r ∂ x (v−1)m+β

and (see proof of Theorem 1.4) (u−1)m+α ∂wu β ∂y = ε eα ∂z v ∂ x (v−1)m+β

(1.14) (u−1)m+α

Remark 1.1 From (1.14), it follows that the Jacobian matrix of functions εβ ∂∂ xy (v−1)m+β    ∂ Du,v ∂ 2 y (u−1)m+α = εγ ∂ x (w−1)m+γ has components of the form D = εγ ∂ x (w−1)m+γ , and there(v−1)m+β ∂x fore, it also satisfies the Scheffers conditions (1.13), i.e. exists the successive derivatives ∂ 2 wu ∂ 3 wu u u 1 r ∂z v ∂z t , ∂z v ∂z t ∂z l , ... of w = w (z , ..., z ). Example 1.1 We note that, in particular, if A2 = C, where C is the complex algebra, then the Scheffers conditions reduce to the Cauchy-Riemann conditions. In fact, by virtue of

6

1 Holomorphic Manifolds Over Algebras

 C1 =

1 C1 C11 12 2 C2 C11 12



 =

     1 C1 C21 10 0 −1 22 , C2 = = 2 C2 C21 01 10 22

from (1.11), we have ∂ y2 ∂ y2 ∂ y1 ∂ y1 = 2, = − 2, 1 1 ∂x ∂x ∂x ∂x where z = x 1 + i x 2 , w = y 1 (x 1 , x 2 ) + i y 2 (x 1 , x 2 ), i 2 = −1. Example 1.2. Let A4 = B(1, i 1 , i 2 , e) be a bicomplex algebra with canonical basis {1, i 1 , i 2 , e}, i 12 = i 22 = −1, i 1 i 2 = i 2 i 1 = e, e2 = 1. The algebra of bicomplex numbers is the first nontrivial complex Clifford algebra (and the only commutative one) and has recently been applied to quantum mechanics and fractal theory. Using the Scheffers  ∂ fα conditions Cα D = DCα , where D = ∂ x β is the Jacobian matrix of bicomplex function F = f 1 (x 1 , x 2 , x 3 , x 4 ) + i 1 f 2 (x 1 , x 2 , x 3 , x 4 ) + i 2 f 3 (x 1 , x 2 , x 3 , x 4 ) + e f 4 (x 1 , x 2 , x 3 , x 4 ) and ⎛

1 ⎜ ⎜0 C1 = ⎜ ⎝0 0 ⎛ 0 ⎜ ⎜0 C3 = ⎜ ⎝1 0

0 1 0 0 0 0 0 1

⎞ ⎛ ⎞ 0 0 −1 0 0 ⎟ ⎜ ⎟ 0⎟ ⎜1 0 0 0 ⎟ ⎟, C2 = ⎜ ⎟, ⎝ 0 0 0 −1 ⎠ 0⎠ 1 00 10 ⎛ ⎞ ⎞ −1 0 00 0 1 ⎜ ⎟ ⎟ 0 −1 ⎟ ⎜ 0 0 −1 0 ⎟ ⎟, C 4 = ⎜ ⎟, ⎝ 0 −1 0 0 ⎠ 0 0 ⎠ 0 0 10 0 0

0 0 1 0

we have: the bicomplex function F is bicomplex-holomorphic if and only if the function F satisfies the following Scheffers conditions: ∂ f2 ∂ f3 ∂ f4 ∂ f1 ∂ f3 ∂ f2 ∂ f4 ∂ f1 = = = , = = − = − , ∂x1 ∂x2 ∂x3 ∂x4 ∂x2 ∂x4 ∂x1 ∂x3 ∂ f2 ∂ f3 ∂ f4 ∂ f1 ∂ f4 ∂ f2 ∂ f3 ∂ f1 = =− 1 =− 2, 4 = =− 3 =− 2 3 4 1 ∂x ∂x ∂x ∂x ∂x ∂x ∂x ∂x We note that the above Scheffers conditions are equivalent to the following bicomplex Cauchy-Riemann equations:

1.3

Algebraic -Structures on Manifolds

7

∂ψ1 ∂ψ2 ∂ψ1 ∂ψ2 = , =− , ∂q1 ∂q2 ∂q2 ∂q1 where F = ψ1 (q1 , q2 ) + i 2 ψ2 (q1 , q2 ), ψ1 (q1 , q2 ) = f 1 + i 1 f 2 , ψ2 (q1 , q2 ) = f 3 + i 1 f 4 , q1 = x 1 + i 1 x 2 , q2 = x 3 + i 1 x 4 .

1.3

Algebraic
-Structures on Manifolds

If a collection of (1,1)-tensor (affinor) fields ϕ , ϕ , ..., ϕ is given on a smooth manifold M m

1 2

of dimension n, then one says that a polyaffinor structure (or -structure) is given on M:



= ϕ ij , α = 1, ..., m; i, j = 1, ..., n. α

If there exists an adapted frame {X i }, i = 1, ..., n such that the each structure affinors ϕ of -structure has constant components ϕ ij = const with respect to this frame, then α

α

the -structure is called a rigid structure [36]. In general, the frame {X i } is not a natural frame. If the adapted frame {X i } is a natural frame, i.e. {X i } = {∂i }, ∂i ϕ ij = 0, then the α

-structure is said to be integrable. It is clear that the integrable -structure is rigid, and the contrary statement is true only under some additional conditions on -structure. For example, if = ϕ, i.e. if the -structure contains only one affinor ϕ, and if there exists a torsion-free connection ∇ on M preserving the rigid ϕ-structure, then the ϕ-structure is integrable [73]. It is well known that for simplest rigid ϕ-structures (almost complex and almost paracomplex structures, etc.), the integrability is equivalent to the vanishing of the Nijenhuis tensor. Definition 1.1 Let ∇ be a linear connection on M. ∇ is called -connection with respect to the -structure, if ∇ϕ = 0 for any ϕ ∈ . Definition 1.2 -structure on M is called almost integrable, if there exists a torsion-free

-connection. We note that for some simplest -structures (regular -structure [27]), the concepts of integrability and almost integrability are equivalent. Let Am be a commutative hypercomplex algebra with the unit e1 = 1. An almost hypercomplex structure on M is a polyaffinor -structure such that γ ϕ ◦ ϕ = Cαβ ϕ, α

β

γ

(1.15)

8

1 Holomorphic Manifolds Over Algebras

i.e. if there exists an isomorphism Am ↔ , where ϕ = I d M , ϕ , ... ϕ are structure 1

affinors corresponding to the base elements e1 = 1, e2 , ..., em ∈ Am .

2

m

Definition 1.3 An almost hypercomplex structure on M is saidto bean r -regular -structure (or for simplicity a regular -structure) if the matrices ϕ = ϕ

n × n simultaneously reduced to the form ⎛

Cα 0   ⎜ 0 Cα ϕ = ϕ ij == ⎜ ⎜ α α ⎝· · · · · · 0 0

α

i α j

, α = 1, ..., m of order

⎞ ··· 0 ⎟ ··· 0 ⎟ ⎟, α = 1, ..., m; i, j = 1, ..., n · · · · · ·⎠ · · · Cα

(1.16)

  γ with respect to the adapted frame {X i }, where Cα = Cαβ is the regular representation of Ar and r is a number of Cα —blocks. From Definition 1.3 immediately follows that the regular -structures are rigid structures. In particular, for almost complex and paracomplex structures, the condition (1.16) immediately follows from (1.15), i.e. almost complex and paracomplex structures (see [10, 19]) on M(dim

M = n = 2r ) automatically are regular structures. Let now = ϕ , α = 1, ..., m be a regular -structure on M. Then from (1.16), we α

have n = mr (n = dim M, m = dim Am ),

(1.17)

where r is a number of the blocks Cα . Thus, the condition (1.17) is a necessary condition for the existence of regular -structures on Mmr , and in this case, we have i = (u − 1)m + α (i = 1, ..., n; u = 1, ..., r ; α = 1, ..., m) or i = uα, j = vβ, k = wγ , ... In other words, the structure affinors ϕ have the components α

ϕ

i σ j

=ϕ σ

uα vβ

α = δvu Cσβ (δvu -Kronecker delta).

(1.18)

Let X i  = Sii X i (D et(Sii ) = 0) be a transformation of adapted frame {X i } with respect to the regular -structure. Then we have the following matrix relationship:

1.3

Algebraic -Structures on Manifolds





ϕ

i j

9

     j = Si ϕ ij S j  , 

i

(1.19)

α

α

   where (Sii ) = (Sii )−1 . If X i  is an adapted frame, then we call transformation X i → X i      

ϕ ij 

the admissible transformation. Since

=

ϕ ij

from (1.19)

for the adapted frames, we have

α

α

S ϕ = ϕ S, α

where S =



j S j



 and ϕ = α

α

(1.20)

 . Thus, we have

ϕ ij α

 Theorem 1.5 Let = ϕ be a regular -structure on Mmr . A transformation S : X i → α

X i  of adapted frames is admissible if and only if the condition (1.20) is true. Using Theorems 1.1 and 1.5, we see that the matrix S has the following special structure: α    Sii = uσ u  C σ α  (i = uα, i = u α )

(1.21)

By similar devices for inverse matrix, we have 





Sii = uu σ Cσαα (i = uα, i  = u  α  ) We can introduce the following matrices in the algebra Am :        uσ  ∗ −1 ∗ ∗ ∗   u u = S u = uu σ eσ S = S u  = u  eσ , S ∗ ∗

−1

∗

= I , where ⎞ ⎛ ⎞ · ·· 0 1 0 · ·· 0 ⎟ ⎜ ⎟   · ·· 0 ⎠ = ⎝ 0 1 · ·· 0 ⎠ = δvu . · ·· e1 0 0 · ·· 1

From here, we easily see that ⎛ e1 0 ∗ ⎜ I = ⎝ 0 e1 0 0

SS

In fact, from (1.21) and (1.22), we obtain ∗

S

u u

∗

S

u v





uε uσ u ε γ = uσ u  eσ v eε = u  v C σ ε eγ 





α uε α i i i u α u u = uσ u  C σ α  v C εβ = Si  S j = δ j = δv C 1β = δv e1 = δv .

(1.22)

10

1 Holomorphic Manifolds Over Algebras

For any vector field ξ = ξ i X i = ξ uα X uα , where {X i } is the adapted frame on Mmr , ∗

we can associate r coordinates ξ

u

(u = 1, ..., r ) in the algebra Am : ∗

ξ

u

= ξ uα eα . ∗





We easily see that, if ξ i = Sii ξ i , then ξ  

u

∗

u u

=S

 

∗

ξ u . In fact, from (1.21), we obtain





u α uα ξ = uu σ Cσαα ξ uα ξ u α = Suα

or ∗

ξ

u

 





= ξ u α eα  = uu σ Cσαα ξ uα eα  ∗



= uu σ eσ ξ uα eα = S

u u

∗

ξ u.

Let now ϕ ∈ . Then ϕ = a α ϕ . The action of operator ϕ for the vector field ξ = α

ξ i X i , i.e. the equation ηi = ϕ ij ξ j , reduces to ∗ u

η

α vβ = ηuα eα = a σ ϕ ij ξ j eα = a σ δvu Cσβ ξ eα σ

∗

= a σ δvu ξ vβ eσ eβ = a σ eσ ξ uβ eβ = a ξ u , where i = uα, j = vβ and a ∈ Am . Thus, we have Theorem 1.6 If is a regular structure on Mmr , then each tangent space Tx (Mmr ), x ∈ Mmr serves as a realization of the module Tr (Am ) over algebra Am . ∗

In particular, for the case, where ϕ = ϕ , from η α

∗ u

η

1.4

u

∗

= a ξ u , we have

∗

= eα ξ u .

Integrable Structures and Holomorphic Manifolds

 Let = ϕ be an integrable regular -structure on Mmr , and let x i = x uα and  xi

  xu α

α

= be an adapted local coordinates in Ux (x ∈ Mmr ). It is well known  the  that affinors ϕ have the constant form (1.18) with respect to the adapted frames ∂∂x i and   α ∂ , and the admissible transformation has the form i ∂x

1.5

Pure Tensors

11

∂ i ∂  = Si  i ∂xi ∂x 





i α uσ α    with Sii = ∂ xi  , Sii = uσ u  C σ α  , Si = u C σ α , i = uα, i = u α (see (1.21) and ∂x (1.22)). Then from Theorem 1.5, we have  uα   uα  ∂x ∂x C = C σ σ  α   u ∂x ∂xu α i



 

for fixed u and u  , i.e. by virtue of (1.13), the hypercomplex functions z u = x u α eα  are the holomorphic functions of z u = x uα eα . Using (1.14) and (1.22), we have 

u  α

 ∗ ∂z u u σ α u σ α ∂x α u σ α  = ε   =   =  = ε e C e δ e α α α u ασ u σ u eσ = S ∂z u ∂ x uα

u u.

By similar devices, we have ∗ ∂z u  = S ∂z u

u u .

Thus in the intersection of any two coordinate charts with local adapted coordinates      x uα and x u α of atlas on Mmr , the transition functions z u = z u (z u ) (z u = x uα eα , z u =   x u α eα  ) are A-holomorphic, i.e. a real C ∞ -manifold Mmr of dimension mr also possesses the structure of a holomorphic A-manifold X r (A) of dimension r. Thus, we have. Theorem 1.7 A realization of a holomorphic A-manifold X r (A) is a real C ∞ -manifold Mmr with integrable regular -structure.

1.5

Pure Tensors

Let now M be a C ∞ -manifold of finite dimension n. We denote by Irs (M) the module over F(M) of all C ∞ -tensor fields of type (r , s) on M, where F(M) is the algebra of C ∞ functions on M. Definition 1.4 Let ϕ be an affinor field on M, i.e. ϕ ∈ I11 (M). A tensor field t of type (r , s) is called pure tensor field with respect to ϕ if     1 2 r 1 2 r t ϕ X 1 , X 2 , ..., X s , ξ , ξ , ..., ξ = t X 1 , ϕ X 2 , ..., X s , ξ , ξ , ..., ξ .. .

12

1 Holomorphic Manifolds Over Algebras

  1 2 r = t X 1 , X 2 , ..., ϕ X s , ξ , ξ , ..., ξ   r 1 2 = t X 1 , X 2 , ..., X s ,ϕ  ξ , ξ , ..., ξ   1 r 2 = t X 1 , X 2 , ..., X s , ξ , ϕ  ξ , ..., ξ .. .



1 2



r



= t X 1 , X 2 , ..., X s , ξ , ξ , ..., ϕ ξ 1 2

r

for any X 1 , X 2 , ..., X s ∈ I10 (M) and ξ , ξ , ..., ξ ∈ I01 (M), where ϕ  is the adjoint operator of ϕ defined by (ϕ  ξ )(X ) = ξ(ϕ X ). Let x 1 , x 2 , ..., x n be a local coordinates in M. By setting X 1 = 1

r

∂ ∂ x i1

, ..., X s =

∂ ∂ x is

and ξ = d x j1 , ..., ξ = d x jr , we see that the condition of pure tensor fields may be j ... j expressed in terms of the components ϕ ij and ti11...isr as follows: j ... j

j ... j

j ... j

m j ... j

j

j m... jr

r tmi1 2 ...ir s ϕim1 = ti11m...ir s ϕim2 = ... = ti11i2 ...m ϕims = ti1 ...i2 s r ϕm1 = ti11...is

j

j j ...m

ϕm2 = ... = ti11...i2 s

j

ϕmr (1.23)

We consider for convenience sake the vector, covector and scalar fields as pure tensor fields. Different problems concerning pure tensor fields have been studied by many authors (see, for example, [6, 7, 13, 17, 18, 20, 22, 23, 45, 50, 54–56, 60, 65–67, 74, 75, 77, 80, 86]). In particular, let now t ∈ I11 (M) be a pure tensor field of type (1.1). Then the purity condition may be written as: ϕ(t X ) = t(ϕ X ). Thus, if t ∈ I11 (M) and ϕ ∈ I11 (M) satisfy the commutativity condition ϕ ◦ t = t ◦ ϕ, 

C



(1.24)

C

where (ϕ ◦ t)X = ϕ ⊗ t X = ϕ(t X ) (⊗ is a tensor product with a contraction C), then t is pure with respect to ϕ, and conversely, ϕ is also pure with respect to t. From (1.24), it follows easily that ϕ itself and the unit affinor field I are examples of the pure tensor field. Also, from (1.24), we have: if ϕ is a regular affinor field, i.e. det(ϕ ij ) = 0, then the affinor field ϕ −1 whose components are given by the elements of the inverse matrix of ϕ is also pure.

1.5

Pure Tensors

13

In particular, being applied to a (1,  1)—tensor field t, the purity condition with respect to the regular -structure =

ϕ γ

means that in the local coordinates, the following

conditions should hold: ϕ tm j γ

i m

We set i = uα, j = vβ, m = wσ and ϕ

= tmi ϕ

m γ j

i γ j

(1.25)

α . Then, from (1.25), we have = δvu Cγβ

wσ u α uα w σ uσ α uα σ δw Cγ σ = twσ δv Cγβ ⇔ tvβ Cγ σ = tvσ Cγβ . tvβ

(1.26)

Using contraction with εβ and (1.5), from (1.26), we have uσ β α uα σ β uα σ uα tvβ ε Cγ σ = tvσ Cγβ ε = tvσ δγ = tvγ

or σ

uα α t ij = tvβ = Iuv Cσβ ,

(1.27)

σ

uσ ε γ . Thus, a pure tensor field t ∈ I1 (M) has the form (1.27). where Iuv = tvγ 1

σ

Conversely, from (1.27), it follows that the tensor field t of type (1.1) is pure if I are arbitrary functions. In fact, substituting (1.27) into (1.25), we find  ε ε  ε σ u α u α w σ u σ α α σ Iw v C εβ δw C γ σ = Iw C εσ δv C γβ ⇔ Iv C εβ C γ σ − C εσ C γβ = 0 .

(1.28)

  σ Cα = Cα Cσ ⇔ C C = C C , we see Since Am is a commutative algebra Cεβ γ ε ε γ γσ εσ γβ ε

that Eq. (1.28) is satisfied for arbitrary functions I uv . Thus, the tensor field t of type (0,2) is pure with respect to the regular -structure if and only if t has the form (1.27). ∗

In the case g ∈ I02 (M), by similar devices, we see that the tensor field g of type (0,2) is pure if and only if g has the form σ gi j = guαvβ = Iuvσ Cαβ

for any functions Iuvσ = guγ vσ εγ (for more details see Chap. 2). In the case the situation is very difficult. The purity condition of G is given by Gm j ϕ γ

In a similar way, we have

i m

= G im ϕ

m . γ j

14

1 Holomorphic Manifolds Over Algebras

G uσ vβ Cγασ = G uαvσ Cγβσ .

(1.29)

After contraction of (1.29) with any covector λα , we find  σ  σ ϕγ σ G uσ vβ = Iuv Cγβσ Iuv = G uαvσ λα ,

(1.30)

  where ϕγ σ = λα Cγασ . If Det ϕγ σ = 0, i.e. if Am is a Frobenius algebra, then from (1.30), we have the following solution: σ

G i j = G uαvβ = I

uv

Cγβσ ϕ γ α .

(1.31) ∗

Conversely, from (1.31) by virtue of (1.7), it follows that the tensor field G ∈ I20 (M) ∗

σ

is pure if I uv are arbitrary functions. Thus, in the case when G ∈ I20 (M), the algebra must be a Frobenius algebra. In the general case, when t ∈ Irs (M), in the space of Am ,we introduce the Kruchkovich tensors [28, 29]: σ

s−2 Bβα1 β2 ...βs = Cβα1 σ1 Cβσ21σ2 · · · Cβs−1 βs (s > 2),

Bβα1 β2 = Cβα1 β2 , Bβα = δβα . If Am is the Frobenius algebra with metric ϕαβ , then we have ···αr Bβα11···β = Bβα1r ···βs λ1 ···λr −1 ϕ λ1 α1 · · · ϕ λr −1 αr −1 , s

B α1 ···αr = Bλα1r ···λr −1 ϕ λ1 α1 · · · ϕ λr −1 αr −1 , Bβ1 ···βs = Bβα1 .···βs−1 ϕαβs . ···αr Now we state some properties of tensors Bβα11···β : s

···αr is a symmetric tensor with respect to the indices α1 , ..., αr and β1 , ..., βs , (B1 )Bβα11···β s

λα1 ···αr 1 ···αr σ B r1 ···rs = B r1 ···rs = Bμβ , Cλμ (B2 )Cσλμ Bβσ1α···β σβ1 ···βs λμβ1 ···βs , s 1 ···βs σ α1 ···αr α1 ···αr σ α1 ···αr (B3 )Bβ1 ···βs λσ = Bσβ1 ···βs ε = Bβ1 ···βs .

Proof of B1 ,B2 ,B3 immediately follows from (1.5), (1.8) and (1.9). In a similar way, the pure tensor field of type (r , s) has the following explicit expression σ

r t ij11 ···i ··· js = I

u 1 ···u r v1 ···vs

α1 ···αr Bσβ , 1 ···βs

(i a = u a αa , jb = vb βb , a = 1, ..., r , b = 1, ..., s)

(1.32)

1.6

Realizations of Holomorphic Tensors

15

σ

···u r where I uv11···v s are arbitrary functions in the adapted coordinate chart U ⊂ Mmr . Using (1.32), we introduce a hypercomplex object in the Frobenius algebra Am : σ

∗ u ...u r 1 v1 ...vs

βs r β1 = t ij11 ...i ... js ε ...ε λα1 ...λαr −1 eαr = I

t

∗ u 1 ...u r v1 ...vs

u 1 ...u r v1 ...vs eσ

(1.33)

are components of hypercomplex tensor field of type (r , s),

We easily see that t i.e.

∗ u 1 ···u r v1 ···vs

t

∗ u 1 u1

=S

∗ u ∗ v r 1 ur S v 1

···S

∗ v ∗ u 1 ···u r s vs t v1 ···vs

···S

For simplicity, we take r = s = 1. In fact, from (1.21) and (1.22), we have 

 

 



vβ



β

uα uα uα t ij  = tvu βα = Suα Sv  β  tvβ = uu σ Cσαα vε v  C εβ  tvβ ,

which implies ∗ u v

t





 



 



vβ

uα uα β = t ij  εβ eα  = tvu βα εβ eα  = Suα Sv  β  tvβ ε eα  





vβ



β



vβ

uα β uσ uα = uu σ Cσαα vε v  C εβ  tvβ ε eα  = u v  eσ eα tvβ ε



vβ

ε

α = uu σ v  eσ eα Iuv Bεβ = uu σ v  Iuv eσ eβ eε ∗

=S

u u

∗

S

∗ v u t  v v

by virtue of (1.27). Thus, we have Theorem 1.8 If is a regular integrable -structure on Mmr , then the pure tensor field t ∗

of type (r,s) on Mmr is a realization tensor of hypercomplex tensor t in the A-holomorphic manifold X r (Am ). ∗

Remark 1.2 It is clear that the hypercomplex tensor field t is not A-holomorphic, in general. ∗

In the next section, we will study a realization of the A-holomorphic tensor t .

1.6

Realizations of Holomorphic Tensors ∗

Let Am be a Frobenius hypercomplex algebra and t ∈ Irs (X r (Am )) be a hypercomplex tensor field on X r (Am ). As previously mentioned, the realization of such a tensor field ∗

is a pure tensor field t ∈ Irs (Mmr ), and t is not holomorphic in general. To investigate ∗

a holomorphic algebraic tensor field t , we consider the Tachibana ϕ -operators on Mmr α

16

1 Holomorphic Manifolds Over Algebras

associated with the regular -structure and applied to a pure tensor field t of type (r , s) [77, 86] (see also [52]):    ϕ t X , Y1 , ..., Ys , ξ 1 , ..., ξ r α       = ϕ X t Y1 , ..., Ys , ξ 1 , ..., ξ r − X t ϕ Y1 , ..., Ys , ξ 1 , ..., ξ r α

+ −

s  λ=1 r 



α





t Y1 , ..., L Yλ ϕ X , ..., Ys , ξ 1 , ..., ξ r



α





μ

t Y1 , ..., Ys , ξ , ..., L ϕ X ξ − L X 1

  r ϕ ξ ◦ , ..., ξ , μ

(1.34)

α

μ=1

where ϕ t ∈ Irs+1 (Mmr ); X , Yλ ∈ I10 (Mmr ), ξ μ ∈ I01 (Mmr ), λ = 1, ..., s; μ = 1, ..., r ; α

and L Y is the Lie derivation with respect to Y . In particular, if t ∈ I1s (Mmr ), that is ϕ(t(Y1 , Y2 , ..., Ys )) = t(ϕY1 , Y2 , ..., Ys ) .. . = t(Y1 , Y2 , ..., ϕYs ), then from (1.34), we have      s     ϕ ϕ X+ X , ..., Ys . ϕ t (X , Y1 , Y2 , ..., Ys ) = − L t(Y1 ,Y2 ,...,Ys ) t Y1 , Y2 , ..., L Yλ α

α

α

λ=1

Also, if ω ∈ I0s (Mmr ), then from (1.34), we have        ϕ ω (X , Y1 , ..., Ys ) = ϕ X (ω(Y1 , ..., Ys )) − X ω ϕ Y1 , ..., Ys α

α

α

    s  + ω Y1 , ..., L Yλ ϕ X , ..., Ys  =

λ=1



Lϕ X ω − L X α

α

 ω ◦ ϕ (Y1 , ..., Ys ), α

where ω ◦ ϕ is defined by (ω ◦ ϕ)(Y1 , ..., Ys ) = ω(ϕY1 , Y2 , ..., Ys ) .. . = ω(Y1 , Y2 , ..., ϕYs ).

1.6

Realizations of Holomorphic Tensors

17

By setting X = ∂k , Yλ = ∂ jλ , ξ μ = d x iμ in Eq. (1.34), we see that the components  i1 ...ir ϕ t of ϕ t with respect to local coordinate system x 1 , ..., x n may be expressed j1 ... js

α

α

as follows:

 i1 ···ir s   ∂ jλ ϕ − ∂k t◦ ϕ +

 i1 ···ir ϕ t =ϕ α

m r ∂m t ij11 ···i ··· js α k

k j1 ··· js

+

r  

∂k ϕ α

μ=1

α

j1 ··· js

λ=1



m α k

r t ij11 ···i ···m··· js

 i μ i 1 ···m···ir − ∂m ϕ k t j1 ··· js

iμ m

.

(1.35)

α

where i 1 ···ir ϕ r (t ◦ ϕ )ij11···i ··· js = tm− js

=

α m···ir ϕ i 1 t j1 ··· js m α

= ···

m r ϕ m = · · · = t ij11 ···i ···m α js α j1 ϕ ir = t ijs1···m ··· jr α m .

We note some important properties of operator  given by (1.34) to our further aims: (i)(ϕ X )Y = −(L X ϕ )Y , (ii)(ϕ ϕ )(X , Y ) = Nϕ ,ϕ (X , Y ) α

α

α

β

α β

= [ϕ X , ϕ Y ] − ϕ [X , ϕ Y ] − ϕ [ϕ X , Y ] α

β

α

β

β α

+ ϕ ϕ [X , Y ], X , Y ∈ I10 (Mmr ) , α β

where Nϕ ,ϕ are the Nijenhuis tensors associated with [28]. α β

The operator  given by (1.35) is first introduced by Tachibana [77]. The Tachibana operators and their generalizations were studied in [28, 52, 74, 86]. One can prove the following theorem (see [28, 60]): Theorem 1.9 Let be an integrable regular -structure on Mmr . The hypercomplex tensor ∗

∗

field t ∈ Irs (X r (Am )) is A-holomorphic if and only if the pure tensor field t ∈ Irs (Mmr ) (the ∗

realization of t ) satisfies the equation. ϕ t = 0, α = 1, ..., m, α

where ϕ t is the Tachibana operator defined by (1.35). α

  Proof Using adapted charts ∂k ϕ ij = 0 , by virtue of (1.18), from (1.35), we have (i a = u a αa , jb = vb βb , k = wγ ; a = 1, ..., r ; b = 1, ..., s).

18

1 Holomorphic Manifolds Over Algebras

 i1 ··· ir ϕ t =ϕ α

j1 ··· js

ir m ∂m t ij11 ··· ··· js α k

 λ μ = Cαγ ∂wμ I

 i1 ··· ir − ∂k t ◦ ϕ α

u 1 ··· u r v1 ··· vs

j1 ··· js

λ − Cασ ∂wγ

σ

I

u 1 ··· u r v1 ··· vs



α1 ··· αr Bλβ = 0. 1 ··· βs

From B3 (see Sect. 1.5), we see that the condition ϕ t = 0 is equivalent to the α

condition λ

μ ∂wμ I Cαγ ∗

u 1 ··· u r v1 ··· vs

σ

λ = Cασ ∂wγ I

u 1 ··· u r v1 ··· vs .

Using (1.13), we see that the last condition is the A-holomorphicity condition of σ

u ··· u u ··· u t v11 ··· vsr = I v11 ··· vsr eσ with respect to the local coordinates z u = x uα eα in X r (Am ). Thus, the proof is complete.

Remark 1.3 Let be a nonintegrable regular -structure on Mmr . Then if t ∈ K er ϕ , α

we say that t is an almost A-holomorphicA− tensor field.

An infinitesimal automorphism of a regular -structure on Mmr is a vector field X such that L X ϕ = 0, α = 1, ..., m, where L X denotes the Lie differentiation with respect α

to X ∈ I10 (Mmr ). From Theorem 1.9 and the property i) of Tachibana operator, we have Theorem 1.10 Let on Mmr be given the integrable regular -structure. A vector field X is an infinitesimal automorphism if and only if X is A-holomorphic. Also we have Theorem 1.11 If ω be an exact 1-form, i.e. ω = d f , f ∈ I00 (Mmr ), then ω is A-holomorphic if and only if the associated 1-forms d f ◦ ϕ are closed. α

Proof Let ω ∈ I01 (M). Using (dω)(X , Y ) =

1 {X (ω(Y )) − Y ω(X ) − ω([X , Y ])} 2

for any X , Y ∈ I10 ∈ (M) and ω ∈ I01 (M), we have

         1 Y ω ϕX − ϕ X (ω(Y )) − ω Y , ϕ X (dω) Y , ϕ X = α α α α 2

       1 Y ω ϕX = − ϕ X (ω(Y )) + ω ϕ X , Y α α α 2

1.6

Realizations of Holomorphic Tensors

19

        1 Y ω ϕX = − ϕ X (ω(Y )) + ω ϕ X , Y − ϕ [X , Y ] α α α α 2   (1.36) +ω ϕ [X , Y ] α

From (1.34), we have           ϕ ϕ ϕ ϕ ϕ ω (X , Y ) = X (ω(Y )) − X ω Y −ω X , Y − [X , Y ] α

α

α

α

α

(1.37)

Substituting (1.37) into (1.36), we have   (dω) Y , ϕ X α

          1 − ϕ ω (X , Y ) + Y ω ϕ X = − X ω ϕY + ω ϕ [X , Y ] α α α α 2

           1 = − ϕ ω (X , Y ) + Y ω ◦ ϕ (X ) − X ω ◦ ϕ (Y ) − ω ◦ ϕ ([Y , X ]) α α α α 2     1 = − ϕ ω (X , Y ) + d ω ◦ ϕ (Y , X ). α 2 α From here, we see that equation ϕ ω = 0 is equivalent to α

     ϕ ϕ d ω◦ (Y , X ) = (dω) Y , X α

α

which for ω = d f turns into the following simple relation:        d d f ◦ ϕ (Y , X ) = d 2 f Y , ϕ X = 0, α

α

i.e. d f ◦ ϕ is a closed 1-form. The proof is completed. α

Let now ω ∈ I01 (Mmr ). Using (1.34), we have          ϕ ω (X , Y ) = ϕ X (ω(Y )) − X ω(ϕ Y ) + ω L Y ϕ X α α α α    = Lϕ X ω − L X ω ◦ ϕ Y α

α

for any X , Y ∈ I10 (Mmr ), where the associated 1-forms ω ◦ ϕ are defined by α

    ω◦ϕ Y =ω ϕY . α

α

(1.38)

20

1 Holomorphic Manifolds Over Algebras

Theorem 1.12 Let ω ∈ I01 (M2r ) and ϕ be a complex or paracomplex structure on M2r , i.e. ϕ 2 = ∓ id M2r . Then the associated 1-form ω ◦ ϕ is holomorphic if and only if ω ∈ Ker ϕ . Proof If we substitute ω ◦ ϕ into ω and ϕ X into X , then Eq. (1.38) may be written as   ϕ (ω ◦ ϕ) (ϕ X , Y ) = (L ϕ 2 X (ω ◦ ϕ) − L ϕ X (ω ◦ ϕ 2 ))Y   = ∓ L X (ω ◦ ϕ) − L ϕ X ω Y   = ± ϕ ω (X , Y ) or 

    ϕ (ω ◦ ϕ) ◦ ϕ (X , Y ) = ± ϕ ω (X , Y ),

from which by virtue of det ϕ = 0, we see that ϕ (ω ◦ ϕ) = 0 if and only if ϕ ω = 0. The proof is completed. Theorem 1.13 Let ϕ 2 = ∓ id M2r . If ω ∈ I01 (M) is a holomorphic 1-form, then ω ◦ Nϕ = 0,     where ω ◦ Nϕ is defined by ω ◦ Nϕ (X , Y ) = ω Nϕ (X , Y ) and Nϕ is the Nijenhuis tensor field associated with ϕ: Nϕ (X , Y ) = [ϕ X , ϕ X ] − ϕ[X , ϕY ] − ϕ[ϕ X , Y ] ∓ [X , Y ], ∀X , Y ∈ I10 (M2r ). Proof immediately follows from Theorem 1.13 and the following formula:   ϕ (ω ◦ ϕ) = ϕ ω ◦ ϕ + ω ◦ Nϕ .

1.7

Pure Connections

In this section, we always assume that the regular -structure is integrable, and we consider only local adapted coordinates with respect to the structure. Let ∇ be a -connection on Mmr , i.e. ∇ ϕ = 0 for any ϕ ∈ . Since the components σ

σ

of ϕ with respect to the local adapted coordinates x 1 , ..., x mr are constant, we have σ

i ϕ ∇ ϕ = 0 ⇔ km σ

m σ j

mϕ = kj σ

i m

.

(1.39)

By the same arguments as developed in Sect. 1.4, we see that the -connection has components of the form

1.7

Pure Connections

21 σ u α wγ v C σβ

uα ki j = wγ vβ = τ

(i = uα, j = vβ, k = wγ ),

(1.40)

σ

uσ β where τ uwγ v = wγ vβ ε are any functions in the adapted chart U ⊂ Mmr . In fact, using β contraction with ε and m = tε, from (1.39), we obtain uα ϕ wγ tε σ

tε vβ

tε ϕ = wγ vβ

uα tε ,

σ

uα t ε tε u α wγ tε δv C σβ = wγ vβ δt C σ ε , ε u α wγ v C εσ ,

uα uε β α wγ vσ = wγ vβ ε C σ ε = τ ε

uε β where τ uwγ v = wγ vβ ε . With -connection of type (2.40), we can associate a hypercomplex objects from Am : ∗ u wv



uα γ β = wγ vβ ε ε eα

(1.41)

σ u γ wγ v ε eσ .

=τ

∗

Definition 1.5 If the hypercomplex objects  ∗   uw v 

u wv

satisfy the following condition:



=

∂z u ∂z w ∂z v ∗  ∂z u ∂z w ∂z v 



u wv

+

∂ 2 z u ∂z u ,   ∂z v ∂z w ∂z u ∗

∗

i.e. if  uwv are the components of the hypercomplex connection ∇ in X r (Am ), then we say that the -connection ∇ is a pure connection. Theorem 1.14 Let be a regular integrable -structure on Mmr . The -connection ∇ is σ pure if and only if τ uwγ ν satisfies the condition α u wγ v

τ σ u wv

where τ

σ u α wv C σ γ ,

=τ

(1.42)

σ u η wηv ε .

=τ

σ

uα α u Proof Let ki j = wγ vβ = τ wγ v C σβ be the components of the -connection ∇. Then, taking transformation {∂i } → {∂i  } of adapted frames with matrix  of the  admissible    account

Sii



=



∂xi ∂xi

=

 

 

∂xu α ∂ x uα

uα w  γ  v β  =

, we have  

 

∂ x u α ∂ x wγ ∂ x vβ uα ∂ 2 x uα ∂xu α  + .         wγ vβ ∂ x uα ∂ x w γ ∂ x v β ∂ x w γ ∂ x v β ∂ x uα

22

1 Holomorphic Manifolds Over Algebras 



After contraction with εγ εβ eα  , by virtue of (1.21), (1.22) and (1.40), we obtain ∗ u w v 



 





uα γ β = w  γ  v  β  ε ε eα  σ u  α ε w γ θ v β uα γ  β u C σ α  w C εγ   v  C θβ  wγ vβ ε ε eα 

=

σ   +  uu Cσαα



ε    u α ε γ ε β eα  C w γ   v  εβ 

∂ ∂x

or ∗ u w v 



σ u ε w θ v γ βω u α u  w  v  eσ eα δε δθ τ wγ v C ωβ

= ∗

=S

u u

∗

S

v v

γ

w θ u w τ wγ v eθ



∗

+S

u u

ε

γ

σ u γ u eσ ε



 α  u eα w γ   v 

∂

+



∂x  α  u eα , w γ   v 

∂ ∂x

where ∗

S



∂z u ∂z u = u , Suu = u  , z u = x uα eα . ∂z ∂z

u u

Using (1.14), we see that εγ





 α  u eα = w γ   v 

∂ ∂x

∂ 2 zu  . ∂z w ∂z v

Thus, we have ∗ u w v 





=

∂z u γ  ∂z u

w w

∂z v θ  τ ∂z v



u wγ v eθ

∗

∗

+

u wv α u τ wγ v

From (1.43), we see that ∇ with components  X r (Am ) if and only if

α τ uwγ v

=

σ u τ wv Cσαγ

∗ u wv



θ τ uwγ v

satisfies the condition

∂ 2 z u ∂z u .   ∂z v ∂z w ∂z u

(1.43)

is hypercomplex connection on σ u α wv C σ γ .

=τ

In fact, substituting

into (1.41) and (1.43), we have σ u γ wγ v ε eσ

uα γ β = wγ vβ ε ε eα = τ

θ u σ γ wv C θγ ε eσ

=τ

θ u σ wv δθ eσ

=τ

and ∗   uw v 



∂z u = u ∂z  ∂z u = u ∂z  ∂z u = u ∂z



∂z v θ u ∂ 2 z u ∂z u τ e +  θ    wγ v ∂z v ∂z v ∂z w ∂z u  γ ∂z v σ u θ ∂ 2 z u ∂z u τ C e + w  θ    w ∂z v wv σ γ ∂z v ∂z w ∂z u  w v 2 ∂z ∂z σ u ∂ z u ∂z u τ e + σ     ∂z w ∂z v wv ∂z v ∂z w ∂z u γ

w w

θ u wv eθ

=τ

1.8 Torsion Tensors of Pure Connections

23



=

∂z u ∂z w ∂z v ∗  ∂z u ∂z w ∂z v 



u wv

+

∂ 2 z u ∂z u .   ∂z v ∂z w ∂z u

Conversely, comparing (1.43) with the last equation, we get γ



γ ∂z w ∗ u ∂z w ∗ u w θ u ⇔    τ wγ v eθ eγ =    wv eγ , wv w ∂z w ∂z w w ∂z σ θ θ σ τ uwγ v eθ = w τ uwηv εη eσ eγ ⇔ τ uwγ v eθ = τ uwηv εη eσ eγ , ∂z

w θ u w τ wγ v eθ

∂z w  ∂z w

θ u wγ v eθ

τ θ

=

σ u η θ wηv ε C σ γ eθ

=τ

σ

θ u wγ v

⇔τ

σ

σ u η θ wηv ε C σ γ

=τ

.

σ

Thus, τ uwγ v = τ uwv Cσθ γ , where τ uwv = τ uwηv εη . The proof is completed. Using (1.42), we get from (1.40) and (1.41), respectively σ u μ α wv C σ γ C μβ

uα ki j = wγ vβ = τ

σ u α wv Bσ γβ

=τ

(1.44)

and ∗ u wv



σ u wv eσ ,

=τ

(1.45)

where Bσαγβ is the Kruchkovich tensor. Thus, we have Theorem 1.15 A pure -connection ∇ has the components (1.44) with respect to the adapted coordinates. ∗

Theorem 1.16 A pure -connection ∇ is a realization of the hypercomplex connection ∇ with components (1.45). From Theorem 1.15 and (1.32), it follows that the pure -connection as a pure tensor fields of type (1,2) is defined by i ϕ km

m α j

= kmj ϕ α

i m

i ϕ = m j

m , α k

α = 1, ..., m

with respect to the adapted charts. But the pure tensor fields of type (1,2) (see (1.23)) are defined by a similiar equation with respect to the arbitrary charts.

1.8

Torsion Tensors of Pure Connections

α Let S be a torsion tensor of pure -connection ∇. Since Bσαγβ = Bσβγ (see Sect. 1.5), from (1.44), we have

24

1 Holomorphic Manifolds Over Algebras

Ski j = ki j −  ijk σ u wv

= (τ

σ u α vw )Bσ γβ ,

−τ

(1.46)

i.e. S is a pure tensor (see (1.32)). Conversely, we now assume that S is a pure torsion tensor of the -connection. Then, by virtue of (1.32), we have λ u α wv Bλγβ .

uα Ski j = Swγ vβ = σ

(1.47)

On the other hand, from (1.40), we have Ski j = ki j −  ijk λ u α wγ v C λβ

=τ

λ u α vβw C λγ .

−τ

(1.48)

From (1.47) and (1.48), by virtue of contraction with εγ , we have   α u λ u λ u γ α τ vβw = τ wγ v ε − σ wv Cλβ , which shows the condition of type (1.42) is true, i.e. the -connection is pure. Thus, we have Theorem 1.17 The -connection ∇ is pure if and only if its torsion tensor is pure. For the pure -connection, by virtue of (1.33), (1.45) and (1.46), we obtain ∗

S

u wv

σ u σ wv − τ ∗ ∗  uwv − 

= (τ

u vw )eσ

=

u vw .

Thus, we have. Theorem 1.18. The pure torsion tensor field S of the -connection ∇ is realization of the ∗

∗

hypercomplex torsion tensor S of hypercomplex connection ∇ . It is well known that the zero tensor field is pure. Therefore, we have Theorem 1.19 A torsion-free -connection ∇ is pure. Also, from Theorems 1.18 and 1.19, we have

1.9

Holomorphic Connections

25 ∗

∗

Theorem 1.20 If ∇ is a torsion-free -connection, then ∇ with components  is also a torsion-free connection.

1.9

u wv

σ u wv eσ

=τ

Holomorphic Connections

Let R be a curvature tensor of the pure -connection ∇. Using (1.44) and the properties of Kruchkovich tensors, we have uα R ijkl = Rvβwγ tδ uα uα uα xε uα xε = ∂vβ wγ tδ − ∂wγ vβtδ + vβxε wγ tδ − wγ xε vβtδ σ  σ  σ σ u θ α α τθ x ε α + τ uvx Bσβε = ∂vβ τ uwt Bσαγ δ − ∂wγ τ uvt Bσβδ wt Bθγ δ − τ wx Bσ γ ε τ       σ σ θ x θ x σ σ α Bσαγ θβδ . = ∂vβ τ uwt Bσαγ δ − ∂wγ τ uvt Bσβδ + τ uvx τ wt − τ uwx τ vt

x ε vt Bθβδ

(1.49)

We now assume that R is a pure tensor. Then by virtue of (1.32), we have λ u α vwt Bλβγ δ

uα ρ R ijkl = Rvβwγ tδ =

(1.50)

From (2.49) and (2.50), we obtain λ u α vwt Bλβγ δ

ρ

 σ = ∂vβ τ

u wt



 σ Bσαγ δ − ∂wγ τ

u vt



 σ α Bσβδ + τ

u τθ x vx wt

σ u θ x wx τ vt

−τ



Bσαγ θβδ . (1.51)

Using contraction with εβ εδ and the properties of Kruchkovich tensors, from (2.51), we have     σ θ σ θ σ α λ u α β u α u u x u x α ρ vwt Cλγ = ε ∂vβ τwt Cσ γ − ∂wγ τvt + τvx τwt − τwx τvt Cσλθ Cλγ  σ εβ ∂vβ τ

u wt

α u vt

= −∂wγ τ

   σ σ θ x θ x α α Cσλθ Cλγ Cσαγ − ∂wγ τ uvt + τ uvx τ wt − τ uwx τ vt   σ u θ x σ u θ x λ λ u β λ α + ε ∂vβ τ wt + τ vx τ wt Cσ θ − τ wx τ vt Cσ θ Cλγ .

From here, it follows that α u vt

∂wγ τ

λ

=P

u α vwt C λγ ,

(1.52)

26

1 Holomorphic Manifolds Over Algebras

where λ

P

u vwt

λ u wt

= εβ ∂vβ τ

σ u θ x λ vx τ wt C σ θ

+τ

σ u θ x λ wx τ vt C σ θ

−τ

λ u vwt .

−ρ

By virtue of (1.12), for fixed u, v, w and t, the condition (1.52) is the A-holomorphity ∗

condition of 

u vt

α u vt eα with respect to the local ∗ u = τα u e is a A-holomorphic  vt vt α

=τ

coordinates z u = x uα eα in X r (Am ).

Conversely, if connection, then from Theorems 1.4 α = Cα ) and (1.12), we obtain the condition of type (1.52). Using (1.52), from and 1.1 (C (1.49), we have uα R ijkl = Rvβwγ tδ    λ λ u σ α = P wvt Cλβ Bσ γ δ − P λ

=P

u α wvt Bλβγ δ

λ

−P

u σ vwt C λγ

u α vwt Bλγβδ



 σ + τ

α Bσβδ

 σ u θ x τ τ − wx vt Bσαγ θβδ  σ θ x α Cσλθ Bλγβδ − τ uwx τ vt 

σ u θ x vx τ wt

+ τ

u τθ x vx wt

λ u α vwt Bλβγ δ ,

=ρ

(1.53)

where λ u vwt

ρ

λ

=P

u wvt

λ

−P

u vwt

 σ + τ

u τθ x vx wt

σ u θ x wx τ vt

−τ

 Cσλθ .

From (1.32) and (1.53), we see that R is a pure curvature tensor. Thus, we have ∗

Theorem 1.21 [28, 82] Let ∇ be a hypercomplex connection on X r (Am ) and ∇ its real∗ ization on Mmr . The curvature tensor R of ∇ is pure if and only if ∇ is an A-holomorphic connection.

1.10

Holomorphic Curvature Tensor

Let now R be a pure tensor of pure -connection ∇. Using (1.3) and (1.49), we have ∗

R

u vwt

uα β γ δ = Rvβwγ tδ ε ε ε eα    α α = εβ ∂vβ τ uwt eα − εγ ∂wγ τ

u vt



 eα +

σ θ x τ uvx τ wt

−

σ θ x τ uwx τ vt

Since, Cσαθ eα = eσ eθ , by virtue of (1.14) and (1.45), we have ∗

R

u vwt

∗ u wt

= ∂v 

∗ u vt

− ∂w 

∗ u ∗ x vx  wt

+

∗ u ∗ x wx  vt ,

−

 Cσαθ eα .

1.10

Holomorphic Curvature Tensor ∗

27

∗

i.e. R is a curvature tensor of  . Thus, we have ∗

Theorem 1.22 Let ∇ be a realization of the A-holomorphic connection ∇ . The hypercom∗

plex components R

u vwt ∗

associated with the pure curvature tensor R are components of ∗

curvature tensor field R of ∇ . ∗

Let now ∇ be an A-holomorphic hypercomplex connection on X r (Am ) and a pure

-connection ∇ its realization on Mmr . From Theorem 1.21 we see that the curvature tensor R of ∇ is pure. Since the curvature tensor R is pure, we can apply the Tachibana operator ϕ to R: α

    ϕ ϕ X + R L Y1 X , Y2 , Y3 (ϕ R)(X , Y1 , Y2 , Y3 ) = − L R(Y1 ,Y2 ,Y3 ) α α α         + R Y1 , L Y2 ϕ X , Y3 + R Y1 , Y2 , L Y3 ϕ X , 

α

α

      where ϕ R(Y1 , Y2 , Y3 ) = R ϕ Y1 , Y2 , Y3 = R Y1 , ϕ Y2 , Y3 = R Y1 , Y2 , ϕ Y3 . α

α

α

α

Now we proved that if a torsion tensor of -connection ∇ is pure, then −(L Y ϕ )X = ∇ϕ X Y − ϕ (∇ X Y ). α

α

α

In fact, from ∇ ϕ = 0 and ϕ S(X , Y ) = S(ϕ X , Y ) = S(X , ϕ Y ), where S(X , Y ) = α

α

α

∇ X Y − ∇Y X − [X , Y ], we obtain     − L Y ϕ X = ϕ X , Y − ϕ [X , Y ] α

α

α



α



= ∇ϕ X Y − ∇Y ϕ X − S ϕ X , Y − ϕ (∇ X Y − ∇Y X − S(X , Y )) α α α α     ϕ ϕ ϕ ϕ = ∇ϕ X Y − (∇ X Y ) − ∇ X, Y (Y , X ) + S(X , Y ) − S α

α

α

α

α

= ∇ϕ X Y − ϕ (∇ X Y ), α

α

Now using the purity conditions of R and S, and also the last formula, we have        ϕ ϕ ϕ R (X , Y1 , Y2 , Y3 ) = − L R(Y1 ,Y2 ,Y3 ) X + R L Y1 X , Y2 , Y3 α α α         + R Y1 , L Y2 ϕ X , Y3 + R Y1 , Y2 , L Y3 ϕ X α

α

28

1 Holomorphic Manifolds Over Algebras

  = ϕ X , R(Y1 , Y2 , Y3 ) − ϕ [X , R(Y1 , Y2 , Y3 )] α

α



  + R L Y1 ϕ X , Y2 , Y3 + R Y1 , L Y2 ϕ X , Y3 + R Y1 , Y2 , L Y3 ϕ X α α α   = ∇ϕ X R(Y1 , Y2 , Y3 ) − ∇ R(Y1 ,Y2 ,Y3 ) ϕ X − S ϕ X , R(Y1 , Y2 , Y3 ) α α α   − ϕ ∇ X R(Y1 , Y2 , Y3 ) − ∇ R(Y1 ,Y2 ,Y3 ) X − S(X , R(Y1 , Y2 , Y3 )) α            + R L Y1 ϕ X , Y2 , Y3 + R Y1 , L Y2 ϕ X , Y3 + R Y1 , Y2 , L Y3 ϕ X α α α       = ∇ϕ X R (Y1 , Y2 , Y3 ) + R ∇ϕ X Y1 , Y2 , Y3 + R Y1 , ∇ϕ X Y2 , Y3 α α α         + R Y1 , Y2 , ∇ϕ X Y3 − ∇ R(Y1 ,Y2 ,Y3 ) ϕ X − ϕ ∇ R(Y1 ,Y2 ,Y3 ) X − S ϕ X , R(Y1 , Y2 , Y3 )) 









α

α







α

α

− ϕ ((∇ X R)(Y1 , Y2 , Y3 ) + R(∇ X Y1 , Y2 , Y3 ) + R(Y1 , ∇ X Y2 , Y3 ) + R(Y1 , Y2 , ∇ X Y3 ) α   ϕ − ∇ R(Y1 ,Y2 ,Y3 ) X − S (X , R(Y1 , Y2 , Y3 ))) + R −∇ϕ X Y1 + (∇ X Y1 ), Y2 , Y3 α α     + R Y1 , −∇ϕ X Y2 + ϕ (∇ X Y2 ), Y3 + R Y1 , Y2 , −∇ϕ X Y3 + ϕ (∇ X Y3 ) α α α α   = ∇ϕ X R (Y1 , Y2 , Y3 ) − ϕ (∇ X R)(Y1 , Y2 , Y3 ), α

α

by virtue of ∇ ϕ = 0. Thus, we have α

    ϕ R (X , Y1 , Y2 , Y3 ) = ∇ϕ X R (Y1 , Y2 , Y3 ) − ϕ (∇ X R)(Y1 , Y2 , Y3 ). α

α

α

(1.54)

For simplicity, let now ∇ be a pure torsion-free -connection. Then, using the purity of R and applying the Bianchi’s 2nd identity to (1.54), we get     ϕ R (X , Y1 , Y2 , Y3 ) = ∇ϕ X R (Y1 , Y2 , Y3 ) − ϕ (∇ X R)(Y1 , Y2 , Y3 ) α α α         ϕ ϕ = − ∇Y1 R Y2 , X , Y3 − ∇Y2 R X , Y1 , Y3 α

α

− ϕ (∇ X R)(Y1 , Y2 , Y3 ). α

On the other hand, using ∇ ϕ = 0, we find α





(∇Y2 R) ϕ X , Y1 , Y3 α          ϕ ϕ ϕ − R ∇Y2 X , Y1 , Y3 X , Y1 , Y3 − R X , ∇Y2 Y1 , Y3 = ∇Y2 R α

α

α

1.10

Holomorphic Curvature Tensor

29

  − R ϕ X , Y1 , ∇Y2 Y3 α      = ∇Y2 ϕ (R(X , Y1 , Y3 )) + ϕ (∇Y2 R(X , Y1 , Y3 )) − R ∇Y2 ϕ X α α α     ϕ ϕ X , ∇Y2 Y1 , Y3 − R X , Y1 , ∇Y2 Y3 + ϕ(∇Y2 X ), Y1 , Y3 ) − R α α      = ϕ ∇Y2 R(X , Y1 , Y3 ) − ϕ R ∇Y2 X , Y1 , Y3 − ϕ (R(X , ∇Y2 Y1 , Y3 )) α

α

α

− ϕ (R(X , Y1 , ∇Y2 Y3 )) α   ϕ = ∇Y2 R (X , Y1 , Y3 )).

(1.55)

α

Similarly, we obtain 

(∇Y1 R) Y2 , ϕ X , Y3



α

= ϕ ((∇Y1 R)(Y2 , X , Y3 )).

(1.56)

α

Substituting (1.55) and (1.56) into (1.54), and using again the Bianchi’s 2nd identity, we obtain   ϕ R (X , Y1 , Y2 , Y3 ) = − ϕ ((∇Y1 R)(Y2 , X , Y3 ) − ϕ ((∇Y2 R)(X , Y1 , Y3 ) α

α

α

− ϕ ((∇ X R)(Y1 , Y2 , Y3 )) α

= − ϕ (σ {(∇ X R)(Y1 , Y2 )}, Y3 ) = 0, α

where σ denotes the cyclic sum with respect to X , Y1 and Y2 . Therefore, by virtue of Theorems 1.9 and 1.21, we have ∗

∗

Theorem 1.23 The curvature tensor R of the A-holomorphic connection ∇ is an A-holomorphic tensor. Example 1.1 Let now take r = 1, i.e. we consider a A-holomorphic manifold X 1 (Am ) of hypercomplex dimension 1. Since u = v = w = t = 1, we have from (1.49). 1α R ijkl = R1β1γ 1δ      σ σ 1 σ 1 α α = ∂1β τ 11 Bσ γ δ − ∂1γ τ 11 Bσβδ + τ     σ σ α = ∂1β τ 111 Bσαγ δ − ∂1γ τ 111 Bσβδ  σ  σ α ε ε α = ∂β τ Cδε Cσ γ − ∂γ τ Cσβ Cεδ   σ σ ε α = Cσε γ ∂β τ −Cσβ ∂γ τ Cδε ,

1 τθ 1 11 11

σ θ − τ 111 τ 111



Bσαγ θβδ

(1.57)

30

1 Holomorphic Manifolds Over Algebras σ

σ

1 . where τ = τ11

We now assume that σ

σ

ε Cσε γ ∂β τ = Cσβ ∂γ τ .

(1.58)

After contraction with εγ , from (1.58), we have ε

σ

ε ∂β τ = εγ (∂γ τ )Cσβ ,

(1.59)

i.e. by virtue of (1.12) τ is an A-holomorphic function of x = x α eα ∈ Am . Conversely, let τ = τ (x) be an A-holomorphic function. Then, from (1.59), we have  σ ε θ ε θ ∂β τ = εγ ∂γ τ Cσβ Cετ Cετ  σ θ = εγ ∂γ τ Cβε Cσε τ  σ θ = εγ ∂γ τ Cσε τ Cβε  ε θ = εγ ∂σ τ Cγσ τ Cβε  ε θ = ∂τ τ Cβε , i.e. the condition (1.58) is true. On the other hand, the condition (1.58) is equivalent to the Scheffers condition (1.11). Thus, if τ = τ (x) is A-holomorphic, then R ijkl = 0. Conversely, if R = 0, then from (1.57), we have 1α δ 0 = R1β1γ 1δ ε   σ σ ε α δ = Cσε γ ∂β τ −Cσβ ∂γ τ Cδε ε σ

σ

α = Cσαγ ∂β τ −Cσβ ∂γ τ ,

i.e. the function τ = τ (x) is A-holomorphic. Thus, we have. ∗

Theorem 1.24 Let Mm be a realization of X 1 (Am ). The connection ∇ with components ∗ σ τ = τ eσ on X 1 (Am ) is A-holomorphic if and only if the real manifold Mm is locally flat. σ

σ

1 = ε σ (1 = ε σ e ∈ A ), then from (1.44), we Remark 1.4 In particular, if τ = τ11 σ m σ

α = C α , i.e. M is the Vranceanu space [44, 83]. Since τ e = ε σ e = 1 is have γβ m σ σ γβ holomorphic, we see that the Vranceanu space is locally flat.

2

Anti-Hermitian Geometry

In this chapter, we study the pseudo-Riemannian metric on holomorphic manifolds. In Sect. 2.1, we give the condition for a hypercomplex anti-Hermitian metric to be holomorphic; also, we prove that there exists a one-to-one correspondence between hypercomplex anti-Kähler manifolds and anti-Hermitian manifolds with an A-holomorphic metrics. In Sect. 2.2, we discuss complex Norden manifolds. We define the twin Norden metric; the main theorem of this section is that the Levi–Civita connection of Kähler-Norden metric coincides with the Levi–Civita connection of twin Norden metric. In Sect. 2.3, we consider Norden-Hessian structures. We give the condition for a Norden-Hessian manifold to be Kähler. Section 2.4 is devoted to the analysis of twin Norden metric connections with torsion. In Sects. 2.5–2.10, we focus our attention to pseudo-Riemannian 4-manifolds of neutral signature. The main purpose of these sections is to study complex Norden metrics on 4-dimensional Walker manifolds. We discuss the integrability and Kahler (holomorphic) conditions for these structures. The curvature properties for Norden-Walker metrics are also investigated, and examples of Norden-Walker metrics are constructed from an arbitrary harmonic function of two variables. We define the isotropic Kähler structures and, moreover, show that a proper almost complex structure on an almost Norden-Walker manifold is isotropic Kähler. We also consider the quasi-Kähler-Norden metric and give the condition for an almost Norden manifold to be quasi-Kähler-Norden. Finally, we give progress to the conjecture of Goldberg under the additional restriction on Norden-Walker metric.

2.1

Equivalence of Holomorphic and Anti-Kähler Conditions

Let Mmr be a pseudo-Riemannian manifold with metric g, and let on Mmr be given the regular hypercomplex -structure:

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Salimov, Applications of Holomorphic Functions in Geometry, Frontiers in Mathematics, https://doi.org/10.1007/978-981-99-1296-4_2

31

32

2 Anti-Hermitian Geometry

   = ϕ ij , ϕ

i σ j

σ

α = δvu Cσβ , i = uα, j = vβ;

i, j = 1, . . . , mr ; α, β, σ = 1, . . . , m; u, v = 1, . . . , r An anti-Hermitian metric with respect to the -structure is a pseudo-Riemannian metric g such that   g ϕ X , Y = g X , ϕ Y , α = 1, ..., m, (2.1) α

α

for any X , Y ∈ I10 (Mmr ), i.e. g is pure with respect to the regular hypercomplex structure. Such metrics were studied in [79], where they were said to be B-metrics, since the metric tensor g with respect to the -structure is B-tensor according to the terminology accepted by Norden [40]. If (Mmr , ) is an almost hypercomplex manifold with anti-Hermitian metric g, we say that the triple (Mmr , , g) is an almost antiHermitian manifold. If the -structure is integrable, we say that the triple (Mmr , , g) is an anti-Hermitian manifold. An anti-Hermitian metric g is called an almost A-holomorphic metric (see Remark 1.3) if         ϕ ϕ ϕ X, Z + g LY ϕ g (X , Y , Z ) = ( X )(g(Y , Z )) − X g Y , Z α α α α     + g Y, LZ ϕ X α    ϕ (2.2) = Lϕ X g − L X g ◦ (Y , Z ) = 0, α = 1, ..., m, α

α

for any X , Y , Z ∈ I10 (Mmr ), where ϕ , α = 1, ..., m are Tachibana operators applied α

to an anti-Hermitian metric (see Sect. 1.6). If (Mmr , , g) is an almost anti-Hermitian manifold with almost A-holomorphic metric g, we say that (Mmr , , g) is an almost Aholomorphic anti-Hermitian manifold. If the -structure is integrable, then we say that the triple (Mmr , , g) is an A-holomorphic anti-Hermitian manifold. Theorem 2.1 An almost anti-Hermitian manifold (Mmr , , g) is an almost A-holomorphic if and only if ∇ ϕ = 0, α = 1, ..., m, where ∇ is the Levi–Civita connection of g. α

Proof Using (2.1) and L X Y = [X , Y ] = ∇ X Y − ∇Y X , from (2.2), we get   ϕ g (X , Z 1 , Z 2 ) α        ϕ ϕ ϕ = Lϕ X g − L X g ◦ Z1, L X Z2 (Z 1 , Z 2 ) + g Z 1 , L X Z 2 − g α

α

α

α

2.1

Equivalence of Holomorphic and Anti-Kähler Conditions

33

      = ϕ X g(Z 1 , Z 2 ) − Xg ϕ Z 1 , Z 2 − g ∇ϕ X Z 1 , Z 2 + g ∇ Z 1 ϕ X , Z 2 α α α α          − g Z 1 , ∇ϕ X Z 2 + g Z 1 , ∇ Z 2 ϕ X + g ϕ (∇ X Z 1 ), Z 2 − g ϕ ∇ Z 1 X , Z 2 α α α α      (2.3) + g ϕ Z 1, ∇X Z 2 − g Z 1, ϕ ∇Z2 X 



α

α

We find

    g ∇ Z 1 ϕ X , Z 2 − g ϕ (∇ Z 1 X ), Z 2 α α      + g Z 1, ∇Z2 ϕ X − g Z 1, ϕ ∇Z2 X α

α

= g((∇ϕ)(X , Z 1 ), Z 2 ) + g(Z 1 , (∇ϕ)(X , Z 2 )).

(2.4)

Substituting (2.4) into (2.3), we have   ϕ g (X , Z 1 , Z 2 ) α        ϕ ϕ ϕ (X , Z 1 ), Z 2 = X g(Z 1 , Z 2 ) − Xg Z1, Z2 + g ∇ α α α         g Z 1 , ∇ ϕ (X , Z 2 ) − g ∇ϕ X Z 1 , Z 2 − g Z 1 , ∇ϕ X Z 2 α α α     + g ϕ (∇ X Z 1 ), Z 2 + g ϕ Z 1 , ∇ X Z 2

(2.5)

On the other hand, with respect to the Levi–Civita connection ∇, we have     ϕ X g(Z 1 , Z 2 ) − g ∇ϕ X Z 1 , Z 2 α α     − g Z 1 , ∇ϕ X Z 2 = ∇ϕ X g (Z 1 , Z 2 ) = 0

(2.6)

α

α

α

and





α

    + g ϕ (∇ X Z 1 ), Z 2 + g ϕ Z 1 , ∇ X Z 2

− Xg ϕ Z 1 , Z 2 α α    = −g ∇ X ϕ Z 1 , Z 2

α

(2.7)

α

By virtue of (2.6) and (2.7), the operator (2.5) reduces to      ϕ g (X , Z 1 , Z 2 ) = −g ∇ X ϕ Z 1 , Z 2 α

α

34

2 Anti-Hermitian Geometry

       + g ∇Z1 ϕ X , Z 2 + g Z 1, ∇Z2 ϕ X , α

(2.8)

α

for any X , Z 1 , Z 2 ∈ I10 (Mmr ). From (2.8), we easily see that if ∇ ϕ = 0, then ϕ g = 0. α

α

Conversely, let now ϕ g = 0. Then, similarly to (2.8), we have α

            ϕ g (Z 2 , Z 1 , X ) = −g ∇ Z 2 ϕ Z 1 , X + g ∇ Z 1 ϕ Z 2 , X + g Z 1 , ∇ X ϕ Z 2 α

α

α

α

(2.9)        Using g Z , ∇Y ϕ X = g ∇Y ϕ Z , X , from the sum of (2.8) and (2.9), we α

find

α

        ϕ Z2 , ϕ g (X , Z 1 , Z 2 ) + ϕ g (Z 2 , Z 1 , X ) = 2g X , ∇ Z 2 α

(2.10)

α

α

for any X , Z 1 , Z 2 ∈ I10 (Mmr ). Now, putting ϕ g = 0 in (2.10), we find ∇ ϕ = 0, α = α

α

1, ..., m. Thus, the proof of Theorem 2.1 is completed. Using the equivalence between the integrability and the almost integrability conditions for regular -structures [27], from Theorem 2.1, we obtain the following useful integrability condition: Theorem 2.2 The regular -structure on almost anti-Hermitian manifold is integrable if g is an almost A-holomorphic metric, i.e. ϕ g = 0, α = 1, ..., m. α

From Theorem 2.2, it follows that the almost anti-Hermitian manifold with an almost A-holomorphic metric has an integrable -structure, and therefore, the manifold of this type is an A-holomorphic anti-Hermitian manifold. Also, from Theorem 1.19 and Theorem 2.1, we have Theorem 2.3 The Levi–Civita connection ∇ of A-holomorphic anti-Hermitian manifolds is a pure connection. An antı-Kähler manifold can be defined as a triple (Mmr , , g) which consists of a manifold Mmr endowed with a hypercomplex integrable regular -structure and a pseudo-Riemannian metric g such that ∇ ϕ = 0, α = 1, ..., m, where ∇ is the Levi– α   Civita connection of g and the metric g is assumed to be Hermitian: g ϕ X , Y = α

2.1

Equivalence of Holomorphic and Anti-Kähler Conditions



35



g X , ϕ Y , α = 1, ..., m. Thus from Theorem 2.1, we see that there exists a one-toα

one correspondence between anti-Kähler manifolds and A-holomorphic anti-Hermitian manifolds. From (2.1), we have σ gi j = guαvβ = G uvσ Cαβ

(2.11) ∗

for arbitrary functions G uvσ (see Sect. 1.5). The corresponding hypercomplex tensor g uv is defined by ∗

g uv = guαvβ εβ eα = G uvα eα ,

(2.12) ∗

α αβ αβ g where  e =  ϕ eβ and ϕ is the Frobenius metric. Since uv is symmetric, nonsingular ∗ ∗ ∗ Det g uv  = 0 and A-holomorphic, it follows that d s 2 = g dz u dz v can be regarded uv

as hypercomplex anti-Kähler metric in X r (A). Since the Levi–Civita connection ∇ of anti-Kähler manifolds is a pure connection, by virtue of (1.44), we obtain σ

uα K ki j = K wyνβ =k

u μ α wv C σ γ C μβ

σ

=k

σ u α u wv Bσ γβ , k wv

σ

=k

σ wv ,

(2.13)

where K ki j are components of ∇ and i = uα, j = vβ, k = wγ . Substituting (2.11) and mg m (2.13) into ∇k gi j = ∂k gi j − K ki m j − K k j gim = 0 , m = tν, we find   σ σ =k ∂wv G uvσ Cαβ

t v μ τ wu C σ γ C vα G tvτ C μβ

σ

+k

μ t v τ wv C σ γ C vβ G tuτ C μα

After contraction with εβ εγ eα , by virtue of (1.5) and (1.9), we obtain   σ εγ ∂wv G uvα eα = k twu G tvμ + kσwv G tuμ eσ eμ By virtue of (1.14), (1.45) and (2.12), we have ∗

∗ ∂ g uv =K w ∂z

∗ t g wu tv

∗

+K

∗ t g wv ut , ∗

from which we easily see that the Christoffel symbol K has components ⎛ ⎞ ∗ ∗ ∗ ∗ 1 ∗ ut ⎝ ∂ g tv ∂ g wt ∂ g wv ⎠ u + − . K wv = g 2 ∂z w ∂z v ∂z t

(2.14)

36

2 Anti-Hermitian Geometry ∗

Since the hypercomplex anti-Kähler metric g is A-holomorphic, there exist the suc∗ ∗ cessive derivatives of g by virtue of Remark 1.1, i.e. from (2.14), it follows that K uwv is A-holomorphic. Thus, we have Theorem 2.4 The Levi–Civita connection of an anti-Kähler manifold is A-holomorphic. Since an anti-Kähler manifold of hypercomplex dimension 1 (i.e. r = 1) is flat (see Theorem 1.24 and Theorem 2.4), we assume in the sequel that mr ≥ 2m, i.e. r ≥ 2. Using Theorem 1.21, we have Theorem 2.5 The Riemannian curvature tensor of an anti-Kähler manifold is pure. From Theorem 1.23, we have Theorem 2.6 The Riemannian curvature tensor of an anti-Kähler manifold is Aholomorphic. From Theorem 1.11, we have: a necessary and sufficient condition for an exact 1-form d f , f ∈ I00 (Mmr ) to be A-holomorphic is that the associated 1-forms d f ◦ ϕ , α = 1, ..., m α

be closed, i.e.   d d f ◦ ϕ = 0, α = 1, ..., m. α

(2.15)

If there exist some functions g , α = 1, ..., m, in a hypercomplex anti-Kähler manifold α such that d f ◦ ϕ = d g , α = 1, ..., m, for a function f , then we shall call f a Aα

α

holomorphic function and g , α = 1, ..., m, associated functions. We notice that Eq. (2.15) α is equivalent to d f ◦ ϕ = d g , α = 1, ..., m, only locally. Hence, the condition for f to α

α

be locally A-holomorphic also is given by. ϕ

m ∂m α i

f = ∂i g . α

Let now (Mmr , , g) be a hypercomplex anti-Kähler manifold. Then, using Theorems 1.21 and 2.1, by virtue of (1.54), we find that in these manifolds, the covariant derivative of the curvature tensor ∇ R is also pure. Therefore, the covariant derivative of the Ricci tensor R ji = Rss ji = g ts Rt jis is pure in all its indices, and hence, ϕ st ∇s R ji = ϕ sj ∇t Rsi , α = 1, . . . , m. α

α

2.2

Kähler-Norden Manifolds

37

Contracting this equation with g ji , we find   ϕ st ∇s r = g ji ϕ sj ∇t Rsi = ∇t g ji ϕ sj Rsi α

α

α

∗

∗

= ∇t r ⇔ ϕ ϕ st ∂s r = ∂t r , α = 1, . . . , m, α

α

α

(2.16)

∗

where r = g i j Ri j is the scalar curvature and r = g ji ϕ sj Rsi . From (2.16),we have

α

α

Theorem 2.7 The scalar curvature r of an anti-Kähler manifold is a locally A-holomorphic function.

2.2

Kähler-Norden Manifolds

If Am = C (m = 2) is a complex algebra and  = {I , ϕ}, ϕ 2 = −I , I = id Mmr , then the anti-Hermitian manifold (M2r , ϕ, g) is called an almost Norden manifold; also, if the -structure is integrable, the triple (M2r , ϕ, g) is called a Norden manifold [20]. It is important that the Norden metric g is necessarily neutral metric. Metrics of this kind have been also studied under the names of pure and B-metrics (see, for example, [6–8, 16–18, 20, 22, 23, 25, 28, 31, 32, 38, 39, 42, 43, 45, 46, 51–59, 63–67, 76, 77, 81]). Let now (M2r , ϕ, g) be an almost Norden manifold. A twin Norden metric of almost Norden manifold is defined by G(X , Y ) = (g ◦ ϕ)(X , Y ) = g(ϕ X , Y ) = g(X , ϕY ) ,

(2.17)

for any X , Y ∈ I10 (M2r ). One can easily prove that G is a pure metric, i.e. DetG  = 0 and G(ϕ X , Y ) = (g ◦ ϕ)(ϕ X , Y ) = g(ϕ(ϕ X ), Y ) = g(ϕ X , ϕY ) . = (g ◦ ϕ)(X , ϕY ) = G(X , ϕY ) The twin metric G is also called the associated (or dual) metric of g, and it plays a role similar to the Kähler form in Hermitian geometry. We shall now apply the Tachibana operator to the pure Riemannian metric G: 

ϕ G (X , Y , Z ) = (L ϕ X G − L X (G ◦ ϕ))(Y , Z ) + G(Y , ϕ L X Z ) − G(ϕY , L X Z )   = ϕ g (X , ϕY , Z ) + g Nϕ (X , Y ), Z (2.18)

where Nϕ is the Nijenhuis tensor defined by Nϕ (X , Y ) = [ϕ X , ϕ X ] − ϕ[X , ϕY ] − ϕ[ϕ X , Y ] − [X , Y ].

38

2 Anti-Hermitian Geometry

It is clear that if ∇ is a torsion-free connection, then   Nϕ (X , Y ) = ∇ϕ X ϕ Y − ∇ϕY ϕ X − ϕ((∇ X ϕ)Y − (∇Y ϕ)X ). Since ϕ g = 0 ⇔ ∇ϕ = 0 ⇒ Nϕ = 0 and ϕ G = 0 ⇔ ∇ϕ = 0 ⇒ Nϕ = 0 (see Theorem 2.1), from (2.18), we have. Theorem 2.8 Let (M2r , ϕ, g) be an almost Norden manifold and G be its twin metric. The following conditions are equivalent:

(a) ϕ g = 0, (b) ϕ G = 0. Let now (M2r , ϕ, g) be an anti-Kähler (or a Kähler-Norden) manifold. We denote by ∇g the covariant differentiation with respect to the Levi–Civita connection of g. Then we have    ∇g G = ∇g g ◦ ϕ + g ◦ ∇g ϕ = g ◦ ∇g ϕ , which implies ∇g G = 0 by virtue of Theorem 2.1. Therefore, we have Theorem 2.9 Let (M2r , ϕ, g) be a Kähler-Norden manifold. Then, the Levi–Civita connection of Norden metric g coincides with the Levi–Civita connection of twin Norden metric G. Now, using Theorem 2.9, we will present the alternative proof of Theorem 2.5. Let R and S be the curvature tensors of g and G, respectively. Then, for a KählerNorden manifold, we have R = S by means of the Theorem 2.9. Applying Ricci’s identity to ϕ:  ∇ X ((∇Y ϕ)Z ) − ∇Y ((∇ X ϕ)Z ) = R(X , Y )ϕ Z − ϕ(R(X , Y )Z ) + ∇[X ,Y ] ϕ Z + (∇Y ϕ)(∇ X Z ) − (∇ X ϕ)(∇Y Z ), we get ϕ(R(X , Y )Z ) = R(X , Y )ϕ Z

(2.19)

by virtue of ∇ϕ = 0. Hence, from (2.19) and R(X 1 , X 2 , X 3 , X 4 ) = g(R(X 1 , X 2 )X 3 , X 4 ), we find that R is pure with respect to X 3 and X 4 : R(X 1 , X 2 , ϕ X 3 , X 4 ) = g(R(X 1 , X 2 )ϕ X 3 , X 4 )

2.2

Kähler-Norden Manifolds

39

= g(ϕ(R(X 1 , X 2 )X 3 ), X 4 ) = g(R(X 1 , X 2 )X 3 , ϕ X 4 ) = R(X 1 , X 2 , X 3 , ϕ X 4 ) On the other hand, since S being the curvature tensor formed by twin metric G and S(X 1 , X 2 , X 3 , X 4 ) = G(S(X 1 , X 2 )X 3 , X 4 ), we have S(X 1 , X 2 , X 3 , X 4 ) = S(X 3 , X 4 , X 1 , X 2 ).

(2.20)

Taking account of (2.17), (2.19) and R = S, we find that S(X 1 , X 2 , X 3 , X 4 ) = G(S(X 1 , X 2 )X 3 , X 4 ) = g(ϕ(S(X 1 , X 2 )X 3 ), X 4 ) = g(S(X 1 , X 2 )X 3 , ϕ X 4 ) = g(R(X 1 , X 2 )X 3 , ϕ X 4 ) = R(X 1 , X 2 , X 3 , ϕ X 4 ) and S(X 3 , X 4 , X 1 , X 2 ) = G(S(X 3 , X 4 )X 1 , X 2 ) = g(ϕ(S(X 3 , X 4 )X 1 ), X 2 ) = g(S(X 3 , X 4 )X 1 , ϕ X 2 ) = g(R(X 3 , X 4 )X 1 , ϕ X 2 ) = R(X 3 , X 4 , X 1 , ϕ X 2 ) = R(X 1 , ϕ X 2 , X 3 , X 4 ). Thus, Eq. (2.20) becomes. R(X 1 , X 2 , X 3 , ϕ X 4 ) = R(X 1 , ϕ X 2 , X 3 , X 4 ), which shows that R = R(X 1 , X 2 , X 3 , X 4 ) is pure with respect to X 2 and X 4 . Since R(X 1 , X 2 , X 3 , X 4 ) = R(X 3 , X 4 , X 1 , X 2 ), therefore R = R(X 1 , X 2 , X 3 , X 4 ) is a pure tensor with respect to all arguments. Also, from (2.16), we see that the curvature scalar r of Kähler-Norden manifold is a ∗ locally C-holomorphic function, and the associated function r coincides with the scalar curvature of the twin metric G.

40

2 Anti-Hermitian Geometry

2.3

Hessian-Norden Structures

Let (M2r , g) be a Riemannian manifold with a metric tensor g. The gradient of a function f ∈ I00 (M2r ) is the vector field metrically equivalent to the differential d f ∈ I01 (M2r )[41]. In terms of a coordinate system,   ∇ f = g i j ∂i f ∂ j Thus, g(∇ f , X ) = g i j (∂i f )X k g jk = X f = (d f )(X ). The Hessian of a function f ∈ I00 (M2r ) is its second covariant differential h = ∇(∇ f ) = ∇ 2 f with respect to the Levi–Civita connection ∇ of g, i.e. h ∈ I02 (M2r ). Since ∇Y f = Y f = (d f )(Y ) and ∇ X Y − ∇Y X − [X , Y ] = 0, we have h(Y , X ) = (∇(∇ f ))(Y , X ) = (∇(d f ))(Y , X ) = X ((d f )(Y )) − (d f )(∇ X Y ) = X Y f − (∇ X Y ) f , h(X , Y ) = (∇(∇ f ))(X , Y ) = (∇(d f ))(X , Y ) = Y ((d f )(X ) − (d f )(∇Y X ) = Y X f − (∇Y X ) f , h(Y , X ) − h(Y , X ) = X Y f − (∇ X Y ) f − (Y X f − (∇Y X ) f ) = [X , Y ] f − (∇ X Y − ∇Y X ) f = [X , Y ] f − [X , Y ] f = 0, i.e. h is a symmetric tensor field. Also, we see that       g(∇ X (∇ f ), Y ) = g(∇ X g i j ∂i f ∂ j , Y ) = g X s ∂s g i j (∂i f )∂ j   + g i j X s (∂s ∂i f )∂ j + g i j (∂i f ) X s smj ∂m , Y k ∂k = −X s g i j ∂s g jk (∂i f )Y k     l + g g i j X s (∂s ∂i f )∂ j + g i j (∂i f )X s smj ∂m , Y k ∂k = −sl j glk − sk g jl X s g i j (∂i f )Y k l i + X s Y k (∂s ∂i f )g i j g jk + smj (∂i f )X s Y k g i j gmk = X s Y k (∂s ∂k f ) − X s sk δl (∂i f )Y k     l = X s ∂s Y k ∂k f − X s ∂s Y k ∂k f − X s sk (∂l f )Y k = X s ∂s (Y f )   l − X s ∂s Y l + sk Y k ∂l f = X Y f − (∇ X Y ) f ,

i.e. h(X , Y ) = g(∇ X (∇ f ), Y ).

2.3

Hessian-Norden Structures

41

If Det h  = 0, then h = ∇ 2 f defines a new indefinite metric on M2r and is called a p Hessian metric. Let i j be the Christoffel symbol and Ri jkm be the components of the curvature tensor fields produced by the Riemannian metric g. If h pk are the contravariant components of the pseudo-Riemannian Hessian metric h, then the components of Levi–  of h are given by the following formula: Civita connection h ∇ hp ij

    1 p = i j + h pk ∇i ∇ j ∇k f + Rik jm + R jkim ∇m f . 2 ∗

Let (M2r , g, ϕ) be a Kähler-Norden manifold. If there exists a function f on M2r such ∗

∗

that d f ◦ ϕ = d f for a function f , then we shall call f a holomorphic function and f its associated function (see Sect. 2.1). If such a function f is defined locally, then we call it a locally holomorphic function.   Remark 2.1 If (M2r , ϕ) is a complex manifold, then in terms of a real coordinates x i , x i , ∗

i = 1, ..., r ; i = r + 1, ..., 2r , the equation d f ◦ ϕ = d f reduces to ⎧ ∗ ⎨ ∂ f = ∂i f , i ∗ ⎩ ∂i f = −∂i f , ∗

which is the Cauchy-Riemann equations for the complex function F = f +i f (see [26, p. 122]). We notice that the condition for f to be locally holomorphic is given also by (see Theorem 1.11).     ϕ d f i j = ϕim ∂m ∂ j f − ∂i ϕ mj ∂m f + ∂ j ϕim ∂m f = 0. If we assume that f is holomorphic, then, from (1.34), we have (ϕ (d f ))(X , Y ) = (ϕ X )((d f )(Y )) − X ((d f )(ϕY )) + (d f )((L Y ϕ)(X )) = ϕ(X )((d f )(Y )) − X ((d f )(ϕY )) + (d f )([Y , ϕ X ] − ϕ([Y , X ])) = ϕ(X )((d f )(Y )) − X ((d f )(ϕY )) + (d f )(∇Y ϕ X − ∇ϕ X Y − ϕ(∇Y X − ∇ X Y )) = (∇ϕ X d f )(Y ) − (∇ X d f )(ϕY ) + (d f )(∇ϕ X Y ) − (d f )(∇ X ϕY ) + (d f )((∇ϕ)(X , Y ) − ∇ϕ X Y + ϕ(∇ X Y )) = (∇ϕ X d f )(Y ) − (∇ X d f )(ϕY ) − (∇ϕ)(Y , X ) = 0

(2.21)

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2 Anti-Hermitian Geometry

We now consider a holomorphic function f on a Kähler-Norden manifold (M2r , g, ϕ). On a Kähler-Norden manifold (M2r , g, ϕ) (∇ϕ = 0), Eq. (2.21) is equivalent to the equation. (∇ 2 f )(Y , ϕ X ) = (∇ 2 f )(ϕY , X ), i.e. h = ∇ 2 f is pure, and a manifold (M2r , ϕ, h = ∇ 2 f ) is a Norden manifold. Thus, h naturally defines a new Norden metric on Kähler-Norden manifold (M2r , g, ϕ). We call it Hessian-Norden metric. Thus, we have. Theorem 2.10 Let (M2r , g, ϕ) be a Kähler-Norden manifold. Then M2r admits a HessianNorden structure (ϕ, h = ∇ 2 f ), if f ∈ I00 (M2r ) is holomorphic. Let (M2r , g, ϕ) be a Kähler-Norden manifold. Then g is holomorphic, and the curvature tensor R of g is pure with respect to the structure ϕ (Theorem 2.5). Let (M2n , h = ∇ 2 f , ϕ) be a Hessian-Norden structure which exists on a Kähler-Norden manifold. Then (∇ 2 f )(ϕ X , Y ) = (∇ 2 f )(X , ϕY ), from which we have (∇ 3 f )(ϕ X , Y , Z ) = (∇ 3 f )(X , ϕY , Z ). Using Ricci equation for (d f )X = X f = ∇ X f , from here we obtain (∇ 3 f )(X , ϕY , Z ) = ∇ Z (∇ϕY (∇ X f )) = ∇ϕY (∇ Z (∇ X f )) − (d f )(R(Z , ϕY )X ) = (∇ 3 f )(X , Z , ϕY ) − (d f )(R(Z , ϕY )X )

(2.22)

and (∇ 3 f )(ϕ X , Y , Z ) = ∇ Z (∇Y (∇ϕ X f )) = ∇Y (∇ Z (∇ϕ X f )) − (d f )(R(Z , Y )ϕ X )

(2.23)

= (∇ f )(ϕ X , Z , Y ) − (d f )(R(Z , Y )ϕ X ) 3

Since h is symmetric and the curvature tensor R of g is pure with respect to ϕ, from (2.22) and (2.23), we have (∇ 3 f )(Z , ϕ X , Y ) = (∇ 3 f )(Z , X , ϕY ), i.e. a tensor field ∇ 3 f is pure in all arguments. On the other hand,  ϕ h (X , Z 1 , Z 2 )

(2.24)

2.4 Twin Norden Metric Connections

43

 = (ϕ X )(h(Z 1 , Z 2 )) − X (h(ϕ Z 1 , Z 2 )) − h ∇ϕ X Z 1 , Z 2  + h((∇ϕ)(X , Z 1 ), Z 2 ) + h(Z 1 , (∇ϕ)(X , Z 2 )) − h Z 1 , ∇ϕ X Z 2 + h(ϕ(∇ X Z 1 ), Z 2 ) + h(ϕ Z 1 , ∇ X Z 2 )  = ∇ϕ X h (Z 1 , Z 2 ) − (∇ X h)(ϕ Z 1 , Z 2 ) + h((∇ϕ)(X , Z 1 ), Z 2 ) + h(Z 1 , (∇ϕ)(X , Z 2 ))

(2.25)

Substituting h(Z 1 , Z 2 ) = ∇ Z 1 ∇ Z 2 f and ∇ϕ = 0 in (2.25), by virtue of (2.24), we have   ϕ h (X , Z 1 , Z 2 ) = ϕ h (X , Z 2 , Z 1 )     = ∇ϕ X ∇ 2 f (Z 2 , Z 1 ) − ∇ X ∇ 2 f (Z 2 , ϕ Z 1 ) ,   = ∇ 3 f (Z 2 , Z 1 , ϕ X ) − ∇ 3 f (Z 2 , ϕ Z 1 , X ) = 0  = 0. Thus, we have i.e. h is holomorphic. Then, using Theorem 2.1, we see that h ∇ϕ Theorem 2.11 The Hessian-Norden triple (M2r , h = ∇ 2 f , J ) is a Kähler-Norden manifold.

2.4

Twin Norden Metric Connections

It is well known that the pair (J , g) of an almost Hermitian structure defines a fundamental 2-form  by (X , Y ) = g(J X , Y ). Let ∇ be the Levi–Civita connection of g. If the skew-symmetric tensor  is a Killing-Yano tensor, i.e. (∇ X )(Y , Z ) + (∇Y )(X , Z ) = 0

(2.26)

or equivalently, if the almost complex structure J satisfies (∇ X J )Y + (∇Y J )X = 0 for any X , Y ∈ I10 (M2r ), then the manifold is called a nearly Kähler manifold (also known as K -spaces or almost Tachibana spaces). Let now (M, g, J ) be an almost Norden manifold. Then the pair (J , g) defines, as usual, the twin Norden metric G(Y , Z ) = (g ◦ J )(Y , Z ) = g(J Y , Z ), but G is symmetric, rather than a 2-form . Thus, the Norden pair (J , g) does not give rise to a 2-form, and the Killing-Yano Eq. (2.26) has no immediate meaning. Therefore, we can replace the Killing-Yano equation by Codazzi equation (∇ X G)(Y , Z ) − (∇Y G)(X , Z ) = 0 .

(2.27)

Equation (2.27) is equivalent to (∇ X J )Y − (∇Y J )X = 0.

(2.28)

44

2 Anti-Hermitian Geometry

Theorem 2.12 Let the triple (M, g, J ) be an almost Norden manifold and G be a twin Norden metric which satisfies the Codazzi Eq. (2.27). Then J is integrable. Proof Using ∇ X Y − ∇Y X = [X , Y ], (2.28) and (∇ X J )(J Y ) = ∇ X (J (J Y )) − J (∇ X J Y ) = −∇ X Y − J ((∇ X J )Y + J (∇ X Y )) = −∇ X Y − J (∇ X J )Y − J 2 (∇ X Y ) = −J (∇ X J )Y ,

(2.29)

we have N J (X , Y ) = [J X , J Y ] − J [X , J Y ] − J [J X , Y ] − [X , Y ] = ∇ J X J Y − ∇ J Y J X − J (∇ X J Y − ∇ J Y X ) − J (∇ J X Y − ∇Y J X ) + J 2 (∇ X Y − ∇Y X ) = −J ((∇ X J )Y − (∇Y J )X ) + (∇ J X J )Y − (∇ J Y J )X = (∇Y J )J X − (∇ X J )J Y = −J ((∇Y J )X − (∇ X J )Y ) = 0, i.e. the Nijenhuis tensor N J vanishes. The proof of Theorem 2.12 is complete. In the above sections, we have given the Norden metric g and considered exclusively the Levi–Civita connection ∇ of g. This is the unique connection which satisfies ∇g = 0  with torsion parallelizing the and has no torsion. But there are many other connections ∇ metric g. We call these connections Norden metric connections with torsion.  by Let (M, g, J ) be an almost Norden manifold. If we introduce a connection ∇ X Y = ∇ X Y + S(X , Y ) ∇ for any X , Y ∈ I10 (M2r ) , where S is a tensor field of type (1.2), then the torsion tensor  is given by T of ∇ X Y − ∇ Y X − [X , Y ] T (X , Y ) = ∇ = ∇ X Y + S(X , Y ) − ∇Y X − S(Y , X ) − [X , Y ] = T ∇ (X , Y ) + S(X , Y ) − S(Y , X ) = S(X , Y ) − S(Y , X )(T ∇ = 0).  of g, we have For the covariant derivative ∇    X g (Y , Z ) = X (g(Y , Z )) − g ∇ X Y , Z − g Y , ∇ X Z ∇ = X (g(Y , Z )) − g(∇ X Y + S(X , Y ), Z ) − g(Y , ∇ X Z + S(X , Z )) = (∇ X g)(Y , Z ) − g(S(X , Y ), Z ) − g(Y , S(X , Z )) = −g(S(X , Y ), Z ) − g(Y , S(X , Z )).

(2.30)

2.4 Twin Norden Metric Connections

45

 = 0, it is necessary and sufficient that Consequently, in order to have ∇g g(S(X , Y ), Z ) + g(Y , S(X , Z )) = 0 . From here, we have  = ∇ + S is Theorem 2.13 Let (M, g, J ) be an almost Norden manifold. A connection ∇  = 0) if and only if a Norden metric connection with torsion T (i.e. ∇g S(X , Y , Z ) + S(X , Z , Y ) = 0 ,

(2.31)

where S(X , Y , Z ) = g(S(X , Y ), Z ). Now putting T (X , Y , Z ) = g(T (X , Y ), Z ), from (2.30), we have T (X , Y , Z ) = S(X , Y , Z ) − S(Y , X , Z ) . Similarly, T (Z , X , Y ) = S(Z , X , Y ) − S(X , Z , Y ) , T (Z , Y , X ) = S(Z , Y , X ) − S(Y , Z , X ) . Using (2.31), from the last equations, we obtain S(X , Y , Z ) = 1/2(T (X , Y , Z ) + T (Z , X , Y ) + T (Z , Y , X )) . Let now G(Y , Z ) = (g ◦ J )(Y , Z ) = g(J Y , Z ) be a twin Norden metric. Then, in  = 0, it is necessary and sufficient that we have order that to have ∇G 

X G (Y , Z ) = (∇ X G)(Y , Z ) − G(S(X , Y ), Z ) − G(Y , S(X , Z )) ∇

= (∇ X G)(Y , Z ) − g(J S(X , Y ), Z ) − g(J Y , S(X , Z )) = (∇ X G)(Y , Z ) − g(S(X , Y ), J Z ) − g(S(X , Z ), J Y ) = (∇ X G)(Y , Z ) − S(X , Y , J Z ) − S(X , Z , J Y ) = 0 which is equivalent to (∇ X G)(Y , Z ) − S J (X , Y , Z ) − S J (X , Z , Y ) = 0,

(2.32)

where S J (X , Y , Z ) = S(X , Y , J Z ).  is not completely determined by (2.31) and (2.32). So we can The connection ∇ introduce some other condition on S. We try to solve the equation with respect to   = 0 (i.e. S. From now on, we assume only ∇G = 0 and make no use of ∇g

46

2 Anti-Hermitian Geometry

S(X , Y , Z ) + S(X , Z , Y ) = 0). This latter equation will be satisfied in special cases as a consequence of the equation introduced in next part of section.  = ∇ + S is called a twin Norden metric connection of type I Case 1. The connection ∇ if  =0 ∇G and S J (X , Y , Z ) − S J (X , Z , Y ) = 0 .

(2.33)

From (2.32) and (2.33), we have S J (X , Y , Z ) = 1/2(∇ X G)(Y , Z ) from which S(X , Y , J Z ) = 1/2g((∇ X J )Y , Z ), g(S(X , Y ), J Z ) = 1/2g((∇ X J )Y , Z ) , g(J S(X , Y ), Z ) = 1/2g((∇ X J )Y , Z ), J S(X , Y ) = 1/2(∇ X J )Y

(2.34)

S(X , Y ) = −1/2 J (∇ X J )Y .

(2.35)

or

If we substitute J Z into Z in the second equation of (2.34), we have S(X , Y , Z ) = −1/2g((∇ X J )Y , J Z )

(2.36)

On the other hand, using g((∇ X J )Z , J Y ) − g(Z , (∇ X J )J Y ) = g(∇ X J Z − J ∇ X Z , J Y ) − g(Z , −∇ X Y − J (∇ X J Y )) = g(∇ X J Z , J Y ) + g(∇ X Z , Y ) + g(Z , ∇ X Y ) + g(J Z , ∇ X J Y ) = Xg(Z , Y ) − (∇ X g)(Z , Y ) + Xg(J Z , J Y ) − (∇ X g)(J Z , J Y ) = Xg(Z , Y ) − Xg(Z , Y ) = 0 and (2.29), we have g((∇ X J )Z , J Y ) = g(Z , (∇ X J )J Y ) = −g(Z , J ((∇ X J )Y )) = −g((∇ X J )Y , J Z ), which follows S(X , Z , Y ) + S(X , Y , Z ) = −1/2(g((∇ X J )Y , J Z ) + g((∇ X J )Z , J Y )) = 0.

2.4 Twin Norden Metric Connections

47

Thus, in an almost Norden manifold, the tensor S in the form (2.36) satisfies Eq. (2.31)  = 0), and consequently, the connection ∇  = ∇ − 1/2 J (∇ J ) is a Norden metric (i.e. ∇g connection with torsion. Thus, we have. Theorem 2.14 Every almost Norden manifold (M, g, J ) admits a twin Norden metric con = ∇ − 1/2 J (∇ J ), and such connection is also a Norden nection of type I in the form ∇ metric connection with torsion.  = ∇ + S is From (2.30) and (2.35), we see that the torsion tensor of the connection ∇ given by T (X , Y ) = −1/2 J ((∇ X J )Y − (∇Y J )X ).

(2.37)

Let now the triple (M, g, J ) be an almost Norden manifold with the Codazzi Eq. (2.27). Then from (2.28) and (2.37), we find that T = 0, i.e. the twin Norden metric connection of type I reduces to Levi–Civita connection. Thus, we have [55] (see also [17]). Theorem 2.15 If an almost Norden manifold satisfies the Codazzi Eq. (2.27), then the twin Norden metric connection of type I coincides with the Levi–Civita connection of g , i.e. the metric g and the twin metric G share the same Levi–Civita connection.  = ∇ + S is called a twin Norden metric connection of type II if Case 2. The connection ∇  =0 ∇G and S J (X , Y , Z ) − S J (Z , Y , X ) = 0 .

(2.38)

From (2.32), we have (∇ X G)(Y , Z ) − S J (X , Y , Z ) − S J (X , Z , Y ) = 0 , (∇Y G)(Z , X ) − S J (Y , Z , X ) − S J (Y , X , Z ) = 0 , (∇ Z G)(X , Y ) − S J (Z , X , Y ) − S J (Z , Y , X ) = 0 , and consequently, taking account of (2.38), we find

(∇ X G)(Y , Z ) − (∇Y G)(Z , X ) + (∇ Z G)(X , Y ) = 2S J (X , Y , Z ) = 2S(X , Y , J Z ) = 2g(S(X , Y ), J Z ) = 2g(J S(X , Y ), Z ). Since the operator  J applied to g reduces to the following form (see (2.8)):

(2.39)

48

2 Anti-Hermitian Geometry

( J g)(Y , Z , X ) = (∇ X G)(Y , Z ) − (∇Y G)(Z , X ) + (∇ Z G)(X , Y ) and the Kähler-Norden condition (∇ J = 0) is equivalent to  J g = 0 (see Theorem 2.1),  reduces to Levi–Civita from (2.39), we have S = 0, the twin Norden metric connection ∇ connection ∇, it is clear that the tensor S = 0 satisfies Eq. (2.31), and consequently, the  = ∇ is a torsion-free Norden metric connection. Thus, we have connection ∇ Theorem 2.16 If an almost Norden manifold (M, g, J ) is Kähler-Norden, then the twin Nor of type II coincides with the torsion-free Norden metric connection, den metric connection ∇ i.e. with the Levi–Civita connection ∇. Let now (∇Y G)(Z , X ) = (∇ Z G)(Y , X ). Then from (2.29) and (2.39), we find 2g(J S(X , Y ), Z ) = (∇ X G)(Y , Z ) − (∇Y G)(Z , X ) + (∇ Z G)(Y , X ) = (∇ X G)(Y , Z ) = g((∇ X J )Y , Z ) or S(X , Y ) = −1/2 J (∇ X J )Y .

(2.40)

By similar devices as above (Case 1, Theorem 2.14), we easily see that the tensor S in the form (2.40) satisfies Eq. (2.31), and consequently, the twin Norden metric connection  of type II is given by ∇  = ∇ − 1/2 J (∇ J ), i.e. coincides with the type of I and also it ∇  = ∇. Thus, we have. reduces to the Levi–Civita connection ∇ Theorem 2.17 If an almost Norden manifold satisfies the Codazzi Eq. (2.27), then the twin Norden metric connection of type II coincides with the Levi–Civita connection ∇ of g . Let now G be a Killing symmetric tensor, i.e.

σ (∇ X G)(Y , Z ) = 0, where σ is the

X ,Y ,Z

cyclic sum with respect to X , Y and Z . This is the class of the quasi-Kähler manifold [32, 38]. The structure J on such manifolds is nonintegrable. Similarly, if (M, g, J ) is quasi-Kähler, i.e. (∇ X G)(Y , Z ) + (∇Y G)(Z , X ) + (∇ Z G)(X , Y ) = 0, then from (2.39), we find 0 = (∇ X G)(Y , Z ) + (∇Y G)(Z , X ) + (∇ Z G)(X , Y )

2.5

Norden-Walker Manifolds

49

= (∇ X G)(Y , Z ) − (∇Y G)(Z , X ) + (∇ Z G)(X , Y ) + 2(∇Y G)(Z , X ) = 2g(J S(X , Y ), Z ) + 2(∇Y G)(Z , X ) = 2g(J S(X , Y ), Z ) + 2(∇Y G)(X , Z ) = 2g(J S(X , Y ), Z ) + 2g((∇Y J )X , Z ), i.e. g(J S(X , Y ), Z ) = −g((∇Y J )X , Z ) ⇒ S(X , Y ) = J (∇Y J )X . It is clear that the tensor S satisfies (2.31) (see Case 1, Theorem 2.14). Thus, we have Theorem 2.18 Every quasi-Kähler manifold (M, g, J ) admits a twin Norden metric con = ∇ + J (∇ J ), and such connection is also a Norden metric nection of type II of the form ∇ connection with torsion. Remark 2.2. Given an almost Hermitian manifold (M, g, J ), there is a unique connection  (known as the Bismut connection) with totally skew torsion which preserves both the ∇  = 0 and ∇  J = 0. For the almost complex structure and the Hermitian metric, i.e. ∇g  = 0 and ∇G  = 0, we have ∇  J = 0; therefore, in some Norden manifolds, from ∇g aspects, Norden metric connections of types I and II introduced in the present section are similar to Bismut connection.

2.5

Norden-Walker Manifolds

The main purpose of the present section is to study the Norden metrics on 4-dimensional Walker manifolds. A neutral metric g on a 4-manifold M4 is said to be a Walker metric if there exists a 2-dimensional null distribution D on M4 , which is parallel with respect to g. From Walker’s theorem [84], there is a system of coordinates (x, y, z, t) with respect to which g takes the following local canonical form ⎛

0 ⎜ ⎜0 g = (gi j ) = ⎜ ⎝1 0

0 0 0 1

1 0 a c

⎞ 0 ⎟ 1⎟ ⎟, c⎠ b

(2.41)

where a, b, c are smooth functions of the coordinates (x, y, z, t). The parallel null 2-plane   D is spanned locally by ∂x , ∂ y , where ∂x = ∂∂x , ∂ y = ∂∂y . Walker manifolds are in focus of many authors for investigations of different geometrical problems (see, for example, [2–4, 6, 11, 12, 33–37, 53, 54, 63, 64, 66]).

50

2 Anti-Hermitian Geometry

In [35], a proper almost complex structure with respect to g is defined as a g-orthogonal almost complex structure J so that J is a standard generator of a positive π2 rotation on D, i.e. J ∂x = ∂ y and J ∂ y = −∂x . Then for the Walker metric g, such a proper almost complex structure J is determined uniquely as ⎛

0 ⎜ ⎜1 J =⎜ ⎝0 0

−1 0 0 0

−c 1 2 (a − b) 0 1

1 2 (a

c −1 0

− b)

⎞ ⎟ ⎟ ⎟ ⎠

(2.42)

In [6], for such a proper almost complex structure J on Walker 4-manifold M, an almost Norden structure (g N + , J ) is constructed, where g N + is a metric on M, with properties g N + (J X , J Y ) = −g N + (X , Y ). In fact, as one of these examples, such a metric takes the form (see Proposition 6 in [6], unfortunately, the calculations of the component N+ in [6] are erroneous): g44 ⎛

gN+

0 ⎜ ⎜ −2 =⎜ ⎝0 −b

−2 0 −a −2c

0 −a 0 1 2 (1 − ab)

⎞ −b ⎟ −2c ⎟ ⎟. 1 ⎠ (1 − ab) 2 −2bc

We may call this an almost Norden-Walker metric. The construction of such a structure in [6] is to find a Norden metric for a given almost complex structure, which is different from the Walker metric. The purpose of the present section is to find also an almost Norden-Walker structure (g, F), where the metric is nothing but the Walker metric g, with an appropriate almost complex structure F, to be determined. That is, for a fixed metric g, we will find an almost complex structure F which satisfies g(F X , FY ) = −g(X , Y ). In [6], for a given almost complex structure, a metric is constructed. Our method is, however, for a given metric, an almost complex structure is constructed. Let F be an almost complex structure on a Walker manifold M4 , which satisfies (i) F 2 = −I , (ii) g(F X , Y ) = g(X , FY ) (iii) F∂x = ∂ y , F∂ y = −∂x . We easily see that these three properties define F nonuniquely, i.e.

2.5

Norden-Walker Manifolds

51

⎧ F∂x = ∂ y , ⎪ ⎪ ⎪ ⎪ ⎨ F∂ = −∂ , y x ⎪ F∂z = α∂x + 21 (a + b)∂ y − ∂t , ⎪ ⎪ ⎪ ⎩ F∂t = − 21 (a + b)∂x + α∂ y + ∂z , and F has the local components ⎛

0 ⎜ 1 ⎜ F = (F ji ) = ⎜ ⎝0 0

−1 0 0 0

⎞ − 21 (a + b) ⎟ 1 ⎟ 2 (a + b) α ⎟ ⎠ 0 1 −1 0 α

  with respect to the natural frame ∂x , ∂ y , ∂z , ∂t , where α = α(x, y, z, t) is an arbitrary function. We must note that the proper almost complex structure J as in (2.42) is determined uniquely. In our case of the almost Norden-Walker structure, the almost complex structure F just obtained contains an arbitrary function α(x, y, z, t). Our purpose is to find a nontrivial almost Norden-Walker structure with the Walker metric g explicitly. Therefore, we now put α = c. Then g defines a unique almost complex structure ⎛

0 ⎜ 1 ⎜ ϕ = (ϕ ij ) = ⎜ ⎝0 0

−1 0 0 0

⎞ − 21 (a + b) ⎟ 1 ⎟ 2 (a + b) c ⎟. ⎠ 0 1 −1 0 c

(2.43)

The triple (M4 , ϕ, g) is called almost Norden-Walker manifold. In conformity with the terminology of [34, 35], we call ϕ the proper almost complex structure. Remark 2.3 From (2.43), we immediately see that in the case a = −b and c = 0, ϕ is integrable. We now consider the general case for integrability. The almost complex structure ϕ on almost Norden-Walker manifolds is integrable if and only if i i (Nϕ )ijk = ϕ mj ∂m ϕki − ϕkm ∂m ϕ ij − ϕm ∂ j ϕkm + ϕm ∂k ϕ mj = 0.

(2.44)

From (2.43) and (2.44), we find the following integrability condition: Theorem 2.19 The proper almost complex structure ϕ on almost Norden-Walker manifolds is integrable if and only if the following PDEs hold:  ax + bx + 2c y = 0, (2.45) a y + b y − 2cx = 0.

52

2 Anti-Hermitian Geometry

From this theorem, we see that if a = −b and c = 0, then ϕ is integrable (see Remark 2.3). Let (M4 , ϕ, g) be an integrable almost Norden-Walker manifold (Nϕ = 0) and a = b. Then Eq. (2.45) reduces to  ax = −c y , (2.46) a y = cx , from which follows ax x + a yy = 0, cx x + c yy = 0, i.e. the functions a and c are harmonic with respect to the arguments x and y. Thus, we have Theorem 2.20 If the triple (M4 , ϕ, g) is an integrable almost Norden-Walker manifold and a = b, then a and c are harmonic with respect to the arguments x,y. Example 2.1 We now apply the Theorem 2.20 to establish the existence of special types of Norden-Walker metrics. In our arguments, the harmonic function plays an important role. Let a = b and h(x, y) be a harmonic function of variables x and y, for example, h(x, y) = e x cos y. We put a = a(x, y, z, t) = h(x, y) + α(z, t) = e x cos y + α(z, t), where α is an arbitrary smooth function of z and t. Then, a is also hormonic with respect to x and y. We have ax = e x cos y, a y = −e x sin y. From (3.46), we have PDEs for c to satisfy as cx = a y = −e x sin y, c y = −ax = −e x cos y. These PDEs have the solutions c = −e x sin y + β(z, t),

2.6

Kähler-Norden-Walker Manifolds

53

where β is arbitrary smooth function of z and t. Thus, the Norden-Walker metric reduces to the following form: ⎛

0 ⎜ ⎜0 g = (gi j ) = ⎜ ⎝1 0

2.6

0 0 0 1

1 0 e x cos y + α(z, t) −e x sin y + β(z, t)

⎞ 0 ⎟ 1 ⎟ ⎟. −e x sin y + β(z, t) ⎠ e x cos y + α(z, t)

Kähler-Norden-Walker Manifolds

Let (M4 , ϕ, g) be an almost Norden-Walker manifold. If  ϕ g ki j = ϕkm ∂m gi j − ϕim ∂k gm j  + gm j ∂i ϕkm − ∂k ϕim + gim ∂ j ϕkm = 0

(2.47)

then by virtue of Theorem 2.1, ϕ is integrable and the triple (M4 , ϕ, g) is called a holomorphic Norden-Walker or a Kähler-Norden-Walker manifold. Taking account of Theorem 2.2, we see that Kähler-Norden-Walker manifold with condition ϕ g = 0 and Nϕ  = 0 does not exist. Substituting (2.41) and (2.43) in (2.47), we see that the nonvanishing components of ϕ g are 

ϕ g x zz  ϕ g x zt  ϕ g xtt  ϕ g yzz  ϕ g yzt  ϕ g ytt  ϕ g zx x  ϕ g zxt  ϕ g zyz 

ϕ g

z,t

 ϕ g zzz 

ϕ g

zzt

= ay ,  1 = ϕ g x zz = (bx − ax ) + c y , 2 = b y − 2cx , = −ax ,  = ϕ g yt z = 21 (b y − a y ) − cx , = −bx − 2c y ,    = ϕ g zx x = ϕ g t xt = ϕ g tt x = cx ,    = ϕ g zt x = − ϕ g t x z = − ϕ g t zx = 21 (ax + bx ),    = ϕ g zzy = ϕ g t yt = ϕ g tt y = c y ,    1 = ϕ g zt y = − ϕ g t zz = − ϕ g t yy = a y + b y , 2 1 = cax − at + 2cz + (a + b)a y , 2  1 = ϕ g zt z = ccx + bz + (a + b)c y , 2

54

2 Anti-Hermitian Geometry

 ϕ g ztt  ϕ g t zz  ϕ g t zt  ϕ g ttt

= cbx + at − 2cz + 21 (a + b)b y , 1 = ca y − bz − (a + b)ax , 2  = ϕ g tt z = cc y − at + 2cz − 21 (a + b)cx , = cb y + bz − 21 (a + b)bx .

(2.48)

From (2.48), we have. Theorem 2.21 The triple (M4 , ϕ, g) is Kähler-Norden-Walker if and only if the following PDEs hold: ax = a y = cx = c y = bx = b y = bz = 0,

at − 2cz = 0.

(2.49)

Remark 2.4 The triple (M4 , ϕ, g) with metric ⎛

0 ⎜ ⎜0 g = (gi j ) = ⎜ ⎝1 0

0 0 0 1

1 0 a(z) 0

⎞ 0 ⎟ 1 ⎟ ⎟ 0 ⎠ b(t)

is always Kähler-Norden-Walker.

2.7

On Curvatures of Norden-Walker Metrics

If R and r are, respectively, the curvature tensor and the scalar curvature of the Walker metric, then the nonvanishing components of R and r have, respectively, expressions (see [34, 35]) 1 1 1 1 Rx x zz = − ax x , Rx xt = − cx x , Rx yzz = − ax y , Rx yyt = − cx y , 2 2 2 2 1 1 1 1 1 1 Rx zzt = axt − cx z − a y bx + cx c y , Rx xtt = − bx x , Rx x yz = − cx y , 2 2 4 4 2 2 1 1 1 1 1 1 1 2 Rx xt y = − bx y , Rxttt = cxt − bx z − (cx ) + ax bx − bx c y + b y cx , 2 2 2 4 4 4 4 1 1 R yzyz = − a yy , R yyyt = − c yy 2 2 1 1 1 1 1 1  2 1 R yzzt = a yt − c yz − ax c y + a y cx − a y b y + c y , R ybtt = − b yy , 2 2 4 4 4 4 2 R yt zt = 21 c yt − 21 b yz − 41 cx c y + 41 a y bx Rzt zt = czt − 21 att − 21 bzz − 41 a(cx )2 + 41 aax bx + 41 cax b y − 21 ccx c y − 21 at cx

2.8

Isotropic Anti-Kähler Manifolds

55

+ 21 ax ct − 41 ax bz + 41 ca y bx + 41 ba y b y − 14 b(c y )2 − 21 bz c y + 41 a y bt + 14 az bx + 21 b y cz − 41 at b y

(2.50)

and r = ax x + 2cx y + b yy .

(2.51)

Suppose that the triple (M4 , ϕ, g) is Kähler-Norden-Walker. Then from (2.49) and (2.50), we see that Rzt zt = czt − 21 att = − 21 (at − 2cz )t = 0. and the other components of R directly all vanish. Thus, we have Theorem 2.22 If a Norden-Walker manifold (M4 , ϕ, g) is Kähler-Norden-Walker, then M4 is flat. Let (M4 , ϕ, g) be a Norden-Walker manifold with the integrable proper structure ϕ, i.e. Nϕ = 0. If a = b, then from the proof of Theorem 2.19, we see that Eq. (2.46) holds. If c = c(y, z, t) and c = c(x, z, t), then cx y = (cx ) y = (c y )x = 0, and by virtue of (2.46), we find a = a(x, z, t) and a = (y, z, t), respectively. Using of cx y = 0 and ax x +a yy = 0, we from (2.51) obtain r = 0. Thus, we have the following result: ˜ are a Theorem 2.23 If (M4 , ϕ, g) and (M4 , ϕ, g) integrable proper structure ϕ and with metrics ⎛ ⎞ ⎛ 001 0 0 ⎜ ⎟ ⎜ 0 0 0 1 ⎜ ⎟ ⎜0 g=⎜ ⎟, g˜ = ⎜ ⎝ 1 0 a(x, z, t) c(y, z, t) ⎠ ⎝1 0 1 c(y, z, t) a(x, z, t) 0

Norden-Walker manifolds with the

0 0 0 1

⎞ 1 0 ⎟ 0 1 ⎟ ⎟, a(y, z, t) c(x, z, t) ⎠ c(x, z, t) a(y, z, t)

˜ are scalar flat. then both (M4 , ϕ, g) and (M4 , ϕ, g)

2.8

Isotropic Anti-Kähler Manifolds

It is well known that the inner product in the vector space can be extended to an inner product in the tensor space. In fact, if t and t are tensors of type (r , s) with components t

1

i 1 ...ir j1 ... js

...kr and t lk11...l , then. s 2

1

2

56

2 Anti-Hermitian Geometry

g(t , t ) = gi1 k1 ...gir kr g j1 l1 ...g js ls 1 2

t i1 ...ir t k1 ...kr 1 j1 ... js 2l1 ...ls

If t = t = ∇ϕ ∈ I12 (M2n ), then the square norm ∇ϕ2 of ∇ϕ is defined by 1

2

m ∇ϕ2 = g i j g kl gms (∇ϕ)ik (∇ϕ)sjl ,

where ∇ is the Levi–Civita connection of g. An almost Norden manifold (M4 , ϕ, g) is said to be isotropic anti-Kähler if ∇ϕ2 = 0 (see [21]). It is clear that if the triple (M4 , ϕ, g) is Kähler-Norden, then it is isotropic anti-Kähler. Our purpose in this section is to show that an almost Norden-Walker manifold (M4 , ϕ, g) is isotropic anti-Kähler. The inverse of the metric tensor (2.41) is given by ⎛

g −1

−a ⎜ −c ⎜ = (g i j ) = ⎜ ⎝1 0

−c −b 0 1

1 0 0 0

⎞ 0 ⎟ 1⎟ ⎟. 0⎠ 0

(2.52)

After some calculations, we see that the nonvanishing components of ∇ϕ are y

y

∇x ϕzx = ∇x ϕt = cx , ∇ y ϕzx = ∇ y ϕt = c y , y

∇z ϕxx = −∇z ϕ y = ∇z ϕzz = −∇z ϕtt = 21 a y + 21 cx , y

∇z ϕx = ∇z ϕ yx = ∇z ϕzt = ∇z ϕtz = − 21 ax + 21 c y , ∇z ϕzx = 2cz + cax − at − 21 cc y − 21 acx + 21 ba y , y

∇z ϕz = az + 41 ac y − 41 bc y + ca y + 43 aax + 41 bax , ∇z ϕtx = 14 aax − 41 bax + ca y + 43 bc y + ccx + 41 ac y , y

∇z ϕt = 2cz + 21 cc y − at + 21 ba y + 21 cax − 21 acx , y

∇t ϕxx = −∇t ϕ y = ∇t ϕzz = −∇t ϕtt = 21 c y + 21 bx , y

∇t ϕx = ∇t ϕ yx = ∇t ϕzt = ∇t ϕtz = − 21 cx + 21 b y , ∇t ϕzx = 23 ccx + bz − 21 cb y − 21 abx + 21 bc y , y

∇t ϕz = 41 ab y − 41 bb y − 41 acx + 41 bcx , ∇t ϕtx = 41 acx − 41 bcx + cc y + 41 bb y + cbx − 41 ab y , y

∇t ϕt = 21 cb y + bz + 21 bc y + 21 ccx − 21 abx Using (2.41), (2.52) and (2.53), we find m ∇ϕ2 = g i j g kl gms (∇ϕ)ik (∇ϕ)sjl = 0.

Thus, we have

(2.53)

2.9

Quasi-Kähler-Norden-Walker Manifolds

57

Theorem 2.24 An almost Norden-Walker manifold (M4 , ϕ, g) is an isotropic anti-Kähler manifold.

2.9

Quasi-Kähler-Norden-Walker Manifolds

The basis class of nonintegrable almost complex manifolds with Norden metric is the class of the quasi-Kähler manifolds. An almost Norden manifold (M2n , ϕ, g) is called a quasi-Kähler, if σ

X ,Y ,Z

g((∇ X ϕ)Y , Z ) = 0,

where σ is the cyclic sum by three arguments. If we add (ϕ g)(X , Y , Z ) and (ϕ g)(Z , Y , X ) (see (2.8) or (2.9)), then by virtue of g(Z , (∇Y ϕ)X ) = g((∇Y ϕ)Z , X )

(2.54)

(ϕ g)(X , Y , Z ) + (ϕ g)(Z , Y , X ) = 2g((∇Y ϕ)Z , X )

(2.55)

we find

Since (ϕ g)(X , Y , Z ) = (ϕ g)(X , Z , Y ), (ϕ g)(Y , Z , X ) = −g((∇Y ϕ)Z , X ) + g((∇ Z ϕ)Y , X ) + g(Z , (∇ X ϕ)Y ), from (2.54) and (2.55), we have (ϕ g)(X , Y , Z ) + (ϕ g)(Y , Z , X ) + (ϕ g)(Z , X , Y ) =

σ

X ,Y ,Z

g((∇ X ϕ)Y , Z ).

Thus, we have. Theorem 2.25 Let (M2n , ϕ, g) be an almost Norden manifold. Then the Norden metric g is a quasi-Kähler-Norden if and only if (ϕ g)(X , Y , Z ) + (ϕ g)(Y , Z , X ) + (ϕ g)(Z , X , Y ) = 0 , for any X , Y , Z ∈ I10 (M2n ). Equation (2.56) also can be written in the form: (ϕ g)(X , Y , Z ) + 2g((∇ X ϕ)Y , Z ) = 0.

(2.56)

58

2 Anti-Hermitian Geometry

From here we see that, if we take a local coordinate system, then a Norden-Walker manifold (M4 , ϕ, g) satisfying the condition of vanishing for k gi j + 2∇k G i j is called a quasi-Kähler manifold, where G is defined by G i j = ϕim gm j . After some calculations, we see that the nonvanishing components of ∇G are ∇x G zz = ∇x G tt = cx , ∇ y G zz = ∇ y G tt = c y , 1 1 ∇z G x z = ∇z G zx = −∇z G yt = −∇z G t y = a y + cx , 2 2 1 1 ∇z G xt = ∇z G t x = ∇z G yz = ∇z G zy = c y − ax , 2 2 1 ∇z G zz = 2cz − at + a y (a + b) + cax , 2 1 1 1 1 ∇z G zt = ∇z G t z = ca y + ccx − ax (a + b) + c y (a + b), 2 2 4 4 1 ∇z G tt = 2cz − at + cc y − cx (a + b), 2 1 1 ∇t G x z = ∇t G zx = −∇t G yt = −∇t G t y = bx + c y , 2 2 1 1 ∇t G xt = ∇t G t x = ∇t G yz = ∇t G zy = b y − cx , 2 2 1 ∇t G zz = bz + ccx + c y (a + b), 2 1 1 1 1 ∇t G zt = ∇t G t = cc y + cbx − cx (a + b) + b y (a + b), 2 2 4 4 ∇t G tt = bz + cb y − 21 bx (a + b).

(2.57)

From (2.48) and (2.57), we have. Theorem 2.26 A triple (M4 , ϕ, g) is a quasi-Kähler-Norden-Walker manifold if and only if the following PDEs hold: bx = b y = bz = 0, a y − 2cx = 0, ax − 2c y = 0, cax − at + 2cz − (a + b)cx = 0.

2.10

The Goldberg Conjecture

Let now (M4 , ϕ, g) be an almost Hermitian manifold. The Goldberg conjecture (see [36]). states that an almost Hermitian manifold (M4 , ϕ, g) must be Kähler (or ϕ must be integrable) if the following three conditions are imposed: (G 1 ) if M is compact, (G 2 ) g is Einstein, and (G 3 ) if the fundamental 2-form  = g ◦ ϕ is closed.

2.10 The Goldberg Conjecture

59

It should be noted that no progress has been made on the Goldberg conjecture, and the original conjecture is still an open problem. Despite many papers by various authors concerning the Goldberg conjecture, there is only Sekigawa paper [72] which obtained substantial results to the original Goldberg conjecture: let (M4 , ϕ, g) be an almost Hermitian manifold, which satisfies the three conditions (G 1 ),(G 2 ) and (G 3 ). If the scalar curvature of M is nonnegative, then ϕ must be integrable. Let now (M4 , ϕ, w g) be an indefinite almost Kähler-Walker-Einstein compact manifold with the proper almost complex structure (2.42). As noted before, many examples of Norden-Walker metrics can be obtained by g N + (J X , J Y ) = −g N + (X , Y ) (see Sect. 2.5), and as one of these examples, such a metric has components ⎛

gN+

0 ⎜ ⎜ −2 =⎜ ⎝0 −b

−2 0 −a −2c

0 −a 0 1 2 (1 − ab)

⎞ −b ⎟ −2c ⎟ ⎟. 1 ⎠ 2 (1 − ab) −2bc

with respect to the Walker coordinates. Using Theorem 2.2, we have Theorem 2.27 The proper almost complex structure ϕ on indefinite almost Kähler-WalkerEinstein compact manifold (M4 , ϕ, w g) is integrable if ϕ g N + = 0, where g N + is the induced Norden-Walker metric on M4 . This resolves the conjecture of Goldberg under the additional restriction on NordenWalker metric (g N + ∈ K er ϕ ).

3

Problems of Lifts

In the first part of this chapter we focus on lifts from a manifold to its tensor bundle. Some introductory material concerning the tensor bundle is provided in Sect. 3.1. Section 3.2 is devoted to the study of the complete lifts of (1,1)-tensor fields along cross-setions in the tensor bundle. In Sect. 3.3 we study holomorphic cross-sections of tensor bundles. In the second part we concentrate our attention to lifts from a manifold to its tangent bundles of order 1 and 2 by using the realization of holomorphic manifolds. The main purpose of Sects. 3.4–3.9 is to study the differential geometrical objects on the tangent bundle of order 1 corresponding to dual-holomorphic objects of the dual-holomorphic manifold. As a result of this approach, we find a new class of lifts, i.e. deformed complete lifts of functions, vector fields, forms, tensor fields and linear connections in the tangent bundle of order 1. Section 3.10 is devoted to the study of holomorphic metrics in the tangent bundle of order 2 (i.e. in the bundle of 2-jets) by using the Tachibana operator. By using the algebraic approach, the problem of deformed lifts of functions, vector fields and 1-forms is solved in Sects. 3.11 and 3.12. In Sect. 3.13, we investigate the complete lift of the almost complex structure to cotangent bundle and prove that it is a transfer by a symplectic isomorphism of complete lift to tangent bundle if the symplectic manifold with almost complex structure is an almost holomorphic A-manifold. Finally, in Sect. 3.14 we transfer via the differential of the musical isomorphism defined by pseudo-Riemannian metrics the complete lifts of vector fields and almost complex structures from the tangent bundle to the cotangent bundle. Throughout the chapter we suppose that all tensor fields on manifolds and their bundles are of class C ∞ .

© The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Salimov, Applications of Holomorphic Functions in Geometry, Frontiers in Mathematics, https://doi.org/10.1007/978-981-99-1296-4_3

61

62

3 Problems of Lifts

3.1

Tensor Bundles

 p p Let Mn be a differentiable manifold of class C ∞ . Then the set Tq (Mn ) = P∈Mn Tq (P) is, by definition, the tensor bundle of type ( p, q) over Mn , where ∪ denotes the disjoint p p union of the tensor spaces Tq (P) for all P ∈ Mn . For any point P˜ of Tq (Mn ) the p surjective correspondence P˜ → P determines the natural projection π : Tq (Mn ) → Mn . p In order to introduce a manifold structure in Tq (Mn ), we define local charts on it as follows: If x j , j = 1, . . . , n are the local coordinates in a neighborhood U of P ∈ Mn , p then a tensor t at P, i.e. a point P˜ = (P, t) which is an element of Tq (Mn ) is expressible i ...i

i ...i

in the form (x j , t j11 ... jpq ), where t j11 ... jpq are the components of t with respect to the natural p

basis in Tq (P). We may consider

i ...i ¯ (x j , t j11 ... jpq ) = (x j , x j ) = (x J ), j = 1, . . . , n, j¯ = n + 1, . . . , n + n p+q ,

J = 1, . . . , n + n p+q . as local coordinates in a neighborhood π −1 (U ) ⊂ Tq (Mn ). p It is straightforward to see that Tq (Mn ) becomes an (n + n p+q )-manifold; indeed if  x j , j  = 1, . . . , n are local coordinates in another neighborhood V of P ∈ Mn , with U ∩ V = ∅, then the change of coordinates is given by ⎧ j  ⎨ x = x j (x j ),    (3.1)  ⎩ x j  = t i1 ...i p = Ai1 . . . Ai p A j1 . . . A jq t i1 ...i p = A(i  ) A( j) x j , i i j ... j (i) (j ) j 1 q j ... j j p 1 p

1

q

=

i Ai11

q

1

where (i  ) ( j) A(i) A( j  )

...

The Jacobian of (3.1) is:  ⎛ ∂ x j   J ∂x j = ⎝ ∂ xj ∂x ∂x J ∂x j

i j Ai pp A j1 1



∂x j ∂x j  ∂x j ∂x j

...

⎞ ⎠=

j A jq , q



 Aii



∂xi ∂x j j = , A =  . j ∂xi ∂x j

j

(i)

Aj

(i  )



0 (k)

(i  )

( j)

t(k) ∂ j (A(i) A( j  ) ) A(i) A( j  ) i ...i

,

(3.2)

(i) where J = ( j, j), J = 1, . . . , n + n p+q , t(k) = tk11 ...kpq . r We denote by Is (Mn ) the module over F(Mn ) (F(Mn ) is the ring of C ∞ -functions q on Mn ) all tensor fields of class C ∞ and of type (r , s) on Mn . If α ∈ I p (Mn ), it is p regarded, in a natural way (by contraction), as a function in Tq (Mn ), which we denote by α(t) = ıα. If α has the local expression j ... j

α = αi11...i pq ∂ j1 ⊗ · · · ⊗ ∂ jq ⊗ dx i1 ⊗ · · · ⊗ dx i p

3.1 Tensor Bundles

63

in a coordinate neighborhood U (x j ) ⊂ Mn , then α(t) = ıα has the local expression j ... j

i ...i

ıα = αi11...i pq t j11 ... jpq with respect to the coordinates (x j , x j ) in π −1 (U ). For later use, we first shall state following theorem [9]: p Theorem 3.1 Let X˜ and Y˜ be vector fields on Tq (Mn ) such that X˜ (ια) = Y˜ (ια), for any q p α ∈ I p (M). Then X˜ = Y˜ , i.e. X˜ ∈ I10 (Tq (M)) is completely determined by its action on functions of type ια. p

p

Suppose that A ∈ Iq (Mn ). Then there is a unique vector field V A ∈ I10 (Tq (Mn )) such q that for α ∈ I p (Mn ) [30] V

A(ıα) = α(A) ◦ π = V (α(A)),

(3.3)

where V (α(A)) is the vertical lift of the function α(A) ∈ F(Mn ). We note that the vertical lift V f = f ◦ π of an arbitrary function f ∈ F(Mn ) is constant along each fibre π −1 (P). p The vector field V A is called the vertical lift of A to Tq (Mn ). If V A = V Ak ∂k + V Ak ∂k , p then from (3.3) we see that the vertical lift V A of A to Tq (Mn ) has the components V V

A=

V

Aj Aj



 =

0

 (3.4)

i ...i

A j11 ... jpq

p

with respect to the coordinates (x j , x j ) in Tq (Mn ). Let now ϕ ∈ I11 (Mn ), i.e. ϕ = ϕ ij

∂ ⊗ dx j . ∂xi

p

p

Two new vector fields γ ϕ ∈ I10 (Tq (Mn )), γ˜ ϕ ∈ I10 (Tq (Mn )) are defined by ⎧ p

⎪ i ...m...i i λ ∂ ⎪ ⎪γϕ = ( t j11 ... jq p ϕm ) , ( p ≥ 1, q ≥ 0), ⎪ ⎪ ⎨ ∂x j λ=1

q ⎪

⎪ ∂ i ...i p m ⎪ ⎪ t j11 ...m... , ( p ≥ 0, q ≥ 1), γ ˜ ϕ = ( ⎪ jq ϕ jμ ) ⎩ ∂x j μ=1 p

with respect to the coordinates (x j , x j ) in Tq (Mn ). From (3.2) we easily see that the p vector fields γ ϕ and γ˜ ϕ determine respectively global vector fields on Tq (Mn ). 1 Let L V be the Lie derivative with respect to a vector field V ∈ I0 (Mn ). The complete p lift c V of V to Tq (Mn ) is defined by [30]

64

3 Problems of Lifts c

V (ıα) = ı(L V α),

(3.5)

q

for any α ∈ I p (M). If c V = c V k ∂k + c V k ∂k , then from (3.5) we see that the complete lift c V of V has components ⎞ ⎛  c j Vj V ⎟ ⎜ P c q  (3.6) V = =⎝ i 1 ...m...i p i 1 ...i p iλ − m⎠ c j t ∂ V t ∂ V m j V j1 ... jq j1 ...m... jq μ λ=1

μ=1

p

with respect to the coordinates (x j , x j ) in Tq (M) (see [31, 48, 49, 61, 62, 68]). Let now ∇V be a covariant derivative with respect to the vector field V ∈ I01 (Mn ), where ∇ is the symmetric affine connection on Mn . We define the horizontal lift H V ∈ p p I10 (Tq (Mn )) of V ∈ I01 (Mn ) to Tq (Mn ) by [30] H

q

V (ıα) = ı(∇V α), ∀α ∈ I p (Mn ). p

From here we see that the horizontal lift H V of V ∈ I01 (Mn ) to Tq (Mn ) has components ⎛ ⎞ Vj ⎜ ⎟ q p H  V =⎝ s  i ...i p i λ i 1 ...m...i p ⎠ V ( smjμ t j11 ...m... − t ) sm j1 ... jq jq μ=1

λ=1

p

with respect to the coordinates (x j , x j ) in Tq (Mn ), where ikj are local components of ∇ (see [49]). p Suppose that there is given a tensor field ξ ∈ Iq (Mn ). If the correspondence x → ξx , p where ξx is the value of ξ at x ∈ Mn , determines a mapping σξ : Mn → Tq (Mn ) such p that π ◦ σξ = id Mn , then the n dimensional submanifold σξ (Mn ) of Tq (Mn ) is called the h ...h

cross-section determined by ξ . If the tensor field ξ has the local components ξk11...kqp (x k ), the cross-section σξ (Mn ) is locally expressed by ⎧ ⎨ xk = xk, (3.7) ⎩ x k = ξ h 1 ...h p (x k ). k1 ...kq Differentiating (3.7) by x j , we see that n tangent vector fields B j to σξ (Mn ) have the components   K  δ kj ∂x K = (3.8) (B j ) = h ...h ∂x j ∂ j ξk11...kqp   p with respect to the natural frame ∂k , ∂k in Tq (Mn ). On the other hand, the fibre is locally expressed by

3.1 Tensor Bundles

65

⎧ ⎨ x k = const, ⎩ x k = t h 1 ...h p , k1 ...kq h ...h

where tk11...kqp are considered as parameters. Thus, by differentiating with respect to x j = i ....i

t j11 ... jqp , we see that n p+q tangent vector fields C j to the fibre have components  (C Kj¯ ) =

∂xK ∂ x j¯



 =



0 j

j

(3.9)

h

δk11 . . . δkqq δih11 . . . δi pp

  p with respect to the natural frame ∂k , ∂k on Tq (Mn ), where δ is the Kronecker symbol. p We consider in π −1 (U ) ⊂ Tq (Mn ), n + n p+q local vector fields B j and C j along   σξ (Mn ). They form a family of local frames B j , C j along σξ (Mn ), which is called the adapted (B, C)-frame of σξ (Mn ) in π −1 (U ). From c V = c V h ∂h + c V h ∂h and c V = cV j B

j

j

j

j

j

+ c V C j we easily obtain c V k = c V B kj + c V C k , c V k = c V B kj + c V C k . Now, j j taking account of (3.6) on the cross-section σξ (Mn ), and of (3.8) and (3.9) also, we firstly j

k

k

h ...h

find c V = V k and therefore c V = −L V ξk11...kqp . Thus, the complete lift c V has along σξ (Mn ) components of the form ⎛ k⎞   c V Vk c ⎠= (3.10) V =⎝ h ...h −L V ξk11...kqp c k V with respect to the adapted (B, C)-frame. From (3.4), (3.8) and (3.9) we see that the vertical lift V A also has components  V k   A¯ 0 V A= = (3.11) h ...h V ¯ k¯ Ak11...kqp A with respect to the adapted (B, C)-frame. In a similar way we see that the horizontal lift H V has along σξ (Mn ) components of the form  H j   V¯ Vj H V = = i ...i H ¯ j¯ −(∇V ξ ) j11 ... jpq V i ...i

with respect to the adapted (B, C)-frame, where (∇V ξ ) j11 ... jpq are local components of ∇V ξ . Let now ϕ ∈ I11 (M). We can easily verify that γ ϕ and γ˜ ϕ have along σξ (M) components

66

3 Problems of Lifts

⎛ p γ ϕ = ((γ ϕ) J ) = ⎝  λ=1

⎛

⎞

0 i ...m....i p i λ ϕm

ξ j11 ... jq

q ⎠, γ˜ ϕ = ((γ˜ ϕ) J ) = ⎜ ⎝ μ=1

⎞

0 i ...i

p m ξ j11 ...m... jq ϕ jμ

⎟ ⎠ (3.12)

with respect to the adapted (B, C)-frame.

3.2

Complete Lifts of (1,1)-Tensor Fields p

Let ϕ ∈ I11 (Mn ). The cross-section σξ : Mn → Tq (Mn ) defined by the tensor field p ξ ∈ Iq (Mn ) is called a pure cross-section if ξ is a pure tensor field with respect to ϕ (See ϕ Chap. 1) and denoted by σξ (Mn ). The complete lift c ϕ of ϕ along the pure cross-section p of Tq (Mn ) is defined as p

Definition 3.1 [31, 50, 68] We define a tensor field c ϕ ∈ I11 (Tq (Mn )) along the pure ϕ cross-section σξ (Mn ) by 

c ϕ(c V ) c ϕ(V

= c (ϕ(V )) − γ (L V ϕ) + V ((L V ϕ) ◦ ξ ), ∀V ∈ I10 (Mn ), (i) p A) = V (ϕ(A)), ∀A ∈ Iq (Mn ), (ii) p

(3.13)

ϕ

and call c ϕ the complete lift of ϕ ∈ I11 (Mn ) to Tq (Mn ), p ≥ 1, q ≥ 0 along σξ (Mn ), where p p ϕ(A) ∈ Iq (Mn ), ((L V ϕ) ◦ ξ )(x1 , . . . , xq , α1 , . . . , α p ) ∈ Iq (Mn ). In particular, if we assume that p = 1, q ≥ 0, then from (3.11) and (3.12) we get γ (L V ϕ) = V ((L V ϕ) ◦ ξ ). Substituting this into (3.13), we find c

ϕ(c V ) = c (ϕ(V )), c ϕ(V A) = V (ϕ(A)) .

We observe that the local vector fields      h ∂ ∂ δj c = c δ hj h = X ( j) = c ∂x j ∂x 0 and V

X ( j) = V (∂ j1 ⊗ · · · ⊗ ∂ j p dx i1 ⊗ · · · ⊗ dx iq ) i

k

= V (δhi11 . . . δhqq δ kj11 . . . δ j pp ∂k1 ⊗ · · · ⊗ ∂k p ⊗ dx h 1 ⊗ · · · ⊗ dx h q )

3.2

Complete Lifts of (1,1)-Tensor Fields

 =

67



0 i

k

δhi11 . . . δhqq δ kj11 . . . δ j pp

,

j = 1, . . . , n, j = n + 1, . . . , n + n p+q span the module of all vector fields in π −1 (U ). Hence any tensor field is determined in π −1 (U ) by its action on c X ( j) and V X ( j) . Now, let us calculate the components of the complete lift of tensor fields of type (1,1) by using the ϕ -operator. ϕ

p

Theorem 3.1 Let ϕ ∈ I11 (Mn ) and σξ (Mn ) be a pure cross-section of Tq (Mn ) with respect p ϕ to ϕ. Then the complete lift C ϕ ∈ I11 (Tq (Mn )) of ϕ has along the pure cross-section σξ (Mn ) the components ⎧ h 1 ...h p ⎨ C ϕ lk = ϕlk , C ϕ k = 0, C ϕ lk = −( ϕ ξ )lk , 1 ...kq l (3.14) ⎩ C ϕ k = ϕ h 1 δ h 2 . . . δ h p δ r1 . . . δ rq s s s p k k 1 2 q 1 l ϕ

with respect to the adapted (B, C)-frame of σξ (Mn ), where ϕ is the Tachibana operator h ...h

s ...s

and x k = tk11...kqp , x l = tr11...rqp . Remark 3.1 The formula (3.14) is a useful expansion of (3.10) to (1,1)-tensor fields along the pure cross-section by using the Tachibana operator instead of the Lie derivative. Proof Let c ϕ LK be components of c ϕ with respect to the adapted (B, C)-frame of the pure ϕ cross-section σξ (Mn ). Then from (3.13) we have 

c ϕ¯ K c V¯ L = c (ϕ(V )) K − (γ (L ϕ)) K V L c ϕ¯ K V A ¯ L = V (ϕ(A)) K , L

+ V ((L V ϕ) ◦ ξ ) K , (i) (ii)

(3.15)

where  ( (ϕ(A)) ) = V

K

mh ...h p

 (γ (L V ϕ)) K =



0

2 ϕmh 1 Ak1 ...k q

0



, ((L V ϕ) ◦ ξ ) = V

K

 mh ...h p

h1 2 )ξk1 ...k ((L V ϕ)m q

0

 h ...m...h p

(L V ϕmh λ )ξk11...kq

,

.

First, consider the case where K = k. In this case, (i) of (3.15) reduces to c kc l ϕl V

l

+ c ϕ lk c V = c (ϕ(V ))k = (ϕ(V ))k = ϕlk V l .

(3.16)

Since the right-hand side of (3.16) are functions depending only on the base coordinates x i , the left-hand sides of (3.16) are too. Then, since c V l = 0 depend on fibre

68

3 Problems of Lifts

coordinates, from (3.16) we obtain c k ϕl

= 0.

(3.17)

l

From (3.16) and (3.17) we have c ϕ lk c V = c ϕ lk V l = ϕlk V l , V i being arbitrary, which implies c k ϕl

= ϕlk .

When K = k, (ii) of (3.15) reduces to c kV ϕl

l

l

A + c ϕ lk V A = V (ϕ(A))k

or c k˜ s1 ...s p ϕ l˜ Ar1 ...rq

mh ...h p

2 = ϕmh 1 Ak1 ...k q

r

h

s s ...s

= δkr11 . . . δkqq ϕsh11 δsh22 . . . δs pp Ar11 r22 ...rqp

p

for all A ∈ Iq (Mn ), which implies c k ϕl h ...h

r

h

= δkr11 . . . δkqq ϕsh11 δsh22 . . . δs pp ,

s ...s

where x k = tk11...kqp , x l = tr11...rqp . When K = k, (i) of (3.15) reduces to c kc l ϕl V

l + c ϕ lk c V

= (ϕ(V )) − c

k

p

λ=2

h ...l...h p

(L V ϕlh λ )ξk11...kq

or c kc l ϕl V

h

r

l

+ ϕsh11 δsh22 . . . δs pp δkr11 . . . δkqq c V +

p

λ=2

h h ...l...h p

(L V ϕlh λ )ξk11...k2 q

= c (ϕ(V ))k .

(3.18)

Now, using the Tachibana operator we will investigate the components c ϕ lk . Let

∗ p I q (Mn )

p

denote module of all the tensor fields ξ ∈ Iq (Mn ) which are pure with respect ∗ p

to ϕ. The Tachibana operator on the pure module I q (Mn ) is given by (see Chap. 1) h ...h

h ...h

∗ h ...h

( ϕ ξ )lk11 ...kpq = ϕlm ∂m ξk11...kqp − ∂l ξ k11...kqp +

q p

  h ...h p h ...m...h ∂ka ϕlm ξk11...m...k + 2 ∂[l ϕ hm]λ ξk11...kq p , q λ=1

a=1

where h ...h

h ...h

h ...h

mh ...h p

p 2 ϕkm1 ξmk1 2 ...kpq = ϕkm2 ξk11m...kpq = · · · = ϕkmq ξk11k2 ...m = ϕmh 1 ξk1 ...k q

h m...h p

= ϕmh 2 ξk11...kq

3.2

Complete Lifts of (1,1)-Tensor Fields

69 ∗ h h ...h

h 2 ...m = · · · = ϕmp ξkh11...k = ξ k11k22...kqp . q h

After some calculations we have V

l

h ...h ( ϕ ξ )lk11 ...kpq

=

∗ h ...h

h ...h (L ϕV ξ )k11...kqp

− (L V ξ )k11...kqp +

p

λ=1

h ...m...h p

(L V ϕmh λ )ξk11...kq

or h ...h p

mh ...h p

h

2 V l ( ϕ ξ )lk1 ...k + ϕm1 (L V ξ )k ...k q q 1 1

mh ...h p

2 + ξk ...k q 1

h

(L V ϕ)m1 −

p

λ=1

h

h ...m...h p

(L V ϕmλ )ξk 1...k 1

q

h ...h p

= (L ϕV ξ )k 1...k 1

q

(3.19)

for any V ∈ I10 (Mn ). Using (3.10), from (3.19) we have h ...h

mh ...h p

2 V l ( ϕ ξ )lk11 ...kpq + ϕmh 1 (L V ξ )k1 ...k q

=V

l

= V c

h ...h ( ϕ ξ )lk11 ...kpq

l

+ ϕsh11 δsh22

h ...h ( ϕ ξ )lk11 ...kpq

mh ...h p

2 + ξk1 ...k q

h . . . δs pp δkr11

− ϕsh11 δsh22

(L V ϕmh 1 ) −

λ=1

r s ...s . . . δkqq (L V ξ )r11 ...rqp

h . . . δs pp δkr11

p

r ¯ . . . δkqq c V l

−

p

λ=2

−

h ...m...h p

(L V ϕmh λ )ξk11...kq

p

λ=2

h h ...m...h p

(L V ϕmh λ )k11...k2 q h h ...m...h p

(L V ϕmh λ )ξk11...k2 q

¯

= −c (ϕ(V ))k

or h ...h

h

r

−( ϕ ξ )lk11 ...kpq c V l + ϕsh11 δsh22 . . . δs pp δkr11 . . . δkqq c V l +

p

λ=2

h h ...m...h p

(L V ϕmh λ )ξk11...k2 q

= c (ϕ(V ))k . (3.20)

Comparing (3.18) and (3.20), we get c k ϕl

h ...h

= −( ϕ ξ )lk11 ...kpq .

This completes the proof. Remark 3.2 We note that the lift c ϕ in the form (3.14) is the unique solution of (3.13). ∗ p Therefore, if ϕ is another element of I11 (Tq (Mn )) such that ∗

∗

ϕ (c V ) = c ϕ(c V ) = c (ϕ(V )) − γ (L V ϕ) + V ((L V ϕ) ◦ ξ ), ϕ (V A) = c ϕ(V A) = V (ϕ(A)), ∗

then ϕ = c ϕ.

70

3 Problems of Lifts

Remark 3.3 If p = 1, q = 0, then (3.14) is the formula of the complete lift of affinor fields ϕ to the tangent bundle along the cross-section σξ (Mn ) (for details, see [87, p. 126]). ϕ

Now, setting B j = C j we write the adapted (B, C)-frame of σξ (Mn ) as B J =  ϕ B j , B j . We define a coframe B˜ J of σξ (Mn ) by B˜ I (B J ) = δ JI . From (3.8), (3.9) and B K B˜ I = δ I we see that covector fields B˜ I have components



J

K

J

B˜ i = ( B˜ Ki ) = (δki , 0), j ... j k j j B˜ i = ( B˜ Ki ) = (−∂k ξi11...iq p , δik11 . . . δiqq δh11 . . . δh pp )

(3.21)

with respect to the natural coframe (dx k , dx k ). Taking account of c K ϕL

= c ϕ(dx K , ∂ L ) = c ϕ IJ B J ⊗ B˜ I (dx K , ∂ L ) = c ϕ IJ dx K (B J ) B˜ I (∂ L ) K ˜I = c ϕ IJ dx K (B JH ∂ H ) B˜ LI = c ϕ IJ B JH δ H B L = c ϕ IJ B JK B˜ LI ,

and also (3.8), (3.9), (3.14) and (3.21), we see that the complete lift c ϕ has along the pure ϕ cross-section σξ (Mn ) components of the form c k ϕl

c k ϕl

h

r

= ϕlk , c ϕlk = 0, c ϕlk = ϕsh11 δsh22 . . . δs pp δkr11 . . . δkqq ,

mh ...h p

2 = (∂l ϕmh 1 )ξk1 ...k q

−

q

μ=1

h ...h

p (∂kμ ϕlm )ξk11...m...k − q

p

λ=1

h ...m...h p

(∂l ϕmh λ − ∂m ϕlh λ )ξk11...kq

  ϕ with respect to the natural frame ∂k , ∂k of σξ (Mn ) in π −1 (U ) [70].

3.3

Holomorphic Cross-Sections

Theorem 3.2 If ϕ is an integrable almost complex structure on Mn , then the complete lift p ϕ of ϕ to Tq (Mn ), p ≥ 1, q ≥ 0 along the pure cross-section σξ (Mn ) is an almost complex structure.

Cϕ

Proof Let ϕ ∈ I11 (Mn ) and S ∈ I12 (Mn ). Using (3.10), (3.12) and (3.14) we have γ (ϕ ± ψ) = γ ϕ ± γ ψ, C ϕ(γ ψ) = γ (ϕ ◦ ψ) = γ (ψ ◦ ϕ), (γ S)C V = γ SV ,

(3.22)

where SV is the tensor field of type (1.1) on Mn defined by SV (W ) = S(V , W ) for any ϕ W ∈ I10 (Mn ) and γ S is the affinor field along σξ (Mn ) with the components

3.3

Holomorphic Cross-Sections

71

⎛ p γS = ⎝  λ=1

0 j

0 j ...m... j p

λ S jm ξi11...iq

0

⎞ ⎠

(3.23)

j

1 are the local components of S. It is clear that with respect to the adapted (B, C)-frame, S jm V (S ◦ ξ ) is also affinor field along σ (M ) with the components ξ n   0 0 V (S ◦ ξ ) = j1 m j2 ... j p S jm ξi1 ...iq 0

with respect to the adapted (B, C)-frame. If V ∈ I10 (Mn ), then from (3.13) and (3.22) we have (C ϕ)2 (C V ) = (C ϕ ◦ C ϕ)C V = C ϕ(C ϕ(C V )) = C ϕ(C (ϕ(V ))) − γ (L V ϕ) + V ((L V ϕ) ◦ ξ ) = C ϕ(C (ϕ(V ))) − C ϕ(γ (L V ϕ)) + C ϕ(V ((L V ϕ) ◦ ξ )) = C (ϕ(ϕ(V ))) − γ (L ϕ(V ) ϕ) − C ϕ(γ (L V ϕ)) + C ϕ(V ((L V ϕ) ◦ ξ )) + V ((L ϕ(V ) ϕ) ◦ ξ ) = C ((ϕ ◦ ϕ)(V )) − γ (L ϕ(V ) ϕ) − γ ((L V ϕ) ◦ ϕ) + V (ϕ((L V ϕ) ◦ ξ ) + (L ϕ(V ) ϕ) ◦ ξ ) = C ((ϕ ◦ ϕ)(V )) − γ (L ϕ(V ) ϕ + (L V ϕ) ◦ ϕ) + V ((L ϕ(V ) ϕ) + (L V ϕ) ◦ ξ ) +

V

((L V (ϕ ◦ ϕ)) ◦ ξ ) = C (ϕ ◦ ϕ)(C V ) + γ (L V (ϕ ◦ ϕ)) − γ (L ϕ(V ) ϕ + (L V ϕ) ◦ ϕ)

= C (ϕ ◦ ϕ)(C V ) − γ (L ϕ(V ) ϕ − ϕ ◦ (L V ϕ)) + V ((L ϕ(V ) ϕ) − ϕ(L V ϕ) ◦ ξ ) = C (ϕ ◦ ϕ)(c V ) − γ N V + V (N V ◦ ξ ) = C (ϕ ◦ ϕ)(C V ) − (γ N )(C V ) + V (N ◦ ξ )(C V ) = (C (ϕ 2 ) − γ N + V (N ◦ ξ ))(C V ),

(3.24)

where N V = L ϕ(V ) ϕ − ϕ ◦ (L V ϕ) and ( ϕ ϕ)(V , W ) = (L ϕ(V ) ϕ − ϕ ◦ (L V ϕ))W = [ϕV , ϕW ] − ϕ[V , ϕW ] − ϕ[ϕV , W ] + ϕ 2 [V , W ] = Nϕ (V , W ) is nothing but the Tachibana operator or the Nijenhuis tensor Nϕ (V , W ) ∈ I12 (M) constructed from ϕ. p Similarly, if A ∈ Iq (M), then from (3.13) we have (C ϕ)2 (V A) = (C ϕ ◦ C ϕ)V A = C ϕ(C ϕ V A) = C ϕ(V (ϕ(A))) = V (ϕ(ϕ(A))) = V ((ϕ ◦ ϕ)(A)) = C (ϕ ◦ ϕ)V A = C (ϕ 2 )V A.

(3.25)

72

3 Problems of Lifts

If we take the integrability condition of ϕ (Nϕ = 0), then by the Remark 3.2, (3.24), (3.25) and the linearity of the complete lift, we have (C ϕ)2 = C (ϕ 2 ) = C (−I ) = −I . Let M2n and N2m be two manifolds with complex structures ϕ and ψ, respectively. A differentiable mapping f : M → N is called a holomorphic (analytic) mapping [46, 47] if at each point P ∈ M2n d f p ◦ ϕ p = ψ f ( p) ◦ d f p .

(3.26)

As the mapping f : M2n → N2m (2m = 2n + (2n) p+q ) we take a pure cross-section p p : M → Tq (M) determined by the pure tensor field ξ ∈ Iq (M) with respect to the p ϕ ϕ-structure. The pure cross-section σξ : M → Tq (M) can be locally expressed by (3.7). In (3.26), if ψ is the complex structure C ϕ (see Theorem 3.2), then the condition that the ϕ pure cross-section σξ be a holomorphic cross-section is locally given by ϕ σξ

K ∂l x M , ϕlm ∂m x K = C ϕ M

(3.27) ϕ

where C ϕ K of C ϕ along the pure cross-section σξ (Mn ) with respect to M are components   the natural frame ∂k , ∂k . In the case K = k, by virtue of (3.7) and (3.14) we get the identity ϕlk = ϕlk . When K = k, by virtue of (3.7) and (3.14), (3.27) reduces to h ...h

h ...h

∗ h ...h

( ϕ ξ )lk11 ...kpq = ϕlm ∂m ξk11...kqp − ∂l ξ k11...kqp +

q

  h ...h p ∂ka ϕlm ξk11...m...k q a=1

+2

p

λ=1

h ...m...h p

∂[l ϕ hm]λ ξk11...kq

= 0,

(3.28)

which is the condition to be holomorphic of tensor field ξ (see Theorem 1.9), where ϕ ξ is the Tachibana operator. From (3.27) and (3.28) we have Theorem 3.3 Let (Mn , ϕ) be a complex manifold. Then the complete lift C ϕ of ϕ to p ϕ the tensor bundle Tq (M) along the pure cross-section σξ (Mn ) leaves the submanifold p ϕ σξ (Mn ) ⊂ Tq (M) invariant if and only if the tensor field ξ is holomorphic with respect to.

3.4

Dual-Holomorphic Functions and Tangent Bundles of Order 1

We consider a two-dimensional dual algebra R(ε), ε2 = 0 (ε is nilpotent) with a standard γ γ basis {e1 , e2 } = {1, ε} and structural constants Cαβ : eα eβ = Cαβ eγ , α, β, γ = 1, 2,

3.4

Dual-Holomorphic Functions and Tangent Bundles of Order 1

73

1 = C 2 = C 2 = 1, C 1 = C 1 = C 1 = C 2 = C 2 = 0 are the components where C11 12 21 12 21 22 11 22 of the (1,2)-tensor C : R(ε) × R(ε) → R(ε). Let Z = x α eα be a variable in R(ε), where x α (α = 1, 2) are real variables. Using a real-valued C ∞ -function f β (x) = f β (x 1 , x 2 ), β = 1, 2, we introduce a dual function F = f β (x)eβ of variable Z ∈ R(ε). It is well known that the dual function F = F(Z ) is holomorphic if and only if the following Scheffers condition holds (see Sect. 1.2):

C2 D = DC2 , 

∂ fα ∂xβ

(3.29) 



f α (x),

is the Jacobian matrix of

=

00



, γ and β 1 0 denotes the row and column numbers of matrix C2 , respectively. The condition (3.29) reduces to the following equations: where D =

C2 =

γ (C2β )

∂ f2 ∂ f1 ∂ f1 = 0, = . 2 2 ∂x ∂x ∂x1 From here it follows that the dual-holomorphic function F = F(Z ) has the following explicit form: F(Z ) = f (x 1 ) + ε(x 2 f  (x 1 ) + g(x 1 )), df 1 ∞ -function. where f (x 1 ) = f 1 (x 1 ), f  (x 1 ) = dx 1 and g = g(x ) is any real C By similar devices, we see that the dual-holomorphic multi-variable function F = F(Z 1 , . . . , Z n ), Z i = x i + εx n+i , i = 1, . . . , n has the form:

F(Z 1 , . . . , Z n ) = f (x 1 , . . . , x n ) + ε(x n+s ∂s f + g(x 1 , . . . , x n )),

(3.30)

where g = g(x 1 , . . . , x n ) is any real multi-variable C ∞ -function, ∂s f = ∂∂xfs . A dual-holomorphic manifold [81] X n (R(ε)) of dimension n is a Hausdorff space with a fixed atlas compatible with a group of R(ε)-holomorphic transformations of space Rn (ε), where Rn (ε) = R(ε) × · · · × R(ε) is the space of n-tupes of dual numbers (z 1 , z 2 , . . . , z n ) with z i = x i + εy i ∈ R(ε), x i , y i ∈ R, i = 1, . . . , n. We shall identify Rn (ε) with R2n , when necessary, by mapping (z 1 , z 2 , . . . , z n ) ∈ Rn (ε) into (x 1 , . . . , x n , y 1 , . . . , y n ) ∈ R2n and therefore the R(ε)-holomorphic manifold X n (R(ε)) is a real manifold M2n of dimension 2n. Let now Mn be a differentiable manifold, T (Mn ) its tangent bundle, and π the projection T (Mn ) → Mn . The tangent bundle T (Mn ) consists of the pairs (x, v), where x ∈ Mn and v ∈ Tx (Mn ) (Tx (Mn ) is the tangent vector space at x ∈ Mn ). Let (U , x = (x 1 , . . . , x n )) be a coordinate chart in Mn . Then it induces the local coordinates (x 1 , . . . , x n , x n+1 , . . . , x 2n ) in π −1 (U ), where x n+1 , . . . , x 2n represent the components of v ∈ Tx (Mn ) with respect to the local frame {∂i }. In the following we use the notation i = i + n for all i = 1, . . . , n.

74

3 Problems of Lifts 



If (U  , x  = (x 1 , . . . , x n )) is another coordinate chart in Mn , then the induced     coordinates (x 1 , . . . , x n , x 1 , . . . , x n ) in π −1 (U  ), will be given by ⎧  i i i ⎪ ⎨ x = x (x ), i = 1, . . . , n,  (3.31)  ∂xi i ⎪ ⎩ xi = x , i = n + 1, . . . , 2n. ∂xi The Jacobian of (3.31) is the matrix ⎛     ∂xi ∂xα i = ⎝ ∂x  S= 2 i ∂xα xs ∂ x

∂xi ∂xs

⎞ 0 ⎠ , α = 1, . . . , 2n . 

∂xi ∂xi

From here follows that there exists a tensor field of type (1,1) ⎞  ⎛    ϕ ij ϕ ij 0 0 α ϕ = ϕβ = ⎝ i i ⎠ = (I = (δ ij ) is an identity matrix of degree n) I 0 ϕj ϕ

(3.32)

j

with properties ϕ 2 = 0 and Sϕ = ϕ S, i.e. the transformation S : {∂α } → {∂α  } preserving ϕ is an admissible dual transformation. Thus, the tangent bundle T (Mn ) of a manifold Mn carries a natural dual structure ϕ, which is integrable (∂k ϕ ij = 0). Therefore, with each induced coordinates (x i , x i ) in π −1 (U ) ⊂ T (Mn ), we associate the local dual coordinates X i = x i + εx i , ε2 = 0. Using (3.31) we see that the local dual coordinates X i = x i + εx i are transformed by 





X i = x i (x i ) + εx s ∂s (x i (x i )). 

(3.33)

Equation (3.33) shows that the quantities X i are the dual-holomorphic functions of i X = x i + εx i (see (3.30) with g(x 1 , . . . , x n ) = 0). Thus the tangent bundle T (Mn ) with a natural integrable ϕ-structure is a real image of the dual-holomorphic manifold X n (R(ε)), dim X n (R(ε)) = n [81]. In such interpretation there exists a one-to-one correspondence between dual tensor fields on X n (R(ε)) and pure tensor fields with respect to the ϕ-structure on T (Mn ) (see Sect. 1.5). It is important that the dual tensor field on X n (R(ε)) corresponding to a pure C ∞ tensor field is not necessarily dual-holomorphic. This tensor field is dual-holomorphic on X n (R(ε)) if and only if the -operator associated with ϕ and applied to a pure tensor field t of type (1, q) or ω of type (0, q) satisfies the following conditions (see Sect. 1.6) ( ϕ t)(Y , X 1 , . . . , X q ) = −(L t(X 1 ,X 2 ,...,X q ) ϕ)Y +

q

λ=1

t(X 1 , X 2 , . . . , (L X λ ϕ)Y , . . . , X q ) = 0

3.5

Deformed Complete Lifts of Vector Fields

75

or ( ϕ ω)(Y , X 1 , . . . , X q ) = (ϕY )(ω(X 1 , X 2 , . . . , X q )) − Y (ω(ϕ X 1 , X 2 , . . . , X q )) +

q

ω(X 1 , X 2 , . . . , ϕ(L Y X λ ), . . . , X q ) = 0,

λ=1

where L Y is the Lie derivation with respect to Y . From (3.30) we immediately have F = V f + ε(C f + V g), where g is any function on Mn , V f = f ◦ π , V g = g ◦ π are the vertical lifts of f , g and C f = x n+s ∂ f is the complete lift of f from M to its tangent bundle T (M ) (see [87]). s n n The study the theory of lifts in the tangent bundles was started by Yano and Kobayashi [85] (see also [69]) and devoleped by many authors (see for example [1, 5, 8, 14, 15, 24, 42, 43, 58, 59, 78]). We call D f = C f + V g the deformed complete lift of a function f to the tangent bundle T (Mn ). Thus we have Theorem 3.4 Let T (Mn ) be a tangent bundle of Mn , which is a real image of the dualholomorphic manifold X n (R(ε)). Then the vertical and the deformed complete lifts to T (Mn ) of any function on Mn are the real and dual part of corresponding dual-holomorphic function on X n (R(ε)), respectively.

3.5

Deformed Complete Lifts of Vector Fields

In a tangent bundle T (Mn ) with dual structure ϕ, a vector field V˜ = (v˜ α ) = (v˜ i , v˜ n+i ) = (v˜ i , v˜ i ) is called a dual-holomorphic vector field if L V˜ ϕ = 0. Such a vector field is the real image of corresponding dual-holomorphic vector field V = (V i ) on X n (R(ε)), where V i = v˜ i + v˜ i ε. The condition of dual-holomorphy of a vector field V˜ on T (Mn ) may be now locally written as follows: L V˜ ϕβα = v˜ σ ∂σ ϕβα − (∂σ v˜ α )ϕβσ + (∂β v˜ σ )ϕσα = 0 .

(3.34)

By virtue of (3.32), we have (a) The case where α = i, β = j, the Eq. (3.34) reduces to L V˜ ϕ ij = v˜ σ ∂σ ϕ ij − (∂σ v˜ i )ϕ σj + (∂ j v˜ σ )ϕσi i i = v˜ m ∂m ϕ ij + v˜ m ∂m ϕ ij − (∂m v˜ i )ϕ mj − (∂m v˜ i )ϕ mj + (∂ j v˜ m )ϕm + (∂ j v˜ m )ϕm

76

3 Problems of Lifts

= −(∂m v˜ i )δ mj = −(∂ j v˜ i ) = 0,

from which it follows v˜ i = v i (x 1 , . . . , x n ).

(3.35)

(b) In the case where α = i, β = j and α = i, β = j, the Eq. (3.34) reduces to 0 = 0. (c) In the case where α = i, β = j, the Eq. (3.34) reduces to L V˜ ϕ ij = v˜ σ ∂σ ϕ ij − (∂σ v˜ i )ϕ σj + (∂ j v˜ σ )ϕσi i i = v˜ m ∂m ϕ ij + v˜ m ∂m ϕ ij − (∂m v˜ i )ϕ mj − (∂m v˜ i )ϕ mj + (∂ j v˜ m )ϕm + (∂ j v˜ m )ϕm i i = −(∂m v˜ i )ϕ mj + (∂ j v˜ m )ϕm = −(∂m v˜ i )δ mj + (∂ j v˜ m )δm = 0,

from which it follows ∂ j v˜ i = ∂ j v i , and after integrating, we find v˜ i = x j ∂ j v i + wi (x 1 , . . . , x n ),

(3.36)

where wi = wi (x 1 , . . . , x n ) are any real multi-variable C ∞ -functions. 

α

Remark 3.4 Using (3.31), (3.35), (3.36) and v˜ α = ∂∂xx α v˜ α , we easily see that v = (v i (x 1 , . . . , x n )) and w = (wi (x 1 , . . . , x n )) are vector fields on Mn . Thus a real dual-holomorphic vector field V˜ on the tangent bundle can be written in the form       i  v˜ vi v i (x 1 , . . . , x n ) 0 α ˜ = = + = C v + V w, V = (v˜ ) = j i i 1 n j i i wi x ∂ j v + w (x , . . . , x ) x ∂jv v˜ where C v and V w are the complete and vertical lifts of vector fields v = (v i ) and w = (wi ) from Mn to the tangent bundle T (Mn ), respectively [87]. Thus we have Theorem 3.5 Let T (Mn ) be the tangent bundle of Mn , which is the real image of dualholomorphic manifold X n (R(ε)). Then the real image of corresponding dual-holomorphic

3.6

Deformed Complete Lifts of Tensor Fields of Type (1,1)

77

vector field V = (V i ) = (v˜ i + v˜ i ε) is a deformed complete lift in the form D V = C v + V w, where C v and V w are the complete and vertical lifts of the vector fields v = (v i ) and w = (wi ) from Mn to T (Mn ), respectively.

3.6

Deformed Complete Lifts of Tensor Fields of Type (1,1)

A tensor field t˜ of type (1,1) on the tangent bundle T (Mn ) is called pure with respect to the dual structure ϕ if t˜(ϕ X ) = ϕ(t˜ X ), for any vector field X on T (Mn ). From here we see that the condition of pure tensor fields may be expressed in terms of the local induced coordinates as follows: t˜σβ ϕασ = t˜ασ ϕσβ . Using (3.32), from the last condition we have     t˜ij 0 α t˜ = t˜β = i i . t˜j t˜j

(3.37)

A pure tensor field t˜ is called a dual-holomorphic tensor field if ϕ t˜ = 0, where ϕ is the Tachibana operator defined by ( ϕ t˜)(X , Y ) = [ϕ X , t˜Y ] − ϕ[X , t˜Y ] − t˜[ϕ X , Y ] + ϕ t˜[X , Y ]. We note that, such a tensor field is the real image of corresponding dual-holomorphic tensor field from X n (R(ε)). Sometimes the tensor ϕ t˜ of type (1,2) is called the Nijenhuis-Shirokov tensor field. It is clear that, if ϕ = t˜, then ϕ t˜ is the Nijenhuis tensor field Nϕ , i.e. ϕ ϕ = Nϕ . The condition of dual-holomorphy of a pure tensor field t˜ on T (Mn ) may be now locally written as follows: ( ϕ t˜)αγβ = ϕγσ ∂σ t˜βα − ϕσα ∂γ t˜βσ − t˜βσ ∂σ ϕγα + t˜σα ∂β ϕγσ = 0.

(3.38)

By virtue of (3.32) and (3.37), the Eq. (3.38) after some calculations reduces to ∂k t˜ij = 0, ∂k t˜ij − ∂k t˜ij = 0. From here it follows that t˜ij = t ij (x 1 , . . . , x n ), t˜ij = x k ∂k t ij + g ij ,

(3.39)

78

3 Problems of Lifts

where g ij = g ij (x 1 , . . . , x n ). 

Remark 3.5 Using (3.31), (3.39) and tβα =



∂xα ∂xβ α ∂ x α ∂ x β  tβ ,

we easily see that t ij (x 1 , . . . , x n )

and g ij (x 1 , . . . , x n ) are components of any tensor fields t and g of type (1,1) on Mn .

Thus, a dual-holomorphic tensor field t˜ on the tangent bundle can be written in the form         t ij 0 t ij 0 0 0 α t˜ = t˜β = = + = C t + V g, g ij 0 x k ∂k t ij + g ij t ij x k ∂k t ij t ij where C t and V g are the complete and vertical lifts of (1,1)-tensor fields t and g from Mn to tangent bundle T (Mn ), respectively (see [87]). Thus we have Theorem 3.6 Let T (Mn ) be the tangent bundle of Mn , which is the real image of the dualholomorphic manifold X n (R(ε)). Then the real image of corresponding dual-holomorphic tensor field T of type (1,1) from X n (R(ε)) is the deformed complete lift in the form D t = C t + V g, where C t and V g are the complete and vertical lifts of the (1,1)-tensor fields t and g from Mn to T (Mn ), respectively. Let (M4n , F, G, H ) be an almost quaternion manifold, i.e. F 2 = −I , G 2 = −I ,

H 2 = −I ,

F = G H = −H G, G = H F = −F H ,

H = F G = −G F.

Then for three tensor fields F, G and H of type (1,1), we now consider the deformed complete lifts: D

F = C F + V G,

D

G = CG + V H,

D

H = C H + V F.

From here, we find    0 F jm 0 Fmi D 2 ( F) = x s ∂s F jm + G mj F jm x s ∂s Fmi + G im Fmi   0 Fmi F jm = x s (∂s Fmi )F jm + G im F jm + Fmi x s ∂s F jm + Fmi G mj Fmi F jm     F2 −I Mn 0 0 = = = −IT (Mn ) . x s ∂s F 2 + G F + F G F 2 0 −I Mn

3.7

Deformed Complete Lifts of 1-Forms

79

Similarly we get ( D G)2 = −IT (Mn ) , ( D H )2 = −IT (Mn ) . Thus we have Theorem 3.7 Let (M4n , F, G, H ) be an almost quaternion manifold. Then the deformed complete lift of each structure F, G and H is an almost complex structure on the tangent bundle T (M4n ).

3.7

Deformed Complete Lifts of 1-Forms

A 1-form ω˜ on the tangent bundle T (Mn ) is called a dual-holomorphic 1-form, if ϕ ω˜ = 0, where ϕ is the Tachibana operator defined by ( ϕ ω)(X ˜ , Y ) = (ϕ X )(ω(Y ˜ )) − X (ω(ϕY ˜ ) + ω((L ˜ Y ϕ)X ) . Such a 1-form is a real image of corresponding dual-holomorphic 1-form from X n (R(ε)). The tensor field ϕ ω˜ of type (0,2) has components ( ϕ ω) ˜ αβ = ϕασ ∂σ ω˜ β − ϕβσ ∂α ω˜ σ − ω˜ σ (∂α ϕβσ − ∂β ϕασ ) with respect to the natural frame {∂α } = {∂i , ∂i }. ˜ αβ = 0 reduces to By virtue of (3.32), the equation ( ϕ ω) ∂i ω˜ j − ∂i ω˜ j = 0, ∂i ω˜ j = 0. From here we have ω˜ j = ω j (x 1 , . . . , x n ), ω˜ j = x i ∂i ω j + θ j (x 1 , . . . , x n ).

(3.40)

∂xβ ˜ β , we easily see that ω j (x 1 , . . . , x n ) and ω ∂xβ 1-forms ω and θ on Mn , respectively.

Remark 3.6 Using (3.31), (3.40) and ω˜ β  = θ j (x 1 , . . . , x n ) are the components of any

Thus, a real dual-holomorphic 1-form ω˜ on tangent bundle can be rewritten in the form ω˜ = (ω˜ j , ω˜ j ) = (x i ∂i ω j + θ j , ω j ) = (x i ∂i ω j , ω j ) + (θ j , 0) = C ω + V θ, where C ω and V θ are the complete and vertical lifts of 1-forms ω = (ω j ) and θ = (θ j ) from Mn to its tangent bundle T (Mn ), respectively (see [87]). Thus we have

80

3 Problems of Lifts

Theorem 3.8 Let T (Mn ) be the tangent bundle of Mn , which is the real image of the dual-holomorphic manifold X n (R(ε)). Then the real image of the corresponding dualholomorphic 1-form from X n (R(ε)) is a deformed complete lift in the form D ω = C ω + V θ , where C ω and V θ are the complete and vertical lifts of 1-forms ω = (ω j ) and θ = (θ j ) from Mn to T (Mn ), respectively.

3.8

Deformed Complete Lifts of Riemannian Metrics

A tensor field g˜ of type (0,2) on the tangent bundle T (Mn ) is called a pure tensor field with respect to the dual structure ϕ if g(ϕ ˜ X , Y ) = g(X ˜ , ϕY ), for any vector fields X and Y on T (Mn ). From here we see that the condition of purity of g˜ may be expressed in terms of the local induced coordinates as follows: g˜ σβ ϕασ = g˜ ασ ϕβσ . Using (3.32), from the last condition we have     g˜i j g˜i j , g˜i j = 0, g˜i j = g˜i j . g˜ = g˜ αβ = g˜i j 0 A pure tensor field g˜ of type (0,2) on the tangent bundle T (Mn ) is called dualholomorphic with respect to ϕ, if ϕ g˜ = 0, where ϕ is the Tachibana operator defined by ( ϕ g)(X ˜ , Y , Z ) = (ϕ X )(g(Y ˜ , Z )) − X (g(ϕY ˜ , Z )) + g((L ˜ ˜ , (L Z ϕ)X ), Y ϕ)X , Z ) + g(Y for every X , Y , Z ∈ I10 (T (Mn )). Such a tensor field is the real image of the corresponding dual-holomorphic tensor field of type (0,2) from X n (R(ε)). It is well known that, if g˜ is a Riemannian metric and ∇ g˜ its Levi–Civita connection, then the condition ϕ g˜ = ˜ ϕ) is a dual 0 is equivalent to the condition ∇ g˜ ϕ = 0 [25], i.e. the triple (T (Mn ), g, anti-Kähler (or Kähler-Norden) manifold. The tensor field ϕ g˜ of type (0,3) has the components ( ϕ g) ˜ αβγ = ϕασ ∂σ g˜ βγ − ϕβσ ∂α g˜ σ γ − g˜ σ γ (∂α ϕβσ − ∂β ϕασ ) + g˜ βσ ∂γ ϕασ with respect to the natural frame {∂α } = {∂i , ∂i }. ˜ αβγ = 0 reduces to By virtue of (3.32), after some calculations, the equation ( ϕ g)

3.9

Deformed Complete Lifts of Connections

81

∂i g˜ jk − ∂i g˜ jk = 0, ∂i g˜ j k = 0, from which we have g˜ j k = g jk (x 1 , . . . , x n ), g˜ jk = x i ∂i g jk + h jk (x 1 , . . . , x n ). Remark 3.7 Using (3.31), (3.41) and g˜ α  β  (x 1 , . . . , x n )

(x 1 , . . . , x n )

gj k and h jk (0,2) on Mn , respectively.

=

∂xα ∂xβ ˜ αβ ,   g ∂xα ∂xβ

(3.41)

we easily see that

are components of any tensor fields g and h of type

Thus a real dual-holomorphic tensor field g˜ of type (0,2) on tangent bundle can be rewritten in the form     x i ∂i g jk + h jk g jk g˜ = g˜ βγ = g jk 0     h jk 0 x i ∂i g jk g jk + = C g + V h, = 0 0 g jk 0 where C g and V h are the complete and vertical lifts of the tensor fields g = (g jk ) and h = (h jk ) of type (0,2) from Mn to the tangent bundle T (Mn ), respectively (see [87]). Therefore we have Theorem 3.9 Let T (Mn ) be the tangent bundle of Mn , which is the real image of the dualholomorphic manifold X n (R(ε)). Then the real image of corresponding dual-holomorphic tensor field of type (0,2) from X n (R(ε)) is a deformed complete lift in the form D g = C g+ V h, where C g and V h are the complete and vertical lifts of g = (g jk ) and h = (h jk ) from Mn to T (Mn ), respectively. Remark 3.8 Let now g be a Riemannian metric, and h be any symmetric (0,2)-tensor field on Mn . It is clear that in such case the tensor field D g = C g + V h is a Riemannian metric on T (Mn ). We note that lifts of this kind have been also studied under the names: the metric I + II [87] if g = h and the synectic lift [78].

3.9

Deformed Complete Lifts of Connections γ

Let ∇˜ be a connection with the components ˜ αβ on the tangent bundle T (Mn ) preserving the structure ϕ. That connection is called a pure connection by definition if γ σ γ γ ϕσ = ˜ σβ ϕασ = ˜ ασ ϕβσ . ˜ αβ

82

3 Problems of Lifts

Using (3.32), from the purity condition we have ˜ ikj = ˜ ikj = ˜ ikj = ˜ ikj = 0.

(3.42)

γ

The pure connection ∇˜ with components ˜ αβ is called a dual-holomorphic connection, if [60] γ

γ

γ

( ϕ )τ αβ = ϕτσ ∂σ ˜ αβ − ϕασ ∂τ ˜ σβ = 0. It is well known that, such a connection is a real image of corresponding dualholomorphic connection from X n (R(ε)). From here, by virtue of (3.32) and (3.42), we have 

k

k m¯ ˜k = ϕtm ∂m ˜ ikj + ϕtm¯ ∂m ˜ ikj − ϕim ∂t ˜ m j − ϕi ∂t m j = 0   ⇔ ˜ iik = iik x 1 , . . . , x n ,  k k m¯ ˜k

ϕ ii j = ϕtm ∂m ˜ ikj + ϕtm¯ ∂m¯ ˜ ikj − ϕim ∂i ˜ m j − ϕi ∂i π j = 0 ⇔ 0 = 0,  k k m ˜k

ϕ t i¯ j = ϕtm ∂m ˜ i¯kj + ϕtm ∂m¯ ˜ i¯kj − ϕi¯m ∂t ˜ m j − ϕi¯ ∂t m j j = 0 ⇔ 0 = 0, k  k k − ϕim ∂t ˜ m = 0 ⇔ 0 = 0,

ϕ ti j¯ = ϕtm ∂mm ˜ ikj¯ + ϕtm ∂m¯ ˜ ikj¯ − ϕim ∂t ˜ m j¯ j  k k m¯ ˜k

ϕ T T T = ϕTm¯ ∂m ˜ ikJ¯ + ϕTm¯¯ ∂m¯ ˜ Tk J¯ − ϕtm ∂T¯ ˜ m j − ϕi ∂T¯ m j = 0 ⇔ 0 = 0,  ¯ ¯ ¯ k¯ m¯ m¯ ˜ k¯ ˜ k¯

ϕ tik j = ϕtm ∂m ˜ ikj + ϕtm¯ ∂m¯ ˜ ikj − ϕim ∂t¯ ˜ m j − ϕi ∂i m j = 0 ⇔ ϕi ∂t¯ m j = 0   ⇔ ˜ Tki = iik x 1 , . . . , x n  k k k

ϕ ii¯ j¯ = ϕTm ∂m ˜ ikj¯ + ϕTm¯ ∂m¯ ˜ ikj¯ − ϕim ∂T ˜ m − ϕim ∂T ˜ m = 0 ⇔ 0 = 0, j¯ j¯ k  m¯ m m¯ ˜k ˜k ˜k ˜k

ϕ t¯i¯ j = ϕtm ¯ ∂m i¯ j + ϕm¯ ∂m¯ i¯ j − ϕi¯ ∂T¯ m j − ϕi¯ ∂T¯ m j j = 0 ⇔ 0 = 0, k  k m ˜k

ϕ ti j = ϕtm ∂m ˜ Tk j + ϕtm ∂min ˜ ikj − ϕTm ∂t ˜ m j j − ϕT ∂t m j = 0 ⇔ 0 = 0,  k¯ ¯ ¯ k¯ m¯ ˜ k¯

ϕ ti j = ϕtm ∂m ˜ i˜kj + ϕtm¯ ∂m¯ ˜ ikj − ϕtm ∂t ˜ m j − ϕt ∂t m j = 0 ⇔ 0 = 0,  k¯ ¯ ¯ k¯ k¯

ϕ ti j = ϕtm ∂m ˜ ikj + ϕtm¯ ∂m¯ ˜ ikj¯ − ϕim ∂t ˜ m − ϕim¯ ∂t ˜ m = 0 ⇔ ϕim¯ ∂ N¯ ˜ ikT¯ = 0 j¯ jj   ⇔ ˜ ikk¯ = iik x 1 , . . . , x n ,

ϕ

ti j

¯ ¯ ¯ k¯ m¯ ˜ k¯ ( ϕ )kti j = ϕtm ∂m ˜ ikj + ϕtm¯ ∂m¯ ˜ ikj − ϕim ∂t ˜ m j − ϕi ∂t m¯ j = 0 ¯ k¯ ˜ k¯ ˜ k¯ ⇔ ϕtm¯ ∂m¯ ˜ ikj − ϕim¯ ∂t ˜ m ¯ j = 0 ⇔ ∂t¯ i j − ∂t i¯ j = 0 ¯ ⇔ ˜ ikj = x t¯∂t ikj + Hikj (x 1 , . . . , x n ), ¯ m¯ m ˜ k¯ m¯ ˜ k¯ ˜ k¯ ˜ k¯ ( ϕ )kt¯i¯ j = ϕtm ¯ ∂m i¯ j + ϕt¯ ∂m¯ i¯ j − ϕi¯ ∂t¯ m j − ϕi¯ ∂t¯ m¯ j = 0 ⇔ 0 = 0, ¯ ¯ ¯ k¯ m¯ ˜ k¯ ( ϕ )kti¯ j¯ = ϕtm ∂m ˜ i¯kj¯ + ϕtm¯ ∂m¯ ˜ i¯kj¯ − ϕi¯m ∂t ˜ m ¯ j − ϕi¯ ∂t m¯ j¯ = 0 ⇔ 0 = 0,

3.9

Deformed Complete Lifts of Connections

83

¯

¯

¯

¯

¯

¯

¯

¯

¯

¯

m¯ m ˜k m¯ ˜k ˜k ˜k ( ϕ )kt¯i j¯ = ϕtm ¯ ∂m i j¯ + ϕt¯ ∂m¯ i j¯ − ϕi ∂t¯ m j¯ − ϕi ∂t¯ m¯ j¯ = 0 ⇔ 0 = 0, m¯ m ˜k m¯ ˜k ˜k ˜k ( ϕ )kt¯i¯ j¯ = ϕtm ¯ ∂m i¯ j¯ + ϕt¯ ∂m¯ i¯ j¯ − ϕi¯ ∂t¯ m j¯ − ϕi¯ ∂t¯ m¯ j¯ = 0 ⇔ 0 = 0. γ

Thus the equation ( ϕ )τ αβ = 0 reduces to ˜ ikj = ikj = ˜ ikj = ikj (x 1 , . . . , x n ), ˜ ikj = x t ∂t ikj + Hikj (x 1 , . . . , x n ).

(3.43)

  ∂xγ ∂xα ∂xβ ˜ γ ∂xγ ∂2xγ ∂ x γ ∂ x α  ∂ x β  αβ + ∂ x γ ∂ x α  ∂ x β  , after straightforward calculations we see that ikj (x 1 , . . . , x n ) and Hikj (x 1 , . . . , x n ) are the components of any connection ∇ and tensor field H of type (1,2) on Mn , respectively. 

γ Remark 3.9 Using (3.31), (3.42), (3.43) and ˜ α  β  =

Taking account of the definition of the complete lift C ∇ of connection ∇ (see [85]), we see that a real dual-holomorphic connection ∇˜ on the tangent bundle can be rewritten in the form ∇˜ = C ∇ + V H , where V H is the vertical lift of the tensor field H = (Hikj ) of type (1,2) from Mn to the tangent bundle T (Mn ). Thus we have Theorem 3.10 Let T (Mn ) be the tangent bundle of Mn , which is the real image of the dual-holomorphic manifold X n (R(ε)). Then the real image of the corresponding dualholomorphic connection from X n (R(ε)) is a deformed complete lift in the form D ∇ = C ∇ + V H , where C ∇ and V H are the complete and vertical lifts of ∇ = ( k ) and ij H = (Hikj ) from Mn to T (Mn ), respectively. Example 3.1 Let (Mn , g) be a Riemannian manifold, and (T (Mn ), ϕ) its tangent bundle with natural dual ϕ-structure:   00 ϕ= . I 0 The complete and vertical lifts of vector and tensor fields from Mn to T (Mn ) have the following properties ϕ C X = V X,

V

X V f = 0,

V

X C f = C X V f = V (X f ),

V

h(V X , C Y ) = 0, C g(V X , C Y ) = V (g(X , Y )),

V

h(C X , C Y ) = V (h(X , Y )), C g(C X , C Y ) = C (g(X , Y )), V h(V X , C Y ) = 0,

[C Y , V X ] = V [Y , X ], [C Y , C X ] = C [Y , X ]

84

3 Problems of Lifts

for any function f on Mn [87]. Using these formulas, we find D

g(ϕ C X , C Y ) = (C g + V h)(ϕ C X , C Y ) = (C g + V h)(V X , C Y ) = C g(V X , C Y ) + V h(V X , C Y ) = C g(V X , C Y ) = V (g(X , Y )) = C g(C X , V Y ) = C g(C X , V Y ) + V h(C X , V Y ) = (C g + V h)(C X , V Y ) = (C g + V h)(C X , ϕ C Y ) =

De f

g(C X , ϕ C Y )

and ( ϕ D g)(C X , C Y , C Z ) = (ϕ C X )( D g(C Y , C Z )) − C X ( D g(ϕ C Y , C Z ) + D g((L C Y ϕ) C X , C Z ) + D g(C Y , (L C Z ϕ) C X ) = V X C (g(Y , Z )) + V X V (h(Y , Z )) − C X V (g(Y , Z )) + D g(L C Y (ϕ C X ) − ϕ(L C Y

C

X ), C Z ) + D g(C Y , L C Z (ϕ C X )

− ϕ(L C Z C X )) = V (X (g(Y , Z ))) − V (X (g(Y , Z ))) + D g(L C Y

V

X − ϕ[C Y , C X ], C Z ) + D g(C Y , L C Z

V

X

− ϕ[ Z , X ]) = g([ Y , X ] − ϕ [Y , X ], Z ) C

C

D

C

V

C

C

+ D g(C Y , [C Z , V X ] − ϕ[C Z , C X ]) = D g(V [Y , X ] − V [Y , X ], C Z ) + D g(C Y , V [Z , X ] − V [Z , X ]) = 0. From here we see that the triple (T (Mn ), D g, ϕ) is a dual anti-Kähler manifold D (∇ ϕ = 0)(see Sect. 2.1). In such manifolds the Levi–Civita connection ∇ g of D g Dg is dual-holomorphic too [25, 52]. Thus the Levi–Civita connection ∇ is the simplest example of deformed complete lift of connection. Dg

3.10

Holomorphic Metrics in the Tangent Bundle of Order 2

Let A3 = R(ε2 ) be an algebra of order 3 with a canonical basis {e1 , e2 , e3 } = {1, ε, ε2 }, ε3 = 0. From 1 2 3 1 2 3 C11 = 1, C11 = 0, C11 = 0, C12 = 0, C12 = 1, C12 = 0, 1 2 3 1 2 3 C13 = 0, C13 = 0, C13 = 1, C22 = 0, C22 = 0, C22 = 1, 1 2 3 1 2 3 C23 = 0, C23 = 0, C23 = 0, C33 = 0, C33 = 0, C33 = 0, γ

we see that the matrices Cσ = (Cσβ ), σ = 1, 2, 3 of the regular representation of the algebra R(ε 2 ) have the following forms

3.10

Holomorphic Metrics in the Tangent Bundle of Order 2

85

⎛

⎞ ⎛ ⎞ ⎛ ⎞ 100 000 000 ⎜ ⎟ ⎜ ⎟ ⎜ ⎟ C 1 = ⎝ 0 1 0 ⎠, C 2 = ⎝ 1 0 0 ⎠, C 3 = ⎝ 0 0 0 ⎠ 001 010 100 For the case of A3 = R(ε2 ), the Scheffers conditions reduce to the following equations: ∂ y1 ∂ y2 ∂ y1 = = = 0, ∂x2 ∂x3 ∂x3 ∂ y2 ∂ y1 ∂ y3 (ii) = 1 = 3, 2 ∂x ∂x ∂x ∂ y3 ∂ y2 (iii) = 1, ∂x2 ∂x (i)

where z = x 1 + εx 2 + ε2 x 3 , w(z) = y 1 (x 1 , x 2 , x 3 ) + εy 2 (x 1 , x 2 , x 3 ) + ε2 y 3 (x 1 , x 2 , x 3 ), ε3 = 0. From (i) and (ii) we have dy 1 y 1 = y 1 (x 1 ), y 2 = y 2 (x 1 , x 2 ), y 2 (x 1 , x 2 ) = x 2 1 + g(x 1 ), dx . 1 dy 3 1 2 3 3 1 2 y (x , x , x ) = x + g(x , x ) dx 1 After substituting the values of y 2 and y 3 into (iii), we find 2 1 ∂g dg 2 d y = x + 1, ∂x2 (dx 1 )2 dx

i.e. g(x 1 , x 2 ) =

1 2 2 d2 y 1 dg + x 2 1 + G(x 1 ), (x ) 2 (dx 1 )2 dx

where g = g(x 1 ) and G = G(x 1 ) are arbitrary functions. Thus the R(ε2 )-holomorphic function w = w(z) has the following expression dy 1 dy 1 + g(x 1 )) + ε2 (x 3 1 1 dx dx dg 1 2 2 d2 y 1 + (x ) + x 2 1 + G(x 1 )). 2 (dx 1 )2 dx

w(z) = y 1 (x 1 ) + ε(x 2

Similarly, if w(z 1 , . . . , z n ) = y 1 (x 1 , . . . , x n ) + εy 2 (x 1 , . . . , x 2 ) + ε2 y 3 (x 1 , . . . , x 2 ), where z i = x i + εx n+i + ε2 x 2n+i , i = 1, . . . , n is a multi-variable R(ε2 )-holomorphic function, then the function w = w(z 1 , . . . , z n ) has the following specific form: w(z 1 , . . . , z n ) = y 1 (x 1 , . . . , x n ) + ε(x n+s ∂s y 1 + g(x 1 , . . . , x n ))

86

3 Problems of Lifts

+ ε2 (x 2n+s

1 ∂ y1 ∂ 2 y1 ∂g + x n+s x n+t s t + x n+s s + G(x 1 , . . . , x n )). s ∂x 2 ∂x ∂x ∂x (3.44)

If g(x 1 , . . . , x n ) = G(x 1 , . . . , x n ) = 0 and y 1 (x 1 , . . . , x n ) = f (x 1 , . . . , x n ), then the function w(z 1 , . . . , z n ) = f (x 1 , . . . , x n ) + εx n+s ∂s f   ∂f ∂2 f 1 + ε2 x 2n+s s + x n+s x n+t s t ∂x 2 ∂x ∂x

(3.45)

is said to be the natural extension of the real C ∞ -functions f = f (x 1 , . . . , x n ) to R(ε2 ). Let now T 2 (Vr ) be the bundle of 2-jets, i.e. the tangent bundle of order 2 over the Riemannian manifold (Vr , g), dim T 2 (Vr ) = 3r and let (x i , x i , x i ) = (x i , x r +i , x 2r +i ), x i = x i (t), xi =

dx i i 1 d2 x i , t ∈ R, i = 1, . . . , r ,x = dt 2 dt 2

be an induced local coordinates system in T 2 (Vr ). It is clear that there exists an affinor field γ in T 2 (Vr ) which has the components of the form ⎛ ⎞ 000 ⎜ ⎟ γ = ⎝ I 0 0⎠ 0I 0   with respect to the natural frame {∂i , ∂i , ∂i } = ∂∂x i , ∂ i , ∂ i , i = 1, . . . , r , where I ∂x ∂x denotes the r × r identity matrix. From here we have ⎛ ⎞ 000 ⎜ ⎟ γ 2 = ⎝ 0 0 0 ⎠, γ 3 = 0, I 00   i.e. T 2 (Vr ) has a natural integrable structure  = I , γ , γ 2 , I = idT 2 (Vr ) , which is an isomorphic representation of the algebra R(ε2 ), ε3 = 0. Using γ ∂i = ∂i , γ 2 ∂i = γ ∂i = ∂i , we have {∂i , ∂i , ∂i } = {∂i , γ ∂i , γ 2 ∂i }. Also, using a frame {∂1 , γ ∂1 , γ 2 ∂1 , ∂2 , γ ∂2 , γ 2 ∂2 , . . . , ∂r , γ ∂r , γ 2 ∂r } = {∂1 , ∂1 , ∂1 , ∂2 , ∂2 , ∂2 , . . . , ∂r , ∂r , ∂r }

3.10

Holomorphic Metrics in the Tangent Bundle of Order 2

87

which is obtained from {∂i , ∂i , ∂i } = {∂i , γ ∂i , γ 2 ∂i } by changing the numbers of frame elements, we see that the structure affinors I , γ and γ 2 have the following components ⎛

C1 0 ⎜ ⎜ 0 C1 I =⎜ ⎝... ... 0 0

⎞ ⎛ ... 0 C2 0 ⎟ ⎜ ... 0 ⎟ ⎜ 0 C2 ⎟, γ = ⎜ ⎝... ... ... ...⎠ . . . C1 0 0

⎛ ⎞ C3 0 ... 0 ⎜ ⎟ ... 0 ⎟ ⎜ 0 C3 ⎟, γ 2 = ⎜ ⎝... ... ... ...⎠ . . . C2 0 0

⎞ ... 0 ⎟ ... 0 ⎟ ⎟ ... ...⎠ . . . C3

with respect to the frame {∂1 , ∂1 , ∂1 , ∂2 , ∂2 , ∂2 , . . . , ∂r , ∂r , ∂r }, where the block matrices Cσ , σ = 1, 2, 3 of order 3 are the regular representation of the algebra R(ε 2 ). Thus the   bundle T 2 (Vr ) has a natural integrable structure  = I , γ , γ 2 , which is an r-regular representation of R(ε2 ). On the other hand, the transformation of induced coordinates (x i , x i , x i ) in T 2 (Vr ) is given by 



x i = x i (x i ), 





dx i ∂ x i dx i ∂xi i x , = = dt ∂ x i dt ∂ x i       1 ∂ x i d2 x i 1 d2 x i 1 d ∂ x i dx i 1 ∂ 2 x i dx i dx j = xi = = + 2 dt 2 2 dt ∂ x i dt 2 ∂ x i dt 2 2 ∂ x i ∂ x j dt dt 

xi =



=



∂ xi i 1 ∂2xi i j x + x x ∂xi 2 ∂xi ∂x j

and its Jacobian matrix by ⎛ i i i ⎞ ∂x

⎜ ∂ x i ⎜ ∂xi A=⎜ ⎜ ∂xi ⎝ i ∂x ∂xi

∂x ∂xi  ∂xi ∂xi ∂xi



∂xi

∂x ∂ x i ∂xi ∂ x i ∂xi ∂xi

⎛ ⎟ ⎟ ⎜ ⎟=⎜ ⎟ ⎝ ⎠

⎞





∂2xi ∂xi ∂xs

∂xi ∂ xi ∂2xi ∂xi ∂xs

xs

+

0



∂xi ∂ xi  ∂3xi s x t ∂2xi x ∂xi ∂xs ∂xt ∂xi ∂xs

xs

0

⎟ 0 ⎟ ⎠.

(3.46)



i x s ∂∂xx i

From here it follows that A−1 γ A = γ , A−1 γ 2 A = γ 2 , i.e. the transformation of local coordinates (x i , x i , x i ) in T 2 (Vr ) is a structure-preserving transformation. Then the transition functions      i   ∂xi i 1 ∂2xi i j i i i i 2 i i i 2 ∂x i x + x x z (z ) = x + εx + ε x = x (x ) + ε i x + ε ∂x ∂xi 2 ∂xi ∂x j of charts on X r (R(ε2 )) are R(ε2 )-holomorphic functions by virtue of (3.45), i.e. the bundle T 2 (Vr ) is a real modeling of the R(ε2 )-holomorphic manifold X r (R(ε2 )). After some calculations, we see that the 2-nd lift I I g [87] (see p. 332) of a Riemannian metric g to T 2 (Vr ), i.e.

88

3 Problems of Lifts

⎞ x s ∂s g ji + 21 x t x s ∂t ∂s g ji x s ∂s g ji g ji ⎟ ⎜ II g=⎝ x s ∂s g ji g ji 0 ⎠ 0 0 g ji ⎛

  is a pure Riemannian metric with respect to the structure  = I , γ , γ 2 and

γ

II

g = γ 2

II

g = 0.

Therefore, the R(ε2 )-holomorphic manifold (T 2 (Vr ), ) is a manifold with R(ε2 )holomorphic metric I I g. From here, using Theorem 2.1, we have Theorem 3.11 [76] If Vr is a Riemannian manifold with metric g, then the triple (T 2 (Vr ), , I I g) is an anti-Kähler type manifold. Finally, we would like to find the local expression of any R(ε2 )-holomorphic pure tensor field t˜ of type (0,2) in T 2 (Vr ). Using t˜M J γ IM = t˜I M γ JM , I = (i, i, i), M = (m, m, m), J = ( j, j, j) and ( γ t˜) K I J = γ KM ∂ M t˜I J − γ IM ∂ K t˜M J = 0, M ∂ M t˜I J − (γ 2 ) M ( γ 2 t˜) K I J = (γ 2 ) K I ∂ K t˜M J = 0,

after straightforward calculations, we find ⎛ ⎞ x s ∂s t ji + 21 x t x s ∂t ∂s t ji + x s ∂s G ji + H ji x s ∂s t ji + G ji t ji ⎜ ⎟ t˜ = ⎝ x s ∂s t ji + G ji t ji 0 ⎠, 0 0 t ji where t ji , G ji , H ji are arbitrary tensor fields of type (0,2) in Vr . If ti j = gi j and G ji , H ji are symmetric tensor fields in Vr , then we have a new R(ε2 )-holomorphic pure Riemannian metric in T 2 (Vr ): ⎛ ⎞ x s ∂s g ji + 21 x t x s ∂t ∂s g ji + x s ∂s G ji + H ji x s ∂s g ji + G ji g ji ⎜ ⎟ g˜ = ⎝ (3.47) x s ∂s g ji + G ji g ji 0 ⎠. 0 0 g ji We denote g˜ by de f ( I I g) and call it a deformed 2-nd lift of a Riemannian metric g to By using again Theorem 2.1, we have

T 2 (Vr ).

3.11

Deformed Lifts of Vector Fields in the Tangent Bundle of Order 2

89

Theorem 3.12 If Vr is a Riemannian manifold with metric g, then the triple (T 2 (Vr ), , de f ( I I g)) is an anti-Kähler type manifold. From (3.47) we see that a general form of the deformed complete lift ( g) =

de f I I

II

de f ( I I g)

is

g + I G + 0H,

where ⎛

⎞ x s ∂s G ji G ji 0 ⎜ ⎟ I G = ⎝ G ji 0 0 ⎠ and 0 0 0

⎛

⎞ H ji 0 0 ⎜ ⎟ 0 H = ⎝ 0 0 0⎠ 0 00

are the 1-st and 0-th lifts [87] of G and H , respectively.

3.11

Deformed Lifts of Vector Fields in the Tangent Bundle of Order 2

Let now T 2 (Mr ) be the tangent bundle of order 2 over a C ∞ -manifold Mr , dim T 2 (Mr ) = 3r and let (x i , x i , x i ) = (x i , x r +i , x 2r +i ), x i = x i (t), x i =

dx i i 1 d2 x i , t ∈ R, i = 1, . . . , r ,x = dt 2 dt 2

be the induced local coordinates in T 2 (Mr ). Since the bundle T 2 (Mr ) is a real modeling of X r (R(ε2 )) and any holomorphic function w(z 1 , . . . , z r ) = f 1 (x 1 , . . . , x r ) + ε f 2 (x 1 , . . . , x r ) + ε2 f 3 (x 1 , . . . , x r ) on X r (R(ε2 )), where z i = x i + εx r +i + ε2 x 2r +i , i = 1, . . . , r , is expressed by (see (3.44)) w(z 1 , . . . , z r ) = f (x 1 , . . . , x r ) + ε(x r +i ∂i f + g(x 1 , . . . , x r )) + ε2 (x 2r +i + x r +i

∂f ∂2 f 1 + x r +i x r + j i j i ∂x 2 ∂x ∂x

∂g + h(x 1 , . . . , x r )), ∂xi

f = f 1,

in the bundle T 2 (Mr ), we introduce the following three functions: V I

f = f (x 1 , . . . , x r ), f = x r +i ∂i f + g(x 1 , . . . , x r ),

90

3 Problems of Lifts C

f = x 2r +i

1 ∂f ∂2 f ∂g + x r +i x r + j i j + x r +i i + h(x 1 , . . . , x r ), i ∂x 2 ∂x ∂x ∂x

where f , g and h are arbitrary functions on Mr . These functions V f , I f and C f are called recpectively the vertical, intermediate and complete lifts of f in Mr to T 2 (Mr ). If g = h = 0, then we have the 0-th f 0 , 1-th f 1 and 2-nd f 2 lifts of f [87], i.e. the lifts I f and C f of f to T 2 (M ) are respectively the deformed lifts of 1-th and 2-nd lifts of f . r Thus we have V

f = f 0,

I

f = f 1 + g0 ,

C

f = f 2 + g1 + h 0 .

(3.48)

We now consider in T 2 (Mr ) two vector fields of Liouville types U and V having components ⎛ ⎛ ⎞ ⎞ 0 0 ⎜ ⎜ ⎟ ⎟ U = ⎝ 0 ⎠, V = ⎝ x r +i ⎠ x r +i x 2r +i Then from (3.48), we immediately have Theorem 3.13 Let U and V be the Liouville vector fields in T 2 (Mr ). Then U (V f ) = 0, U ( I f ) = 0, U (C f ) = f 1 , V (V f ) = 0, V ( I f ) = f 1 , V (C f ) = f 2 + g 1 for any function f in Mr . Let X˜ = X˜ I ∂ ∂x I = X˜ i ∂∂x i + X˜ r +i ∂ x∂r +i + X˜ 2r +i ∂ x 2r∂ +i be a vector field in T 2 (Mr ), and    = I , γ , γ 2 , I = idT 2 (Mr ) be a -structure naturally existing in T 2 (Mr ). We would like to find the local expression of any vector field X˜ = ( X˜ I ) in T 2 (Mr ) corresponding ∗

∗

to the R(ε2 )-holomorphic vector field X = ( X u ) = (X uα eα ), u = 1, . . . , r ; α = 1, 2, 3 in X r (R(ε2 )). Using Theorem 1.9, we obtain ( γ X˜ ) I = γ KM ∂ M X˜ I − γ MI ∂ K X˜ M = 0, M I ( γ 2 X˜ ) I = (γ 2 ) K ∂ M X˜ I − (γ 2 ) M ∂ K X˜ M = 0.

From here, after straightforward calculations (see Sect. 3.10), we find the following vector field ⎛ ⎞ Xh ⎜ ⎟ C X = ⎝ x r +i ∂i X h + G h (3.49) ⎠, h 2 h h 1 r +i r + j ∂ X ∂X ∂G 2r +i r +i h x + 2x x +x +H ∂xi ∂xi ∂x j ∂xi

3.11

Deformed Lifts of Vector Fields in the Tangent Bundle of Order 2

91

where G = (G h (x 1 , . . . , x r )), H = (H h (x 1 , . . . , x r )) are two arbitrary vector fields in Mr . In fact, by means of (3.46), we easily see that C X determine some vector fields in T 2 (Mr ), which are called the complete lifts of X from Mr to T 2 (Mr ). From (3.49), we have C

X = X 2 + G1 + H 0,

(3.50)

where ⎛

⎛ ⎞ ⎞ ⎛ ⎞ Xh 0 0 ⎜ ⎜ ⎟ ⎟ ⎜ ⎟ H 0 = ⎝ 0 ⎠, G 1 = ⎝ G h ⎠, X = ⎝ x r +i ∂i X h ⎠ h 2 h x r +i ∂i G h Hh x 2r +i ∂∂Xx i + 21 x r +i x r + j ∂∂x i X∂ x j are respectively the 0-th, 1-th and 2-nd lifts of H , G and X [87]. Putting G = X , H = G and using (3.50), we see that ⎛ ⎞ 0 ⎜ ⎟ I X = X 1 + G0 = C X − X 2 = ⎝ X h (3.51) ⎠, r +i h h x ∂i X + G determines a new vector field in T 2 (Mr ), which is called the intermediate lift of a vector field X from Mr to T 2 (Mr ). We note that the intermediate and complete lifts I X and C X of X to T 2 (Mr ) are respectively the deformed lifts of 1-th and 2-th lifts of X. Thus we have V

X = X 0,

I

X = X 1 + G0,

C

X = X 2 + G1 + H 0,

(3.52)

where ⎛

⎞ 0 ⎜ ⎟ V X = ⎝ 0 ⎠. Xh

(3.53)

Using the local expressions of γ and γ 2 (see Sect. 3.10) and also (3.49), (3.51) and (3.53) we have I

X = γ (C X ),

V

X = γ 2 (C X ).

In particular, for the vector fields X = ∂i , G = ∂ j , H = ∂k , from (3.49) to (3.53) we obtain V

(∂i ) = ∂2r +i , I (∂i ) = ∂r +i + ∂2r + j , C (∂i ) = ∂i + ∂r + j + ∂2r +k .

By (3.48), (3.49) and (3.50) we have

92

3 Problems of Lifts

Theorem 3.14 Let X , G, H and f , g be respectively any vector fields and functions in Mr . Then V

( f X ) = f 0 X 0, I ( f X ) = f 1 X 0 + f 0 X 1 + G0,

C

( f X ) = f 2 X 0 + f 1 X 1 + f 0 X 0 + G1 + H 0,

V

X V f = 0, V X I f = 0, V X C f = (X f )0 ,

I

X V f = 0, I X I f = (X f )0 , I X C f = (X f )1 + (Xg)0 + (G f )0 ,

C

X V f = (X f )0 , C X I f = (X f )1 + (Xg)0 + (G f )0 ,

C

X C f = (X f )2 + (Xg)1 + (X f )0 + (G f )1 + (Gg)0 + (H f )0 .

Theorem 3.15 Let X , Y , G, Q, H , K be any vector vector fields in Mr . Then for the Lie bracket of complete lifts we have [C X , C Y ] = [X , Y ]2 + [X , Q]1 + [X , K ]0 + [G, Y ]1 + [G, Q]0 + [H , Y ]0 , where C

X = X 2 + G1 + H 0,

C

Y = Y 2 + Q1 + K 0.

Proof If f is any function in Mr , then by virtue of (3.52), we have [C X , C Y ]( f 2 ) = [X 2 , Y 2 ]( f 2 ) + [X 2 , Q 1 ]( f 2 ) + [X 2 , K 0 ]( f 2 ) + [G 1 , Y 2 ]( f 2 ) + [G 1 , Q 1 ]( f 2 ) + [G 1 , K 0 ]( f 2 ) + [H 0 , Y 2 ]( f 2 ) + [H 0 , Q 1 ]( f 2 ) + [H 0 , K 0 ]( f 2 ) Using the formulas (see [87, p. 322]) [X 2 , Y 2 ] = [X , Y ]2 , [X 2 , Q 1 ] = [X , Q]1 , [X 2 , K 0 ] = [X , K ]0 , [G 1 , Y 2 ] = [G, Y ]1 , [G 1 , Q 1 ] = [G, Q]0 , [G 1 , K 0 ] = 0, [H 0 , Y 2 ] = [H , Y ]0 , [H 0 , Q 1 ] = 0, [H 0 , K 0 ] = 0 we have [C X , C Y ]( f 2 ) = [X , Y ]2 ( f 2 ) + [X , Q]1 ( f 2 ) + [X , K ]0 ( f 2 ) + [G, Y ]1 ( f 2 ) + [G, Q]0 ( f 2 ) + [H , Y ]0 ( f 2 ) = ([X , Y ]2 + [X , Q]1 + [X , K ]0 + [G, Y ]1 + [G, Q]0 + [H , Y ]0 )( f 2 ).

Since the vector field X˜ in T 2 (Mr ) is completely determined by the action of X˜ on the functions f 2 (see [87, p. 320]), i.e. if X˜ f 2 = Y˜ f 2 for any f , then X˜ = Y˜ , we have

3.12

Deformed Complete and Intermediate Lifts of 1-Forms in the Tangent …

93

[C X , C Y ] = [X , Y ]2 + [X , Q]1 + [X , K ]0 + [G, Y ]1 + [G, Q]0 + [H , Y ]0 . Thus the proof is completed. From Theorem 3.15 we have [C X , C Y ] = [X , Y ]2 + [X , Q]1 + [X , K ]0 + [G, Y ]1 + [G, Q]0 + [H , Y ]0 = C [X , Y ] + [X , Q]1 + [X , K ]0 + [G, Y ]1 + [G, Q]0 + [H , Y ]0 − [X , Y ]1 − [X , Y ]0 , i.e. the correspondence X → C X is not an isomorphism of the Lie algebras of vector fields on Mr and T 2 (Mr ) with respect to constant coefficients.

3.12

Deformed Complete and Intermediate Lifts of 1-Forms in the Tangent Bundle of Order 2

Let ω˜ = ω˜ I dx I = ω˜ i dx i + ω˜ r +i dx r +i + ω˜ 2r +i dx 2r +i be an 1-form in T 2 (Mr ), and    = I , γ , γ 2 , I = idT 2 (Mr ) be a -structure naturally existing in T 2 (Mr ). We would like to find local expression of ω˜ = (ω˜ I ) in T 2 (Mr ) which is corresponding to the R(ε2 )∗ ∗ ∗ holomorphic 1-form ω = (ωu ) = (ωuα eα ), eα = ϕ αβ eβ , u = 1, . . . , r ; α, β = 1, 2, 3 in X r (R(ε2 )). Using Theorem 1.9, we obtain ( γ ω) ˜ J I = γ JH ∂ H ω˜ I − γ IH ∂ J ω˜ H = 0, ˜ J I = (γ 2 ) H ˜ I − (γ 2 ) IH ∂ J ω˜ H = 0. ( γ 2 ω) J ∂H ω From here, after straightforward calculations, we find 1 2 ω˜ = (ω˜ I ) = (x 2r +h ∂h ωi + x r +h x r +m ∂hm ωi + x h+i ∂h G i + Hi , x r +h ∂h ωi + G i , ωi ), 2 (3.54) where G = (G i (x 1 , . . . , x r )), H = (Hi (x 1 , . . . , x r )) are any covector fields in Mr . In fact, by means of (3.46), we easily see that ω˜ = (ω˜ I ) determine the 1-form in T 2 (Mr ) which are called the deformed complete lifts of ω from Mr to T 2 (Mr ) and denoted by C ω = (C ω ). From (3.54), we have I C

ω = ω2 + G 1 + H 0 ,

where H 0 = (Hi , 0, 0), G 1 = (x r +h ∂h G i , G i , 0),

(3.55)

94

3 Problems of Lifts

1 2 ω2 = (x 2r +h ∂h ωi + x r +h x r +m ∂hm ωi , x r +h ∂h ωi , ωi ) 2 are respectively the 0-th (vertical), 1-th and 2-nd (complete) lifts of H , G and ω [87]. Thus we have Theorem 3.16 Let ω = ωi dx i be a 1-form on Mr . The deformed complete lift C ω of ω to the bundle of 2-jets T 2 (Mr ) have the following expression C

ω = ω2 + G 1 + H 0 ,

where H 0 , G 1 and ω2 are respectively the 0-th, 1-th and 2-th lifts of any 1-forms H , G and ω. Putting ω = G in (3.55), we see that ω1 + H 0 = C ω − ω2 = (x r +i ∂i ωh + Hh , ωh , 0)

(3.56)

determine a new 1-form in T 2 (Mr ), which are called the deformed intermediate lift of 1-form ω from Mr to T 2 (Mr ) and denoted by I ω = ω1 + H 0 . We note that the deformed intermediate lift I ω of ω to T 2 (Mr ) is deformation of 1-th lift of ω. Thus we have V

ω = ω0 ,

I

ω = ω1 + H 0 ,

C

ω = ω2 + G 1 + H 0 ,

(3.57)

where V

ω = (ωh , 0, 0).

(3.58)

Now we can state the following Theorem 3.17 Let ω = ωi dx i be a 1-form on Mr . The deformed intermediate lift I ω of ω to the bundle of 2-jets T 2 (Mr ) have the following expression I

ω = ω1 + H 0 ,

where H 0 is the 0-th lift of 1-form H . Using the local expressions of γ and γ 2 (see Sect. 3.10), from (3.54), (3.56) and (3.58) we have Theorem 3.18 The deformed complete lifts satisfy the following matrix formulas C

ωγ = ω1 + G 0 ,

C

ωγ 2 = ω0 = V ω,

3.12

Deformed Complete and Intermediate Lifts of 1-Forms in the Tangent …

95

where ⎛

⎞ ⎛ ⎞ 000 000 ⎜ ⎟ ⎜ ⎟ γ = ⎝ I 0 0 ⎠, γ 2 = ⎝ 0 0 0 ⎠. 0I 0 I 00 Let now ω = dx i , G = dx j , H = dx k , i, j, k = 1, . . . , r . Then from (3.56) to (3.58) we have Theorem 3.19 Deformed complete, intermediate and vertical lifts of differentials dx i are the following linear combinations of differentials in T 2 (Mr ): V

(dx i ) = dx i , I (dx i ) = dx r +i + dx k , C (dx i ) = dx 2r +i + dx r + j + dx k .

Let now X be a vector field in Mr . It is well known that the vertical and deformed lifts I X , C X of X have the following expressions (see Sect. 3.11): ⎛ ⎞ 0 ⎜ ⎟ V X = X 0 = ⎝ 0 ⎠, Xh ⎛ ⎞ ⎛ ⎞⎛ ⎞ 0 0 0 ⎜ ⎟ ⎜ ⎟⎜ h ⎟ I X = X1 + Y 0 = ⎝ Xh ⎠ + ⎝ 0 ⎠⎝ X ⎠, x r +i ∂i X h x r +i ∂i X h + Y h Yh ⎛ ⎞ ⎛ ⎞ ⎛ ⎞ Xh 0 0 ⎜ ⎟ ⎜ h ⎟ ⎜ ⎟ C X = X 2 + Y 1 + Z 0 = ⎝ x r +i ∂i X h ⎠ + ⎝Y ⎠ + ⎝0 ⎠ x 2r +i ∂i X h + 21 x r +i x r + j ∂i2j X h Zh x r +i ∂i Y h ⎛ ⎞ Xh ⎜ r +i ⎟ = ⎝ x ∂i X h + Y h ⎠ x 2r +i ∂i X h + 21 x r +i x r + j ∂i2j X h + x r +i ∂i Y h + Z h for any vector fields Y , Z in Mr . Using the last formulas and also (3.48), (3.56)–(3.58) we have Theorem 3.20 Let X , ω and f be respectively any vector field, 1-form and function in Mr . Then V ( f ω) = f 0 ω0 , I ( f ω) = f 1 ω0 + f 0 ω1 + G 0 , C ( f ω) = ( f 2 + f 0 )ω0 + f 1 ω1 + G 1 + H 0 , V ω(V X ) = 0, V ω( I X ) = 0, V ω(C X ) = (ω(X ))0 , I ω(V X ) = 0, I ω( I X ) = (ω(X ))0 , I ω(C X ) = (ω(X ))1 + (ω(Y ))0 + (H (X ))0 , C ω(V X ) = (ω(X ))0 , C ω( I X ) = (ω(X ))1 + (ω(Y ))0 + (G(X ))0 ,

96

3 Problems of Lifts C ω(C X ) = (ω(X ))2 + (ω(Y ))1 + (ω(Z ))0 + (G(X ))1 + (G(Y ))0 + (H (X ))0 .

Let now  be a tensor field of type (0,2) in Mr . We define a 1-form γY  by (γY )X = (X , Y ) for any vector fields X and Y . If  has the local components i j , then γY  has the local components i j Y j . It is well known that the deformed intermediate and complete lifts of  has respectively components (see Sect. 3.10) ⎛ ⎞ x r +s ∂s  ji + π ji  ji 0 ⎜ ⎟ I =⎝ 0 0 ⎠ = 1 + π 0 ,  ji 0 0 0 ⎛ ⎞ 1 r +t r +s 2 2r +s ∂s  ji + 2 x x ∂ts  ji + x r +s ∂s  ji + π ji x r +s ∂s  ji +  ji  ji x ⎜ ⎟ C =⎝ x r +s ∂s  ji +  ji  ji 0 ⎠ 0 0  ji = 2 + 1 + π 0 , where ⎛

⎞ π ji 0 0 ⎜ ⎟ V π = 0π = ⎝ 0 0 0 ⎠ 0 00 is the vertical lift of any tensor field π of type (0,2). Using the expression of lifts V π, I , C  and (3.54), (3.56)–(3.58) we have γ X 2 V  = (i j X j , 0, 0) = ((γ X )i , 0, 0) = V (γ X ), γ X 2 I  = ((x r +s ∂s i j + πi j )X j + i j (x r +s ∂s X j ), i j X j , 0) = (x r +s ∂s (γ X )i + (γ X π )i , (γ X )i , 0) = (γ X )1 + (γ X π )0 = I (γ X ), 1 γ X 2 C  = ((x 2r +s ∂s i j + x r +s x r +t ∂st2 i j + x r +s ∂s i j + πi j )X j 2 + (x r +s ∂s i j + i j )x r +t ∂t X j 1 + i j (x 2r +s ∂s X j + x r +s x r +t ∂st2 X j , (x r +s ∂s i j + i j )X j 2 + i j x r +t ∂t X j , i j X j ) 1 = (x 2r +s ∂s (γ X )i + x r +s x r +t ∂st2 (γ X )i + x r +s ∂s (γ X )i 2

3.12

Deformed Complete and Intermediate Lifts of 1-Forms in the Tangent …

97

+ (γ X π )i , x r +s ∂s (γ X )i + (γ X )i , (γ X )i ) = (γ X )2 + (γ X )1 + (γ X π )0 = C (γ X ). Thus we have Theorem 3.21 Let  be a tensor field of type (0,2) on Mr . Then γ X 2 V  = (γ X )0 = V (γ X ), γ X 2 I  = (γ X )1 + (γ X π )0 = I (γ X ), γ X 2 C  = (γ X )2 + (γ X )1 + (γ X π )0 = C (γ X ). We shall now study the deformed lifts of exterior differentials of 1-forms ω = ωi dx i , i = 1, . . . , r . Using [X 2 , Y 2 ] = [X , Y ]2 and linearity of mappings X → X 0 , X → X 1 , X → X 2 , from Theorems 3.20 and 3.21 we have 2(d I ω)(X 2 , Y 2 ) = X 2 ( I ω(Y 2 )) − Y 2 ( I ω(X 2 )) − I ω([X 2 , Y 2 ]) = X 2 ((ω(Y ))1 + (H (Y ))0 ) − Y 2 ((ω(X ))1 + (H (X ))0 ) − I ω([X , Y ]2 ) = (X ω(Y ))1 + (X H (Y ))0 − (Y ω(X ))1 − (Y H (X ))0 − (ω([X , Y ]))1 − (H ([X , Y ]))0 = (X ω(Y ) − Y ω(X ) − ω([X , Y ]))1 + (X H (Y ) − Y H (X ) − H ([X , Y ]))1 = 2((dω)(X , Y ))1 + 2((dH )(X , Y ))0 = 2(γY (dω)(X ))1 + 2(γY (dH )(X ))0 = 2(γY (dω))1 (X 2 ) + 2(γY (dH ))0 (X 2 ) = 2(γY 2 (dω)1 )(X 2 ) + 2(γY 2 (dH )0 )(X 2 ) = 2((dω)1 + (dH )0 )(X 2 , Y 2 ). By similar devices, we have 2(dC ω)(X 2 , Y 2 ) = 2((dω)2 + (dG)1 + (dH )0 )(X 2 , Y 2 ). Since the arbitrary tensor field  of type (0.2) in T 2 (Mr ) is completely determined by ˜ 2 , Y 2 ) for any its action on the lifts X 2 , Y 2 (see [87, p. 324]), i.e. if (X 2 , Y 2 ) = (X ˜ X , Y , then  = , and we have Theorem 3.22 Let ω, G and H be 1-forms in Mr . Then the exterior differentials of the deformed intermediate and complete lifts of ω to T 2 (Mr ) satisfy the following formulas: d I ω = (dω)1 + (dH )0 , dC ω = (dω)2 + (dG)1 + (dH )0 .

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3 Problems of Lifts

3.13

Problems of Lifts in Symplectic Geometry

A manifold M is symplectic if it possesses a nondegenerate 2-form ω which is closed (i.e. dω = 0). For any manifold M of dimension n, the cotangent bundle T ∗ M is a natural symplectic 2n-manifold with symplectic 2-form ω˜ = −d p = dx i ∧ d pi , where p = pi dx i is the Liouville form (basic 1-form) on T ∗ M.  ∗ Let now (M, ω) be a symplectic manifold and T M = P∈M T P (M) (T M =  ∗ P∈M T P (M)) its tangent (cotangent) bundle. Suppose that the base space M is covered by a system of coordinate neighborhoods (U , x i ), where x i , i = 1, . . . , n are the local coordinates in the neighborhood U . We introduce a system of local induced coordinates (x i , x i ) = (x i , v i )((x i , x˜ i ) = (x i , pi )), i = n + 1, . . . , 2n in the open set π −1 (U ) ⊂ T M (π −1 (U ) ⊂ T ∗ M). Then the symplectic isomorphisms ωb : T M → T ∗ M and ω : T ∗ M → T M are given by ωb : x I = (x i , x i ) = (x i , v i ) → x˜ K = (x k , x˜ k ) = (x k = δik x i , pk = ωki v i ) and ω : x˜ K = (x k , x˜ k ) = (x k , pk ) → x I = (x i , x i ) = (x i = δki x k , v i = ωik pk ), where ωik ωk j = δ ij , δ ij is the Kronecker symbol. The Jacobian matrices of ωb and ω are given respectively by       A˜ ik A˜ k ∂ x˜ K δik 0 b K i ˜ ˜ = (3.59) = (ω )∗ = A = ( A I ) = ∂x I v s ∂i ωks ωki A˜ ik A˜ k i

and  (ω )∗ = A = (A KI ) =

Aik Ai



k

Aik Ai

 =

k

∂x I ∂ x˜ K



 =

 δki 0 . ps ∂k ωis ωik

(3.60)

Let f be any function on a symplectic manifold (M, ω). If C X T is the complete lift of a vector field X from manifold M to its tangent bundle T M which is defined by C

then

CX

T

XT

C

f = C (X f ),

C

f = v s ∂s f ,

has the components [87, p. 15]  C

XT =

Xi v s ∂s X i

 (3.61)

3.13

Problems of Lifts in Symplectic Geometry

99

with respect to the coordinates (x i , x i ) = (x i , v i ). Using (3.59) and (3.61) we have    Xi 0 δik b C K C I ˜ (ω )∗ X T = ( A I X T ) = v s ∂s X i v s ∂i ωks ωki   Xk = i s X v ∂i ωks + ωki v s ∂s X i   Xk = v s (X i ∂i ωks + ωis ∂k X i + ωki ∂s X i ) − v s ωis ∂k X i   Xk , = v s L X ωks − pi ∂k X i

(3.62)

where L X denotes the Lie derivative. On other hand, the complete lift C X T ∗ of a vector field X from manifold M to its cotangent bundle T ∗ M is defined by C

X T ∗ (γ Z ) = γ (L X Y ),

where γ Z and γ (L X Y ) are functions in T ∗ M with the local expressions γ Z = pi Z i , γ (L X Z ) = pi [X , Z ]i and the complete lift

CX

T∗

has the components [87, p. 236]   Xk C XT∗ = − pi ∂k X i

with respect to the coordinates (x i , x i ) = (x i , pi ). From (3.62) we obtain  (ω )∗ X T = X T ∗ + b

C

C

 0 v s L X ωks

,

i.e. if L X ωks = 0, then (ωb )∗ C X T = C X T ∗ . A symplectic vector field X is a vector field on (M, ω) which preserves the symplectic form, i.e. L X ω = 0. Thus we have Theorem 3.23 Let (M, ω) be a symplectic manifold, C X T and C X T ∗ the complete lifts of a vector field X to tangent bundle T M and cotangent bundle T ∗ M, respectively. If X is a symplectic vector field, then C X T and C X T ∗ are ωb -related, i.e. (ωb )∗ C X T = C X T ∗ .

100

3 Problems of Lifts

Since every Hamiltonian vector field X H (ı X H ω = dH ) is a symplectic vector field (L X H ω = d ◦ ı X H ω + dı X H ◦ dω = d2 H = 0), from Theorem 3.23 we immediately have Theorem 3.24 If ωb -related.

C (X

X H is a Hamiltonian vector field, then

H )T

and

C (X

H )T ∗

are

Let (M, ω) be a symplectic manifold of dimension n = 2m. It is well known that in the cotangent bundle T ∗ M there exists a closed 2-form ω˜ = d p = d pi ∧ dx i , where p = pi dx i , i.e. T ∗ M is a symplectic 4m-manifold. If we write ω˜ = 21 ω˜ K L dx K ∧ dx L , then we have   0 δkl ω˜ = (ω˜ K L ) = . −δlk 0 The complete lift C ωT of ω to the tangent bundle T M is a 2-form and has the components of the form [87, p. 38]   v s ∂s ωi j ωi j C (3.63) ωT = ωi j 0 with respect to the coordinates (x i , x i ) = (x i , v i ). We now consider the symplectic isomorphism ω : T ∗ M → T M . Using (dω)skl = 13 (∂s ωkl + ∂k ωls + ∂l ωsk ) = 0, ωi j = −ω ji , ωi j = −ω ji , ωis ωs j = δ ij ,

(3.64)

from (3.60) and (3.63) we see that the pullback of C ω by ω is a 2-form (ω )∗ C ωT on T ∗ M and has the components ¯

j¯

((ω )∗ C ωT )kl = AkI AlJ (C ωT ) I J = Aik Al (C ωT )i j + Aik Al (C ωT )i¯ j + Aik Al (C ωT )i j

= =

j

j j δki δl v s ∂s ωi j + ps (∂k ωis )δl ωi j + δki ps (∂l ω js )ωi j v s ∂s ωkl + ps ((∂k ωis )ωi l − (∂l ωs j )ωk j )

= pt ωts ∂s ωkl − ps (ωsi ∂k ωi l − ωs j ∂l ω jk ) = pt ωts (∂s ωkl − ∂k ωsl + ∂l ωsk ) = 3 pt ωts (dω)skl = 0, j¯

((ω )∗ C ωT )k l¯ = Aik Al¯ (C ωT )i j¯ = δki ω jl ωi j = δki δil = δkl , ¯

((ω )∗ C ωT )k¯ l = Aik¯ Al (C ωT )i¯ j = ωik δl ωi j = −δ kj δl = −δlk , ((ω )∗ C ωT )k¯ l¯ = 0

j

j

j

j¯

3.13

Problems of Lifts in Symplectic Geometry

or

101

  ∗C

(ω )

 ∗C

ωT = (((ω )

ωT ) K L ) =

0 δkl −δlk 0

 .

From here it follows that the pullback (ω )∗ C ωT coincides with the symplectic form ω˜ = d p = d pi ∧ dx i . Thus we have Theorem 3.25 Let (M, ω) be a symplectic manifold. The natural symplectic structure d p = d pi ∧ dx i on the cotangent bundle T ∗ M is the pullback by ω of the complete lift of ω to the tangent bundle T M, i.e.(ω )∗ C ωT = d p . A diffeomorphism between any two symplectic manifods f : (M, ω) → (N , ω ) is called symplectomorphism if f ∗ ω = ω, where f ∗ is the pullback of f . Since d C ωT = C (dω) = 0 [87, p. 25], from Theorem 3.25 we have that the symplectic isomorphism T ω : (T ∗ M, d p) → (T M, C ωT ) is a symplectomorphism. Let now (M, ω) be a symplectic manifold with almost complex structure ϕ (ϕ 2 = −I ). If the 2-form ω satisfies the purity condition ω(ϕ X , Y ) = ω(X , ϕY ), i.e. (ω ◦ ϕ)(X , Y ) = −(ω ◦ ϕ)(Y , X ), then the triple (M, ω, ϕ) is called A-manifold according to the terminology accepted in [40] (also, see [81, p. 31]). We call (X , Y ) = (ω ◦ ϕ)(X , Y ) = ω(ϕY , X ) the twin 2-form associated with ω. ∗ ∗ Let C be a complex algebra and ω = (ωv1 v2 ), v1 , v2 = 1, . . . , r be a complex tensor field of type (0,2) on the holomorphic (analytic) complex manifold Xr (C). Then the real ∗ model of ω is a tensor field ω = (ω j1 j2 ), j1 , j2 = 1, . . . , 2r on M such that ω(ϕ X 1 , X 2 ) = ω(X 1 , ϕ X 2 ) for any vector fields X 1 , X 2 . The ϕ -operator applied to a pure tensor field ω is defined by ( ϕ ω)(X , Y1 , Y2 ) = (ϕ X )(ω(Y1 , Y2 )) − X (ω(ϕY1 , Y2 )) + ω((L Y1 ϕ)X , Y2 ) + ω(Y1 , (L Y2 ϕ)X ) and has the local expression ( ϕ ω)ki j = ϕkm ∂m ωi j − ∂k (ω ◦ ϕ)i j + ωm j ∂i ϕkm + ωim ∂ j ϕkm ,

(3.65)

where ϕ ω is a tensor field of type (0,3), L X is the Lie derivative with respect to X and (ω ◦ ϕ)i j = ϕim ωm j . Let on M be given the integrable almost complex structure ϕ. For a complex tensor ∗ field ω of type (0,2) on Xr (C) to be a C-holomorphic tensor field it is necessary and

102

3 Problems of Lifts

sufficient that ϕ ω = 0. Let now M be a manifold with non-integrable almost complex structure ϕ. In this case, when ϕ ω = 0, ω is said to be almost holomorphic. If the symplectic 2-form ω of the A-manifold (M, ω, J ) satisfies the almost holomorphicity condition ϕ ω = 0, then it is called an almost holomorphic symplectic 2-form. We call an A-manifold admitting such a 2-form an almost holomorphic A-manifold. Let ϕ = ϕ ij ∂i ⊗ dx j be a tensor field of type (1,1) in U ⊂ M. The complete lift C ϕT M of ϕ to the tangent bundle is completely determined by C ϕT M (C X ) = C (ϕ(X ))T M . In an analogous way, the complete lift C ϕT ∗ M of ϕ to the cotangent bundle is completely determined by C ϕT ∗ M (C X ) = C (ϕ(X ))T ∗ M +γ (L X ϕ), where γ (L X ϕ) is a vertical vector n field on T ∗ M with the components γ (L X ϕ) = i=1 ps (L X ϕ)is ∂i . The complete lift of ϕ to the tangent and cotangent bundles are given respectively by [87]   ϕ ij 0 C C I ϕT M = (( ϕT M ) J ) = v s ∂s ϕ ij ϕ ij and

 C

ϕT ∗ M = ((C ϕT ∗ M ) IJ ) =

ϕ ij

0



j

ps (∂ j ϕis − ∂i ϕ sj ) ϕi

with respect to the induced coordinates (x j , x j ) = (x j , v j ) and (x j , x j ) = (x j , p j ). Using (3.59), (3.60), (3.64) and ϕ mj ωmk = ϕkm ω jm to transfer C ϕT M by ω : T ∗ M → T M we have (ω )∗C ϕT M = (( ϕT ∗ M ) LJ ) = ( A˜ IJ A LK (C ϕT M ) KI ) or j

j

j l

j l

( ϕT ∗ M )l = ϕl , ( ϕT ∗ M ) = 0, ( ϕT ∗ M ) = ω ji ωkl ϕki = ϕ lj , j

( ϕT ∗ M )l = v s (∂i ω js )ϕli + ω ji v s ∂s ϕli + ω ji ps (∂l ωks )ϕki   = v s ( ϕ ω)l js + ∂l (ω ◦ ϕ) js − ωis ∂ j ϕli + ω ji ps (∂l ωks )ϕki   = v s ( ϕ ω)l js − pi ∂ j ϕli + v s ∂l ϕ mj ωms + ω ji ps (∂l ωks )ϕki   = v s ( ϕ ω)l js − pi ∂ j ϕli + v s ∂l ϕ mj ωms + v s ϕ mj (∂l ωms ) + ω jm ps (∂l ωks )ϕkm = v s ( ϕ ω)l js − pi ∂ j ϕli + pm ∂l ϕ mj + v s (∂l ωms )ϕ mj + ωmk ps (∂l ωks )ϕ mj = v s ( ϕ ω)l js + pm (∂l ϕ mj − ∂ j ϕlm ) + v s (∂l ωms )ϕ mj − ωks ps (∂l ωmk )ϕ mj = v s ( ϕ ω)l js + pm (∂l ϕ mj − ∂ j ϕlm ) + v s (∂l ωms )ϕ mj − v k (∂l ωmk )ϕ mj = v s ( ϕ ω)l js + pm (∂l ϕ mj − ∂ j ϕlm ).

3.13

Problems of Lifts in Symplectic Geometry

103

Thus, if ϕ ω = 0, then the transfer (ω )∗C ϕT M of C ϕT M coincides with C ϕT ∗ M . Thus we have Theorem 3.26 Let (M, ω, ϕ) be a symplectic A-manifold and ω : T ∗ M → T M be a symplectic isomorphism between cotangent and tangent bundles. If the symplectic Amanifold is an almost holomorphic ( ϕ ω = 0), then the complete lift C ϕT ∗ M is a transfer of C ϕT M by ω , i.e. (ω )∗C ϕT M = C ϕT ∗ M . In the case of integrability of ϕ, the complete lifts C ϕT M and C ϕT ∗ M are complex structures on the tangent and cotangent bundles, respectively (see [87, p. 37, p. 256]), i.e. (T ∗ M, C ϕT ∗ M ) and (T M, C ϕT M ) are complex manifolds. Since A˜ −1 = A (see (3.59), (3.60)), the condition (ω )∗C ϕT M = ( A˜ IJ A LK (C ϕT M ) KI ) = ((C ϕT ∗ M ) LJ ) = C ϕT ∗ M , in Theorem 3.26 can be written in the following form C

ϕT M ◦ (ω )∗ = (ω )∗ ◦ C ϕT ∗ M ,

where (ω )∗ = (A IJ ) . From here it is clear that the mapping ω : T ∗ M → T M is holomorphic. Thus we have Theorem 3.27 Let (M, ω, ϕ) be a holomorphic symplectic A-manifold. If ϕ is an integrable almost complex structure, then the symplectic isomorphism ω (or ωb ) is a holomorphic mapping. On the other hand, from (3.65) we obtain ( ϕ ω)ki j = ϕkm ∂m ωi j − ∂k (ω ◦ ϕ)i j + ωm j ∂i ϕkm + ωim ∂ j ϕkm = ϕkm (∂m ωi j − ∂i ωm j − ∂ j ωim ) + (∂i ωm j )ϕkm + (∂ j ωim )ϕkm   + ωm j ∂i ϕkm + ωim ∂ j ϕkm − ∂k ϕim ωm j = ϕkm (∂m ωi j + ∂i ω jm + ∂ j ωmi )   + ∂i (ϕkm ωm j ) + ∂ j (ϕkm ωim ) − ∂k ϕim ωm j = 3ϕkm (dω)mi j + ∂i k j + ∂ j (ϕim ωmk ) − ∂k i j = 3ϕkm (dω)mi j + ∂i k j + ∂ j ik + ∂k  ji = 3(ϕkm (dω)mi j + (d)ik j ), which on symplectic A-manifold (dω = 0) has the form ( ϕ ω)(X , Y1 , Y2 ) = 3(d)(Y1 , X , Y2 ), where  = ω ◦ ϕ is the twin 2-form. Thus we have Theorem 3.28 A symplectic A-manifold (M, ω, ϕ) is holomorfic if and only if the twin 2-form  = ω ◦ ϕ is closed.

104

3 Problems of Lifts

From Theorems 3.27 and 3.28 we have Theorem 3.29 If  = ω ◦ ϕ is a closed twin 2-form on the A-manifold (M, ω, ϕ), then C  ∗ T ∗ M is a transfer of ϕT M by the symplectic isomorphism ω : T M → T M .

Cϕ

3.14

Anti-Kähler Manifolds and Musical Isomorphisms

In pseudo-Riemannian geometry the musical (bemolle and diesis) isomorphism is an isomorphism between the tangent and cotangent bundles. Let (M, g) be a smooth pseudo-Riemannian manifold of dimension n. A very important feature of any pseudoRiemannian metric g is that it provides the musical isomorphisms g b : T M → T ∗ M and g  : T ∗ M → T M between the tangent and cotangent bundles. The musical isomorphisms g b and g  are expressed by g b : x I = (x i , x i ) = (x i , y i ) → x˜ K = (x k , x˜ k ) = (δik x i , pk = gki y i ) and g  : x˜ K = (x k , x˜ k ) = (x k , pk ) → x I = (x i , x i ) = (δki x k , y i = g ik pk ) with respect to the local coordinates in T M and T ∗ M respectively. The Jacobian matrices of g b and g  are given by   K  k ∂ x ˜ δ 0 b K i = (g∗ ) = ( A˜ I ) = (3.66) ∂x I y s ∂i gks gki and  (g∗ )

 = (A KI ) =

∂x I ∂ x˜ K



 =

δki 0 is ps ∂k g g ik

 (3.67)

respectively, where δ is the Kronecker delta. Let X = X i ∂i be the local expression in U ⊂ M of a vector field X ∈ I10 (M). Then the complete lift C X T of X to the tangent bundle T M is given by C

X T = X i ∂i + y s ∂s X i ∂i

with respect to the natural frame {∂i , ∂i }. Using (3.66) and (3.68), we have

(3.68)

3.14

Anti-Kähler Manifolds and Musical Isomorphisms

 g∗b C X T = 

δik

y s ∂∂gxksi

0 gki



Xi y s ∂s X i



 =

105

 Xk X i y s ∂i gks + gki y s ∂s X i 

Xk      i (L X g)sk − ∂k X gis − ∂s X i gki + gik y s ∂s X i   Xk   , = y s (L X g)sk − pi ∂k X i =

ys



(3.69)

where L X is the Lie derivative of g with respect to the vector field X :     (L X g)sk = X i ∂i gsk + ∂s X i gik + ∂k X i gsi . In a manifold (M, g), a vector field X is called a Killing vector field if L X g = 0. It is well known that the complete lift C X T ∗ of X to the cotangent bundle T ∗ M is given by C

X T ∗ = X k ∂k − ps ∂k X s ∂k .

From (3.69) we find g∗b C X T =C X T ∗ + γ (L X g), where γ (L X g) is defined by 

 γ (L X g) =

0 s y (L X g)sk

Thus we have Theorem 3.30 Let (M, g) be a pseudo-Riemannian manifold, and let C X T and C X T ∗ be the complete lifts of a vector field X to the tangent and cotangent bundles, respectively. Then the differential (pushforward) of C X T by g b coincides with C X T ∗ , i.e. g∗b C X T =C X T ∗ if and only if X is a Killing vector field. Let X and Y be Killing vector fields on M. Then we have L [X ,Y ] g = [L X , L Y ]g = L X ◦ L Y g − L Y ◦ L X g = 0, i.e. [X , Y ] is a Killing vector field. Since C [X , Y ]T = [C X T , C YT ] and [C X T ∗ , C YT ∗ ], from Theorem 3.30 we have Theorem 3.31 If X and Y are Killing vector fields on M, then g∗b [C X T , C YT ] = [C X T ∗ ,C YT ∗ ],

C [X , Y ]

T∗

=

106

3 Problems of Lifts

where g∗b is the differential (pushforward) of musical isomorphism g b . Let now (M, g, ϕ) be an anti-Kähler manifold, where ϕ denote its almost complex structure. If ϕ = ϕ ij ∂i ⊗ dx j is the local expression in U ⊂ M of an almost complex strucure ϕ, then it is well known that the complete lift C ϕT of ϕ to the tangent bundle T M is given by   i 0 ϕ C j (3.70) ϕT = (C ϕ JI ) = y s ∂s ϕ ij ϕ ij with respect to the induced coordinates (x i , x i ) = (x i , y i ) in T M. It is also well known that C ϕT defines an almost complex structure on T M, if and only if so does ϕ on M. Using (3.66), (3.67) and (3.70), we have LK C ϕ KI ) g∗b C ϕT = ( ϕ LJ ) = (A IJ A  =

 j 0 ϕl . y s (∂i g js )ϕli + g ji y s ∂s ϕli + g ji ps (∂l g ks )ϕki g ji g kl ϕki

(3.71)

Since g = (gi j ) and g −1 = (g i j ) are pure tensor fields with respect to ϕ, we find g ji g kl ϕki = g ji g ik ϕkl = δ kj ϕkl = ϕ lj

(3.72)

and y s (∂i g js )ϕli + g ji y s ∂s ϕli + g ji ps (∂l g ks )ϕki   = y s l g js + ∂l (g ◦ ϕ) js − gis ∂ j ϕli + g ji ps (∂l g ks )ϕki = y s l gs j + y s ∂l (g ◦ ϕ) js − pi ∂ j ϕli + g ji ps (∂l g ks )ϕki = y s l gs j − pi ∂ j ϕli + y s ∂l (g ◦ ϕ) js + g ji ps (∂l g ks )ϕki   = y s l gs j − pi ∂ j ϕli + y s ∂l gsm ϕ mj + g ji ps (∂l g ks )ϕki   = y s l gs j − pi ∂ j ϕli + y s (∂l gsm )ϕ mj + y s ∂l ϕ mj gsm + g jm ps (∂l g ks )ϕkm   = y s l gs j − pi ∂ j ϕli + y s (∂l gsm )ϕ mj + y s ∂l ϕ mj gsm + gmk ps (∂l g ks )ϕ mj   = y s l gs j − pi ∂ j ϕli + y s (∂l gsm )ϕ mj + y s ∂l ϕ mj gsm − g ks ps (∂l gmk )ϕ mj   = y s l gs j − pi ∂ j ϕli + y s (∂l gsm )ϕ mj + pm ∂l ϕ mj − y k (∂l gmk )ϕ mj = y s l gs j + ps (∂l ϕ sj − ∂ j ϕls ), where

(3.73)

3.14

Anti-Kähler Manifolds and Musical Isomorphisms

107

k gi j = ϕkm ∂m gi j − ∂k (g ◦ ϕ)i j + gm j ∂i ϕkm + gim ∂ j ϕkm . Substituting (3.72) and (3.73) into (3.71), we obtain   j 0 ϕl bC . g∗ ϕT = y s l gs j + ps (∂l ϕ sj − ∂ j ϕls ) ϕ lj It is well known that the complete lift is given by 

T ∗M

C

ϕT ∗ =

Cϕ

T∗

of ϕ ∈ I11 (M) to the cotangent bundle

j

0 ϕl s s ps (∂l ϕ j − ∂ j ϕl ) ϕ lj



with respect to the induced cordinates in T ∗ M. Thus we obtain g∗b C ϕT =C ϕT ∗ + γ ( ϕ g), where

 γ ( ϕ g) =

 0 0 . y s l gs j 0

From here, we have Theorem 3.32 Let (M, g, ϕ) be an almost anti-Hermitian manifold, and let C ϕT and C ϕT ∗ be the complete lifts of an almost complex structure ϕ to the tangent and cotangent bundles, respectively. Then the differential of C ϕT by g b coincides with C ϕT ∗ , i.e. g∗b C ϕT =C ϕT ∗ if and only if (M, g, ϕ) is an anti-Kähler ( ϕ g = 0) manifold.

References

1. Bejan, C., Gul, I. Sasaki metric on the tangent bundle of a Weyl manifold. Publ. Inst. Math. (Beograd) (N.S.) 103 (117) (2018), 25–32. 2. Bejan, C.L., Dru¸ta˘ -Romaniuc, S.L.: Harmonic functions and quadratic harmonic morphisms on Walker spaces. Turk. J. Math. 40(5), 1004–1019 (2016) 3. Bejan, C.L., Dru¸ta˘ -Romaniuc, S.L.: Structures which are harmonic with respect to Walker metrics. Mediterr. J. Math. 12(2), 481–496 (2015) 4. Bejan, C.L., Dru¸ta˘ -Romaniuc, S.L.: Walker manifolds and Killing magnetic curves. Dif. Geom. Appl. 35, 106–116 (2014) 5. Bejan, C.L., Crasmareanu, M.: Weakly-symmetry of the Sasakian lifts on tangent bundles. 83(1–2), 63–69 (2013) 6. Bonome, A., Castro, R., Hervella, L.M., Matsushita, Y.: Construction of Norden structures on neutral 4-manifolds. JP J. Geom. Topol. 5(2), 121–140 (2005) 7. Borowiec, A., Francaviglia, M., Volovich, I.: Anti-Kählerian manifolds. Dif. Geom. Appl. 12(3), 281–289 (2000) 8. Cakan, R., Akbulut, K., Salimov, A.: Musical isomorphisms and problems of lifts. Chinese Ann. Math. Ser. B 37(3), 323–330 (2016) 9. Cengiz, N., Salimov, A.A.: Complete lifts of derivations to tensor bundles. Bol. Soc. Mat. Mexicana. 8(3), 75–82 (2002) 10. Cruceanu, V., Fortuny, P., Gadea, P.M.: A survey on paracomplex geometry. Rocky Mountain J. Math. 26(1), 83–115 (1996) 11. Davidov, J., Díaz-Ramos, J.C., García-Río, E., Matsushita, Y., Muškarov, O., Vázquez-Lorenzo, R.: Almost Kähler Walker 4-manifolds. J. Geom. Phys. 57, 1075–1088 (2007) 12. Davidov, J., Díaz-Ramos, J.C., García-Río, E., Matsushita, Y., Muškarov, O., Vázquez-Lorenzo, R.: Hermitian-Walker 4-manifolds. J. Geom. Phys. 58, 307–323 (2008) 13. Dragomir, S., Francaviglia, M.: On Norden metrics which are locally conformal to antiKählerian metrics. Acta Appl. Math. 60(2), 115–135 (2000) 14. Dru¸ta˘ -Romaniuc, S.L. General natural α-structures parallel with respect to the Schouten-Van Kampen connection on the tangent bundle. Mediterr. J. Math. 19 (4) (2022), Paper No. 195, 21 pp. 15. Dru¸ta˘ -Romaniuc, S.L.: General natural Riemannian almost product and para-Hermitian structures on tangent bundles. Taiwanese J. Math. 16(2), 497–510 (2012) 16. Dru¸ta˘ -Romaniuc, S.L.: Classes of general natural almost anti-Hermitian structures on the cotangent bundles. Mediterr. J. Math. 8(2), 161–179 (2011) 17. Etayo, F., deFrancisco, A., Santamaría, R. There are no genuine Kähler-Codazzi manifolds. Int. J. Geom. Methods Mod. Phys. 17 (2020), no. 3, 2050044, 12 pp. © The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Salimov, Applications of Holomorphic Functions in Geometry, Frontiers in Mathematics, https://doi.org/10.1007/978-981-99-1296-4

109

110

References

18. Etayo, F., Santamaría, R.: metric manifolds. Publ. Math. Debrecen. 57(3–4), 435–444 (2000) 19. Gadea, P.M., Grifone, J., Munoz Masque, J.: Manifolds modelled over free modules over the double numbers. Acta Math. Hungar. 100(3), 187–203 (2003) 20. Ganchev, G.T., Borisov, A.V.: Note on the almost complex manifolds with Norden metric. Compt. Rend. Acad. Bulg. Sci. 39, 31–34 (1986) 21. García-Río, E., Matsushita, Y.: Isotropic Kähler structures on Engel 4-manifolds. J. Geom. Phys. 33, 288–294 (2000) 22. Gribachev, K., Mekerov, D., Djelepov, G.: Generalized B-manifolds. C. R. Acad. Bulgare Sci. 38(3), 299–302 (1985) 23. Gribachev, K., Mekerov, D., Djelepov, G.: On the geometry of almost B-manifolds. C. R. Acad. Bulgare Sci. 38(5), 563–566 (1985) 24. Gudmundsson, S., Kappos, E.: On the Geometry of the Tangent Bundles. Expo. Math. 20(1), 1–41 (2002) 25. Iscan, M., Salimov, A.A. On Kahler-Norden manifolds. Proc. Indian Acad. Sci. (Math. Sci.) 119 (2009), no.1, 71–80. 26. Kobayashi, S., Nomizu, K.: Foundations of differential geometry, vol. II. Interscience Publishers, New York-London-Sydney (1969) 27. Kruchkovich, G.I.: Conditions for the integrability of a regular hypercomplex structure on a manifold. Ukrain. Geometr. Sb. 9, 67–75 (1970) 28. Kruchkovich, G.I.: Hypercomplex structures on manifolds. I. Trudy Sem. Vektor. Tenzor. Anal. 16, 174–201 (1972) 29. Kruchkovich, G.I.: Hypercomplex structures on manifolds. II. Trudy Sem. Vektor. Tenzor. Anal. 17, 218–227 (1974) 30. Ledger, A.J., Yano, K.: Almost complex structures on tensor bundles. J. Dif. Geom. 1, 355–368 (1967) 31. Magden, A., Salimov, A.A.: Complete lifts of tensor fields on a pure cross-section in the tensor bundle. J. Geom. 93(1–2), 128–138 (2009) 32. Manev, M., Mekerov, D.: On Lie groups as quasi-Kähler manifolds with Killing Norden metric. Adv. Geom. 8(3), 343–352 (2008) 33. Matsushita, Y., Law, P.: Hitchin-Thorpe-type inequalities for pseudo-Riemannian 4-manifolds of metric signature. Geom. Ded. 87, 65–89 (2001) 34. Matsushita, Y.: Four-dimensional Walker metrics and symplectic structure. J. Geom. Phys. 52, 89–99 (2004) 35. Matsushita, Y.: Walker 4-manifolds with proper almost complex structure. J. Geom. Phys. 55, 385–398 (2005) 36. Matsushita, Y. Counterexamples of compact type to the Goldberg conjecture and various version of the conjecture. Proceedings of The 8th International Workshop on Complex Structures and Vector Fields, Sofia, Bulgaria, August 20 - 26, (2004), ed. S. Dimiev and K. Sekigawa, World Scientific (2007). 37. Matsushita, Y., Haze, S., Law, P.R.: Almost Kähler-Einstein structure on 8-dimensional walker manifolds. Monatsh. Math. 150, 41–48 (2007) 38. Mekerov, D.: A connection with skew symmetric torsion and Kähler curvature tensor on quasiKähler manifolds with Norden metric. C. R. Acad. Bulgare Sci. 61(10), 1249–1256 (2008) 39. Mekerov, D.: Connection with parallel totally skew-symmetric torsion on almost complex manifolds with Norden metric. C. R. Acad. Bulgare Sci. 62(12), 1501–1508 (2009) 40. Norden, A.P.: On a class of four-dimensional A-spaces. Izv. Vyssh. Uchebn. Zaved. Matematika. 17(4), 145–157 (1960) 41. O’Neill, B.: Semi-Riemannian geometry. Academic Press, New York (1983)

References

111

42. Oproiu, V., Papaghiuc, N.: Some classes of almost anti-Hermitian structures on the tangent bundle. Mediterr. J. Math. 1(3), 269–282 (2004) 43. Oproiu, V., Papaghiuc, N.: An anti-Kählerian Einstein structure on the tangent bundle of a space form. Colloq. Math. 103(1), 41–46 (2005) 44. Popovici, I.: Contributions à l’étude des espaces à connexion constante. Rev. Roumaine Math. Pures Appl. 23(8), 1211–1225 (1978) 45. Salimov, A.A. Almost analyticity of a Riemannian metric and integrability of a structure.Trudy Geom. Sem. Kazan. Univ. 15 (1983), 72–78. 46. Salimov, A.A. Quasiholomorphic mapping and a tensor bundle.Translation in Soviet Math. (Iz. VUZ) 33 (1989), no. 12, 89–92. 47. Salimov, A.A. Almost -holomorphic tensors and their properties. Translation in Russian Acad. Sci. Dokl. Math. 45 (1992), no. 3, 602–605. 48. Salimov, A.A. A new method in the theory of liftings of tensor fields in a tensor bundle. Translation in Russian Math. (Iz. VUZ) 38 (1994), no. 3, 67–73. 49. Salimov, A.A. Generalized Yano-Ako operator and the complete lift of tensor fields.Tensor (N.S.) 55 (1994), no. 2, 142–146. 50. Salimov, A.A. Lifts of poly-affinor structures on pure sections of a tensor bundle. Translation in Russian Math. (Iz. VUZ) 40 (1996), no. 10, 52–59. 51. Salimov, A.A.: Non-existence of Para-Kahler-Norden warped metrics. Int. J. Geom. Methods Mod. Phys. 6(7), 1097–1102 (2009) 52. Salimov, A.: On operators associated with tensor fields. J. Geom. 99(1–2), 107–145 (2010) 53. Salimov, A.A.: A note on the Goldberg conjecture of Walker manifolds. Int. J. Geom. Methods Mod. Phys. 8(5), 925–928 (2011) 54. Salimov, A. Tensor operators and their applications. Mathematics Research Developments. Nova Science Publishers, Inc., New York, 2013. xii+186 pp. 55. Salimov, A.: On anti-Hermitian metric connections. C. R. Math. Acad. Sci. Paris. 352(9), 731– 735 (2014) 56. Salimov, A.: On structure-preserving connections. Period. Math. Hungar. 77(1), 69–76 (2018) 57. Salimov, A.A., Agca, F.: On para-Nordenian structures. Ann. Polon. Math. 99(2), 193–200 (2010) 58. Salimov, A., Aslanci, S., Jabrailzade, F.: Dual-holomorphic functions and problems of lifts. Chinese Ann. Math. Ser. B 43(2), 223–232 (2022) 59. Salimov, A., Behboudi Asl, M., Kazimova, S.: Problems of lifts in symplectic geometry. Chinese Ann. Math. Ser. B 40(3), 321–330 (2019) 60. Salimov, A.A., Cengiz, N., Behboudi Asl, M.: On holomorphic hypercomplex connections. Adv. Appl. Clifford Algebr. 23(1), 179–207 (2013) 61. Salimov, A., Gezer, A.: On the geometry of the (1,1)-tensor bundle with Sasaki type metric. Chin. Ann. Math. Ser. B 32(3), 369–386 (2011) 62. Salimov, A.A., Gezer, A., Akbulut, K.: Geodesics of Sasakian metrics on tensor bundles. Mediterr. J. Math. 6(2), 135–147 (2009) 63. Salimov, A., Iscan, M. Some properties of Norden-Walker metrics.Kodai Math. J. 33 (2010), no.2, 283–293. 64. Salimov, A.A., Iscan, M.: On Norden-Walker 4-manifolds. Note Mat. 30, 111–128 (2010) 65. Salimov, A., Iscan, M., Akbulut, K.: Some remarks concerning hyperholomorphic B-manifolds. Chin. Ann. Math. Ser. B 29(6), 631–640 (2008) 66. Salimov, A.A., Iscan, M., Akbulut, K. Notes on para-Norden-Walker 4-manifolds.Int. J. Geom. Methods Mod. Phys. 7 (2010), no.8, 1331–1347. 67. Salimov, A.A., Iscan, M., Etayo, F. Paraholomorphic B-manifold and its properties.Topology Appl. 154 (2007), no. 4, 925–933.

112

References

68. Salimov, A.A., Magden, A.: Complete lifts of tensor fields on a pure cross-section in the tensor bundle. Note Mat. 18(1), 27–37 (1998) 69. Sasaki, S.: On the Differantial geometry of tangent bundles of Riemannian manifols. Tohoku Math. J. 10, 338–358 (1958) 70. Sato, I. Almost analytic tensor fields in almost complex manifolds. Tensor (N.S.) 17 (1966), 105–119. 71. Scheffers, G. Generalization of the foundations of ordinary complex functions. I, II. (Verallgemeinerung der Grundlagen der gewöhnlich complexen Functionen. I. II.) (German) Leipz. Ber. XLV. (1893), 828–848. 72. Sekigawa, K.: On some 4-dimensional compact Einstein almost Kähler manifolds. Math. Ann. 271(3), 333–337 (1985) 73. Shirokov, A.P. On a property of covariantly constant affinors. Dokl. Akad. Nauk SSSR (N.S.) 102 (1955), 461–464. 74. Shirokov, A.P. On the question of pure tensors and invariant subspaces in manifolds with almost algebraic structure. Kazan. Gos. Univ. Uˇcen. Zap. 126 (1966), kn. 1, 81–89. 75. Shirokov, A.P. Spaces over algebras and their applications. Geometry, 7. J. Math. Sci. (New York) 108 (2002), no. 2, 232–248. 76. Sultanova, T., Salimov, A. On holomorphic metrics of 2-jet bundles. Mediterr. J. Math. 19 (2022), no. 1, Paper No. 29, 12 pp. 77. Tachibana, S.: Analytic tensor and its generalization. Tohoku Math. J. 12, 208–221 (1960) 78. Talantova, N.V., Shirokov, A.P.: A remark on a certain metric in the tangent bundle. Izv. Vysš. Uˇcebn. Zaved. Matematika 157(6), 143–146 (1975) 79. Vishnevskii, V.V. On the complex structure of B-spaces.Kazan. Gos. Univ. Uchen. Zap. 123 (1963), kn. 1, 24–48. 80. Vishnevskii, V.V.: A certain class of spaces over plural algebras. Izv. Vysš. Uˇcebn. Zaved. Matematika 81(2), 14–22 (1969) 81. Vishnevskii, V.V., Shirokov, A.P., Shurygin, V.V.: Spaces over algebras. Kazanskii Gosudarstvennii Universitet, Kazan (1985) 82. Vishnevskii, V.V. Integrable affinor structures and their plural interpretations. Geometry, 7. J. Math. Sci. (New York) 108 (2002), no. 2, 151–187. 83. Vranceanu, G. Spazi a connessione affine e le algebre di numeri ipercomplessi. Ann. Scuola Norm. Sup. Pisa (3) 12 (1958), 5–20. 84. Walker, A.G.: Canonical form for a Riemannian space with a paralel field of null planes. Quart. J. Math. Oxford 1(2), 69–79 (1950) 85. Yano, K., Kobayashi, Sh. Prolongations of tensor fields and connections to tangent bundles. I. General theory. J. Math. Soc. Japan 18 (1966), 194–210. 86. Yano, K., Ako, M.: On certain operators associated with tensor fields. Kodai Math. Sem. Rep. 20, 414–436 (1968) 87. Yano, K., Ishihara, S. Tangent and cotangent bundles: differential geometry. Pure and Applied Mathematics, No. 16. Marcel Dekker, Inc., New York, 1973.

Index

A Adapted charts, 23 frame, 7 Almost anti-Hermitian manifold, 32 hypercomplex structure, 7 holomorphic, 18 A holomorphic metric, 32 A holomorphic anti-Hermitian manifold, 32 Norden manifold, 37 Norden-Walker manifold, 51 Norden-Walker metric, 50 A-manifold, 102 Anti-Hermitian manifold, 32 metric, 32 Anti-Kähler manifold, 34 Arbitrary charts, 23 Associated function, 41

C Codazzi equation, 43 Complete lift, 64, 90, 91 Cross-section, 64

D Deformed complete lift, 75, 93 intermediate lift, 94 lift, 90, 91

Derivative, 4 Dual-holomorphic manifold, 73 tensor field, 77 vector field, 75

F Frobenius algebras, 3

G Gradient of a function, 40

H Hessian metric, 41 -Norden metric, 42 Hessian of a function, 40 Holomorphic A-manifold, 102 function, 4, 41 mapping, 72 Holomorphic anti-Hermitian manifold, 32 A holomorphic anti-Hermitian manifold, 32 A holomorphic function, 36 Horizontal lift, 64

I Infinitesimal automorphism, 18 Integrable, 7 Intermediate lift, 90, 91

© The Editor(s) (if applicable) and The Author(s), under exclusive license to Springer Nature Singapore Pte Ltd. 2023 A. Salimov, Applications of Holomorphic Functions in Geometry, Frontiers in Mathematics, https://doi.org/10.1007/978-981-99-1296-4

113

114 Isotropic anti-Kähler, 56

K Killing vector field, 105

L 0-th lift, 90, 94 1-th lift, 90, 94 2-th lift, 90, 94 Liouville types, 90 Locally holomorphic function, 41

M Musical (bemolle and diesis) isomorphism, 104

N Non-integrable, 18 Norden manifold, 37 metric connections with torsion, 44

P Polyafinor structure, 7 Proper almost complex structure, 51 Pure, 77 connection, 21

Index cross-section, 66 tensors, 11

Q Quasi-Kähler, 57

R Regular representation of type I, 2 Rigid, 7

S Symplectic, 98 vector field, 99 Symplectomorphism, 101

T Transpose regular representations, 3 Twin Norden metric, 37 connection of type I, 46 connection of type II, 47

V Vertical lift, 60, 93

W Walker metric, 49