Morse Theory. (AM-51), Volume 51 9781400881802

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Annals of Mathematics Studies Number 5 1

MORSE T H E O R Y BY

J. Milnor Based on lecture notes by M. SPIVAK and R. WELLS

PRINCETON, NEW JERSEY PRINCETON UNIVERSITY PRESS

Copyright © 1963, © 1969, by Princeton University Press All Rights Reserved L.C. Card 63-13729 ISBN 0-691-08008-9 Third Printing, with corrections and a new Preface, 1969 Fourth Printing, 1970 Fifth Printing, 1973

Printed in the United States of America 19

18

17

16

15

14

PREFACE This book gives a present-day account of Marston Morse's theory of the calculus of variations in the large.

However, there have been im­

portant developments during the past few years which are not mentioned Let me describe three of these R. Palais and S. 3male have studied Morse theory for a real-valued function on- an infinite dimensional manifold and have given direct proofs of the main theorems, without making any use of finite dimensional ap­ proximations.

The manifolds in question must be locally diffeomorphic

to Hilbert space, and the function must satisfy a weak compactness con­ dition. M

As an example, to study paths on a finite dimensional manifold

one considers the Hilbert manifold consisting of all absolutely con­

tinuous paths

or. [o,l 3 -> M with square integrable first derivative. Ac­

counts of this work are contained in R. Palais, Morse Theory on Hilbert Manifolds, Topology, Vol. 2 (1963), pp. 299-31*0; and in S. Smale, Morse Theory and a Non-linear Generalization of the Dirichlet Problem, Annals of Mathematics, Vol. 80 (196I+), pp. 382-396. The Bott periodicity theorems were originally inspired by Morse theory

(see part IV). However, more elementary proofs, which do not in­

volve Morse theory at all, have recently been given.

See M. Atiyah and

R. Bott, On the Periodicity Theorem for Complex Vector Bundles,Acta Mathematica, Vol. 112 (1964), pp. 229-2t7, as well as R. Wood, Banach Algebras and Bott Periodicity, Topology, b (1965-66), pp. 3 7 1 -3 8 9 Morse theory has provided the inspiration for exciting developments in differential topology by S. Smale, A. Wallace, and others, including a proof of the generalized Poincare hypothesis in high dimensions.

I

have tried to describe some of this work in Lectures on the h-cobordism theorem, notes by L. Siebenmann and J. Sondow, Princeton University Press, 1965.

Let me take this opportunity to clarify one term which may cause fusion.

In §12 I use the word "energy" for the integral v

con­

vi

PREFACE

E=y

0

along a path a>(t) . V. Arnol'd points out to me that mathematicians for the past 200 years have called E the "action”integral. This discrepancy in terminology is caused by the fact that the integral can be interpreted, in terms of a physical model, in more than one way. Think of a particle P which moves along a surface M during the time interval 0< t < 1 . The action of the particle during is defined

this time interval

to be a certain constant times the integral E.

If no forces

act on P (except for the constraining forces which hold it within M), then the "principle of least action" asserts that E will be minimized within the class of all paths joining a>(o) to 03(1), or at least that the first variation of E will be zero.

Hence P must traverse a geodesic.

But a quite different physical model is possible.

Think of a rubber

band which is stretched between two points of a slippery curved surface. If the band is described parametrically by the equation x = o)(t), 0 < t < 1 , then the potential energy arising from tension will be proportional to our integral E (at least to a first order of approximation). For an equilibrium position this energy must be minimized, and hence the rubber band will describe a geodesic. The text which follows is identical with that of the first printing except for a few corrections.

I am grateful to V. Arnol'd, D. Epstein

and W. B. Houston, Jr. for pointing out corrections. J.W.M.

Los Angeles, June 1968.

CONTENTS PREFACE

v

PART I.

NON-DEGENERATE SMOOTH FUNCTIONS ON A MANIFOLD § 1. Introduction.........................................

1

§2. Definitions and Lemmas...............; ..............

k

§3 . Homotopy Type in Terms of Critical Values.............

12

§^. Examples............................................

25

§5 . The Morse Inequalities...............................

28

§6. Manifolds in Euclidean Space:

The Existence of

Non-degenerate Functions

32

§7 . The Lefschetz Theorem on Hyperplane Sections...........

PART II.

39

A RAPID COURSE IN RIEMANNIAN GEOMETRY §8. Covariant Differentiation............................

^3

§9. The Curvature Tensor.................................

51

§10. Geodesics and Completeness............................

55

PART III. THE CALCULUS OF VARIATIONS APPLIED TO GEODESICS .§11. The Path Space of a Smooth Manifold...................

67

§12. The Energy of a Path.................................

70

§13. The Hessian of the Energy Function at a Critical Path . .

7^

§D.

77

Jacobi Fields:

The Null-space of

E**................

§1 5 . The Index Theorem...................................

83

§1 6 . A Finite Dimensional Approximation to Qc .............

88

§1 7 . The Topology of the Full Path Space...................

93

§1 8 . Existence of Non-conjugate Points ....................

98

§19. Some Relations Between Topology and Curvature ..........

100

vii

CONTENTS PART IV.

APPLICATIONS TO LIE GROUPS AND SYMMETRIC SPACES §20.

Symmetric S p a c e s ........................

§21.

Lie Groups as Symmetric S p a c e s

§22.

Whole Manifolds of Minimal Geodesics.

§23.

The Bott Periodicity Theorem for theUnitary Group

§2k .

The Periodicity Theorem for the OrthogonalGroup.. . . .

APPENDIX.

........ 109

................... 112 ............. 118 ...

12k 133

THE H0 M0 T0 PY TYPE OF A MONOTONE U N I O N .................... 1U9

viii

PART I NON-DEGENERATE SMOOTH FUNCTIONS ON A MANIFOLD.

§1 . Introduction. In this section we will illustrate by a specific example the situ­ ation that we will investigate later for arbitrary manifolds. sider a torus

M,

tangent to the plane

V,

Let us con­

as indicated in Diagram 1.

Diagram 1 . Let above the V f(x) < a.

f: M —►R

(R always denotes the real numbers) be the height

plane, and let

M8, be the set of all points

x e M

such that

Then the following things are true: (1)

If a < 0 = f (p), then

(2)

If f(p) < a < f(q),

then

M? is homeomorphic to

a 2-cell.

(3 ) *

If f(q) < a < f(r), then

M8,is homeomorphic to

a cylinder:

(i*)

M8, is vacuous.

If f(r) < a < f(s), then M8, is homeomorphic to a compact manifold of genus one having a circle as boundary:

1

2

I. NON-DEGENERATE FUNCTIONS

(5 )

If

f(s) < a,

then

M8

is the full torus.

In order to describe the change in of the points

f(p),f(q),f(r),f(s)

as

a

passes through one

it is convenient to consider homotopy

type rather than homeomorphism type. (1) —► (2)

M8

In terms of homotopy types:

is the operation of attaching a o-cell.

homotopy type is concerned, the space

For as far as

M8, f(p) < a < f(q), cannot be dis­

tinguished from a o-cell:

■ Here

"«"

means "is of the same homotopy type as." (2) -► (3)

(3)

is the operation of attaching a

1-cell:

(J+) is again the operation of attaching a 1-cell:

(U) -*• (5)

is the operation of attaching a 2_cell.

The precise definition of "attaching a k-cell" can be given as follows.

Let

Y

be any topological space, and let ek

=

{x e R k : ||x|| < 1}

be the k-cell consisting of all vectors in Euclidean k-space with length
- 0 • The real number

f(p)

is called a critical value of

We denote by If

a

M8, the set of all points

is not a critical value of

function theorem that f~1(a)

f

f.

x € M

such that

then it follows from the implicit

M8, is a smooth manifold-with-boundary.

is a smooth submanifold of A critical point

p

f(x) < a.

The boundary

M.

is called non-degenerate if and only if the

matrix

.o a2f (p)) dx^dx^

is non-singular.

It can be checked directly that non-degeneracy does not

depend on the coordinate system.

This will follow also from the following

intrinsic definition. If functional then

v

p

f**

and

w

is a critical point on

TMp,

where

f we define a symmetric bilinear

called the Hessian of

have extensions

f**(v,w) = vp(w(f)),

of

vp

v

and

w

f

p.

If

to vector fields.

is, of course, just

this is symmetric and well-defined.

at

v.

v,w € TMp We let

We must show that

It is symmetric because

vp (w(f)) - ftpWf)) = [v,w]p (f) = 0 where

[v,w]

Here w(f)

is the Poisson bracket of

v

and

w,

denotes the directional derivative of

and where [v,w] (f) = 0 f

in the direction w.

§2. since

f has

p

as a critical point.

Therefore

f**

Vp(w(f)) = v(w(f)) Wp(v(f))

DEFINITIONS AND LEMMAS

is symmetric.

It is now clearly well-defined since

is independent of the extension

v

of

v,

while

is independent of w. If

(x"*,...,xn) is a local coordinate system and v = E a. —^-r-l , ^ d 1 w = E b. — - _ we can take w = E b. — s- where b. now denotes a conJ p J J n

stant function.

Then

f**(v,w) = v(w(f))(p) = v(E b.

=

3 8xJ /

so the matrix

\

respect to the basis

—^r-

a. b. — jMU- (p) ; 1 J 8x 8x

E

ij

represents the bilinear function

f**

with

.. ., — _ p' ^ x n p

We can now talk about the index and the nullity of the bilinear functional tor space

f** on V,

on which

H

TM^.

The index of a bilinear functional

for every

is negative definite; the nullity is the dimension of the null-

w e V. f

index of

f**

on a vec­

is defined to be the maximal dimension of a subspace of V

space, i.e., the subspace consisting of all

point of

H,

The point

if and only if on

TMp

p f**

v e V

such that

H(v,w) = 0

is obviously a non-degenerate critical on

TM^

has nullity equal to

0.

will be referred to simply as the index of

The Lemma of Morse shows that the behaviour of described by this index.

f

at p

The f

can be completely

Before stating this lemma we first prove the

following: LEMMA 2.1 . Let f be a C°° function in a convex neigh­ borhood V of 0 in R n, with f(0) =0. Then n f(x1,...,xn) = ^ x ^ C x , , . . . ^ ) i= 1

for some suitable

C°° functions

defined in V,

with

si(0) = H i(o) * PROOF:

. P df(tx1,...,tx )

f (X1,•••,xn) =

J

0

at

-n r V d t - J ^ ( t X i , . . . , t x n)-Xidt. 0 i=i 1 1

Therefore we can let

gi(x1,... ,xn) =

J 0

( t x ^ ,... ,txn) dt .

1

at

p.

6

I. NON-DEGENERATE FUNCTIONS LEMMA 2.2 (Lemma of Morse) . Let p be a non-degenerate critical point for f. Then there is a local coordinate system (y1,*--,yn) in a neighborhood U of p with yi(p) = o for all i and such that the identity f . f(p) . (yV -

holds throughout

PROOF:

... - ( y V

U,

where

\

- (yV

is the index of

f

at

p.

We first show that if there is any such expression for

X must be the index of

then

♦ (yx+1)2 ♦

f

at

p.

f,

For any coordinate system

(z1,..., zn), if f(q) = f(p) - (z’(q))2- ... - (zx (q))2 + (zX+1 (q))2 + ... + (zn (q))2

then we have

—dzi^dzJ r(P) =

X ’ otherwise ,

which shows that the matrix representing I 3 7

IP’

± |

i =J < x , ,

f**

with respect to the basis

is

’ Sz11 lp

/

>

("'■.) Therefore there is a subspace of

TMp

tive definite, and a subspace V

of dimension

definite.

If there were a subspace of

on which

f** were

which is

clearlyimpossible.

x

of dimension

TM^

negativedefinite then

n-x

where

where

p

is positive x

this subspace would intersect V, of f**.

We now show that a suitable coordinate system Obviously we can assume that

f**

is nega­

of dimension greater than

xis the index

Therefore

f**

is the origin of R n

(y1,...,yn) exists. and that

f(p) = f(o) = o*

By 2.1 we can write n f(Xl ,...,xn ) = £

xjgj(x1,...,xn )

j= 1 for

(x1,...,xn) in some neighborhood of

0.

Since

critical point: sj(0) = ^ r r throughout

U 1; where the matrices

a linear change in the last Let

n-r+i

(H^ (u1,... ,un)) are symmetric. After coordinates we may assume that

g(u1,...,un) denote the square root of

be a smooth, non-zero function of u-j,...,^ borhood

U2 C Ui

vi = ui

of

0.

13^ ( 0 ) 4 0.

|Hpr(u1,...,un)|. This will throughout some smaller neigh­

Now introduce new coordinates

v1,...,vn

by

for 1 ^ r + Y ulH Lr(u1,...,un)/Hpi>(u1,...,un)l. 1> r

vr (u1,...,un) =

It follows from the inverse function theorem that

v1,...,vn will serve as

coordinate functions within some sufficiently small neighborhood It is easily verified that f ■ Y

± = = + 1.

Thus the corresponhence t — f( 0. Then, for all sufficiently small e, the set Mc+e has the homotopy type of Mc”e with a x-cell attached.

The idea of the proof of this theorem is indicated in Diagram 4 , for the special case of the height function on a torus.

The region

Mc-e = f-i is heavily shaded.

We will introduce a new function

coincides with the height function borhood of

p.

Thus the region

gether with a region

H

near

p.

f

F: M -*■ R

except that F < f

F”1(-oo,c-s]

Diagram 4.

in a small neigh­

will consist of Mc_e

In Diagram 4 , H

shaded region.

which

to­

is the horizontally

§3. Choosing a suitable cell

HOMOTOPY TYPE e

C H,

15

a direct argument

ing in along the horizontal lines) will show that retract of Mc"e u H.

(i.e., push­

Mc”eu e*' is a deformation

Finally, by applying 3.1 to the function F

region F”1[c-e,c+s] we will see that

Mc_e u H

and the

is a deformation retract

of Mc+e. This will complete the proof. Choose hoose a c coordinate o< system u1,...,un

in a neighborhood

U

of

p

so that the identity f = c - (u1)2- ... - (ux)2 + (ux+1)2+... + (Un)2 holds throughout

U.

Thus the critical point

p

will have coordinates

u1(p) - ... « un (p) = 0 . Choose

e > 0 (1)

sufficiently small so that The region

points other than (2)

f~1[c-e,c+e]

is compact and contains no critical

p.

The image of

U

under the diffeomorphic

(u\...,un):

U

imbedding

►R

contains the closed ball. ((u1,....u11) : Z ( u 1)2 s ii(r) = 0 -1 < where

n ’(r) =

neighborhood

U,

for

r > 2e

ii*(r) < 0

. Now let

for all r,

F coincide with

f

outside of the coordinate

and let

F = f - n((u1)2+... +(uX)2 + 2(uX+1)2+...+2(un)2) within this coordinate neighborhood.

It is easily verified that

well defined smooth function throughout

F

is a

M.

It is convenient to define two functions 1 ,1: TJ

► t0,»)

by I = ( u V + ... + (ux)2 t, = (uX+1)2 + ... + (u11)2 Then

f = c - £ + t^;

so that:

F(q) = o - i(q) + n(q) - ix(|(q) + Sn(q))

for all

q e U. ASSERTION 1. Mo+e = f-i(_ PROOF:

The region

F”1(

Outside of the ellipsoid

c+e] coincides with the region

| + 2t^ < 2e the functions

f and

ade­

§3. F

coincide.

HOMOTOPY TYPE

17

Within this ellipsoid we have F < f = c-t+n < c+ l|+n < c+e .

This completes the proof. ASSERTION 2. PROOF:

The critical points of F

are the same as those of

f.

Note that || = -1 - n'(! +2t)) < 0 9P ■Sri =

1 - 2u'(t +2t|) > 1 .

Since

^ =H where the covectors it follows that

F

d|

and

+ ^ d"

dti are simultaneously zero only at the origin,

has no critical points in

U

other than the origin.

Now consider the region F“1[c-e,c+e].

By Assertion 1 together

with the inequality F < f we see that F”1[c-e,c+e] C f“1[c-e,c+e] . Therefore

this region is compact.

except possibly

p.

It can contain no criticalpointsof F

But F(p) = c - n(0) < c - e .

Hence

F“1[c-e,c+e]

contains no critical points.

Together with 3.1 this

proves the following. ASSERTION 3 . The region

F~1(-oo,c-e]

is a deformation retract of

Mc+e. It will be convenient to denote this region Mc~e u H;

where

REMARK: described

as

H

F"1(-0. CASE 2.

rQ maps the entire region into

F~1(-°o,c-s3

maps

Within the region

e\

The

into itself, follows from the in-

6 < |
* < V ' % Z > which completes the proof.

■

I

(W

^

+

TT )

■ m < V ’W > -

II. RIEMANNIAN GEOMETRY COROLLARY Q.k. vector Xp €

For any vector fields

Xp

and completes the proof that

P

P

is parallel along

Henceforth we will assume that

M

is identically zero; s.]

is a Riemannian manifold, pro­

vided with the unique symmetric connection which is compatible with its metric.

In conclusion we will prove that the tensor

R

satisfies four

symmetry relations.

LEMMA 9.3. The curvature tensor of a Riemannian manifold satisfies: (1 ) R(X,Y)Z + R(Y,X)Z = 0 (2 ) R(X,Y)Z + R(Y,Z)X + R(Z,X)Y = 0 (3) + = 0 (1*) = .

The skew-symmetry relation (1 ) follows immediately from the

PROOF:

definition of R. Since all three terms of (2) prove (2) when the bracket products zero.

are tensors, it is sufficient to [X,Y], [X,Z]

and

[Y,Z]

are all

Under this hypothesis we must verify the identity - X b (Y b Z) +

Y b (X b Z)

-Yb(ZbX)

Zb(YbX)

+

- Z b (X b Y) +

X b (Z b Y)

=

0 .

But the symmetry of the connection implies that Y b Z - Z b Y

=

= 0

[Y,Z]

Thus the upper left term cancels thelower right maining terms cancel in pairs.

.

term.Similarly the re­

This proves (2).

To prove (3 ) we must show that the expression skew-symmetric in

Z and W.

This is clearly equivalent to the assertion

that

for all

X,Y,Z.

is

Again we may assumethat

=

0

[X,Yl

= 0,

so that

is equal to < - X h (Y b Z) + Y I- (X H Z),Z>

•

54

II.

RIEMANNIAN GEOMETRY

In other words we must prove that the expression < Y b (X b Z),Z> is symmetric in X Since and

Y.

and

[X,Y]

Y.

=

0

the expression YX

is symmetric in X

Since the connection is compatible with the metric, we have X




2

hence YX < Z , Z > = 2 < Y H X I- Z),Z> + 2 < X I- Z,Y h Z > . But the right hand term is clearly symmetric in X

is symmetric in X

and

and

Y.

Therefore

Y; which proves property (3 ).

Property (4 ) may be proved from (1 ), (2), and (3 ) as follows.

►

r< R (Z , X)Y,

< R (X,W )Y,Z>

Formula (2) asserts that the sum of the quantities at the vertices of shaded triangle W

is zero.

Similarly (making use of (1 ) and (3 )) the

sum of the vertices of each of the other shaded triangles is zero.

Adding

these identities for the top two shaded triangles, and subtracting the identities for the bottom ones, this means that twice the top vertex minus twice the bottom vertex is zero.

This proves (4 ), and completes the proof

§1 0 . GEODESICS AND COMPLETENESS

§10. Let

M

Geodesics and Completeness

be a connected Riemannian manifold.

DEFINITION.

A parametrized path 7:

where

I

55

I - M,

denotes any interval of real numbers, is called a geodesic

acceleration vectorfield vector field identity

^ must

^

^ isidentically

beparallelalong

d . d,

dr .

y.

2 . Ddr

y.

Thus thevelocity

If y is ageodesic,

of the velocity vector is

Introducing the arc-length function

s(t)

=

^

+ constant

This statement can be rephrased as follows:

The parameter t

geodesic is a linear function of the arc-length. ally equal to the arc-length if and only if

along

The parameter

||^|| =

t -*• r(t) e M

The equation -pjr ^

determines

n

smooth functions

is actu­

u1,--*,un

u 1(t),...,un(t).

for a geodesic then takes the form

■d2uk . y

rk

, 1

L

rij

(u ' •• •’U 3“TO ~ 5Z

+

dt

t

a

1

In terms of a local coordinate system with coordinates a curve

then the

dr .

shows that the length ||^|| = ! - ' ( « I2 where equality

holdsonly if

-^

b 1 a

= 0;

henceonly if

-^ =

0.

Thus

b |[|^||dt > y

|r'(t) |dt > |r(b) - r(a) 1 a

where equality holds only if

r(t)

is monotone and

v(t) is constant.

This completes the proof. The proof of Theorem 10.1* piecewise smooth path

©

from

q

qf ||v||

expq (rv) e Uq ;

=

o < r < s,

tain a

segment joining the spherical shell of radius r,

segment will be

> r - 5;

will be

> r.

If

1.

Then for any

8 > 0the path

hence letting

tend to

8

does not coincide with

easily obtain a strict inequality.

COROLLARY 10.7 .

Suppose that a path

by arc-length,

the length of any

must con­

0

The length of this the length of

r(Co,l]),

©

then we

This completes the proof of 10.1*.

An important consequence of Theorem 10.1*

metrized

©

5 to the spherical

and lying between these two shells.

©([0,1])

Consider any

to a point

where

shell of radius

=

±3 now straightforward.

is the following.

cd: [o,jU -*■ M, para­

haslength

less than or equal

other path from

©(0) tocd(j£) . Then

to ©

is a geodesic. PROOF:

Consider any segment of

above, and having length 10.1*.

< e.

Hence the entire path DEFINITION.

©

lying within an open set

W, as

This segment must be a geodesic by Theorem ©

A geodesic

is a geodesic. 7:

[a,b]

-► M

will be called minimal

if

62

II. RIEMANNIAN GEOMETRY

its length is less than or equal to the length of any other piecewise smooth path joining its endpoints. Theorem 1 0 . 4 asserts that any sufficiently small segment of a geodesic is minimal.

On the other hand a long geodesic may not be minimal.

For example we will see shortly that a great circle arc on the unit sphere is a geodesic.

If such an arc has length greater than

*,

it is certainly

not minimal. In general, minimal geodesics are not unique.

For example two anti­

podal points on a unit sphere are joined by infinitely many minimalgeodesics. However, the following assertion is true. Define the distance

p(p,q)

p,q e M

between two points

to be the

greatest lower bound for the arc-lengths of piecewise smooth paths joining these points.

This clearly makes

M

into a metric space.

It follows

easily from 10.4 that this metric is compatible with the usual topology of M. COROLLARY 10.8. Given a compact set K C M there exists a number 5 > 0 so that any two points of K with dis­ tance less than & are joined by a unique geodesic of length less than &. Furthermore this geodesic is minimal; and depends differentiably on its endpoints. PROOF.

Cover

K

by open sets Wa, as in 1 0 .3 , and let

small enough so that any two points in

K

with distance less than

5

be 5

lie

in a common Wa . This completes the proof. Recall that the manifold

M

is geodesically complete if every geo­

desic segment can be extended indifinitely. THEOREM 10.9 (Hopf and Rinow*). If M is geodesically complete, then any two points can be joined by a minimal geodesic. PROOF. U

.r'

*

Given

p,q e M

as in Lemma 10.3. Let

S C U

with distance

r > 0,

choose a neighborhood

denote a spherical shell of radius

5 < e

Compare p. 341 of G. de Rham, Sur la r§ductibilite d ’un espace de Rlemann, Commentarii Math. Helvetici, Vol. 26 (1952); as well as H. Hopf and W. Rinow, Ueber den Begriff der•vollstandigen differentialgeometrischen Flache, Commentarii,Vol. 3 (1 9 3 1 ), pp. 209-225.

§1 0 . GEODESICS AND COMPLETENESS about

p.

Since

S

is compact, there exists a point p0

on

S

63

I v ||

=, expp (5v),

for which the distance to

q

=

1,

is minimized.

expp(rv)

=

q.

This implies that the geodesic segment

t -►

is actually a minimal geodesic from

to

p

We will prove that

7(t) = expp (tv),

0 < t < r,

q.

The proof will amount to showing that a point which moves along the

y

geodesic

must get closer and closer

to

q.

In fact for each

t € [5 ,r]

we will prove that O t)

p(r(t),q)

This identity, for

t = r,

=

r-t

.

will complete the proof.

First we will show that the equality path from

p

to

q

must pass through

p(p,q) = Min

S,

(1 )

is true.

we have

(p(p,s) + p(s,q)) = 5 + p(p0 ,q)

s€S

p(pQ,q) = r - 5.

Therefore

tQ € [s,r]

Let (1.)

is true.

z

If

tQ < r

point of

S'

.

0 Since

this proves (1 &).

pQ = 7(5),

denote thesupremum of those numbers

Then by continuity the equality

we will obtain a contradiction.

cal shell of radius

Since every

s!

p(r(tQ ),q) = Min

s €S '

(p(7 (t

0

for which

(1. ) is true also. S’

denote a small spheri­

about the point

po €

with minimum distance from

0

Let

t

q.

a

(Compare Diagram 10.)

),s) + p(s,q)) = 5' + p(pi,q)

0

Then

,

hence (2)

p(Po,q) = (r - tQ) - 5 * We claim that

p^

is equal to

.

7 ( ^ + 5').

In fact the triangle

inequality states that p (P,Pq )

(making use of (2)). p^

>

p (P^)

“

p (Pq

^)

= ^

+ 5’

But a path of length precisely

is obtained by following

a minimal geodesic from

7

y(tQ)

from to

p^.

p

to

y(tQ),

tQ + 5 f

from

p

and then following

Since this broken geodesic has

minimal length, it follows from Corollary 10.7 that it is an (unbroken)

to

64

II. REIMANNIAN GEOMETRY

geodesic, and hence coincides with Thus

7(tQ + 5 1) = p^.

7.

Now the equality (2) becomes

p(7 (^0 + 5 '),q) = r - (t0 + 8')

(ito+5l)

This contradicts the definition of

.

tQ; and completes the proof.

Diagram 10. As a consequence one has the following. COROLLARY 10.10. If M is geodesically complete then every bounded subset of M has compact closure. Con­ sequently M is complete as a metric space (i.e., every Cauchy sequence converges). PROOF. exppi

TMp-*’M

If X C M

has diameter

maps the disk of radius

d d

then for any in TMp

of M which (making use of Theorem 10.9) contains of X

p €X

the map

onto a compact subset X.

Hence the closure

is compact. Conversely, if M

is complete as a metric space, then it is not

difficult, using Lemma 10.3, to prove that M

is geodesically complete.

For details the reader is referred to Hopf and Rinow.

Henceforth we will

not distinguish between geodesic completeness and metric completeness, but will refer simply to a complete Riemannian manifold.

§1 0 . GEODESICS AND COMPLETENESS FAMILIAR EXAMPLES OF GEODESICS. the usual coordinate system dx1 0 dx1 + ...+ dxn 0 dxR desic

7,

given by

X j ,...,xn

In Euclidean n-space,

Rn,

with

and the usual Riemannian metric

b-

we have

65

= 0

t -► (x^ (t),...,xR (t)

and the equations for a geo­

become

d 2x.

whose solutions are the straight lines. follows:

This could also have been seen as

it is easy to show that the formula for arc length

i-1 coincides with the usual definition of aru length as the least upper bound of the lengths of inscribed polygons;

from this definition it is clear that

straight lines have minimal length, and are therefore geodesics. The geodesics on intersections of

PROOF.

Sn

are precisely the great circles, that is, the

with the planes through the center of

Reflection through a plane

whose fixed point set is with a unique geodesic

C = Sn (1 E 2 . C*

is an isometry, the curve between

Sn

I(x) = x

and

Let

E2 x

Sn .

is an isometry and

y

I(y) = y.

Then, since

is a geodesic of the same length as Therefore

Sn -*■ Sn

be two points of

of minimal length between them. I(O')

I:

C' = I(C’).

C I

C*

This implies that

C 1 C C. Finally, since there, is a great circle through any point of

Sn

in

any given direction, these are all the geodesics. Antipodal points on the sphere have a continium of geodesics of minimal length between them.

All other pairs of points have a unique geo­

desic of minimal length between them, but an infinite family of non-minimal geodesics-, depending on how many times the geodesic goes around the sphere and in which direction it starts. By the same reasoning every meridian line on a surface of revolution is a geodesic. The geodesics on a right circular cylinder

Z

are the generating

lines, the circles cut by planes perpendicular to the generating lines, and

66

II.

the helices on

Z.

PROOF:

If

isometry

I:

The geodesics on in R

2

.

L

is a generating line of

- L -► R

Z

RIEMANNIAN GEOMETRY

2

by rolling

Z

onto

are just the images under

Two points on

Z

Z

then we can set up an 2

R :

I

of the straight lines

have infinitely many geodesics between them.

PART III THE CALCULUS OF VARIATIONS APPLIED TO GEODESICS

§11.The Path Space of a Smooth Manifold. M

Let

be a smooth manifold and let

sarily distinct) points of will be meant a map 1) [0,1 ]

co:

cd(o )

[0,1 ] -► M

o>| [t^_.j,t^]

* p

and

be

two (not neces­

path

from p

to

q

0 = tQ < t1
*

dd x d d 'd t > = Therefore dE(a(u)) du

1 f / da 8a ^ J Xdd'dd > o

d = du

By Lemma 8 . 7 we cansubstitute Choose each strip

=

T\

jA

(-e,e) x [t^_1,t^].

8 / 8a

in this last formula.

so that

Then we can

a

is differentiable on

"integrate by parts" on

The identity

8a v

/ D 8a

8a v

/ 8a D 8a x

dd X d u 'd t > = x d d du'db > + Xdu'dd dd > implies that t. f / D 8a 8a \ J X dd dCLdd > Li-1

t=t. / 8a 8a v < dd'dd >

=

t. f /

"3

*

/y

for

0 = tQ < t1 at H-i

‘

72

III. CALCULUS OF VARIATIONS

Adding up the corresponding formulas for that

= °

for

1 dE(a(u)) I -- Hu

k-1 V / =

• Z
* J < H u ’ HI HI

1=1 Setting

i = l,...,k;

t = 0 or 1, this gives

1

...

> dt

•

o

a = 0, we now obtain the required formula 1

(o)

■

-i < w >Atv > - a t 0

< w ’a >

dt

•

This completes the proof. Intuitively, the first term in the expression for that varying the path decrease

oi in thedirection of decreasing

E; see Diagram

— (o)

"kink,” tends

shows to

11.

Diagram 11 . The second term shows that varying the curve in the direction of its acceleration vector ^

(^)

tends to reduce

E.

Recall that the path oi e ft is called a geodesic oi

is C°° on the whole interval [o,1],

of

ai is identically zero along

if and only if

and the acceleration vector

oi.

COROLLARY 12.3. The path oi is a critical point for the function E if and only if oi is a geodesic.

§1 2 . THE ENERGY OF A PATH PROOF: critical point. f(t)

73

Clearly a geodesic is a critical point. There is a variation of

co with

is positive except that it vanishes at the

This is zero if and only if

=

where

t.. Then 1

< A(t),A(t)

0 A(t) = o

co be a

W(t) = f(t)A(t)

1

- ? H (0)

Let

for all

t.

> dt . Hence each

^|[ti,ti + 1 3

is a geodesic. Now pick a variation such that W(t.) = A. V. Then V' i \ a§(0) = - 2^ . If this is zero then all a^V co is differentiable of class

C1, even at the points

t^.

from the uniqueness theorem for differential equations that everywhere:

thus

co is an unbroken geodesic.

are

o,

and

Now it follows co is

C°°

71*

III.

CALCULUS OF VARIATIONS

§13 . The Hessian of the Energy Function at a Critical Path. Continuing with the analogy developed in the preceding section, we now wish to define a bilinear functional E**: Tnr x Tnr when

7

R

is a critical point of the function

E,

i.e., a geodesic.

bilinear functional will be called the Hessian of E If point

p,

f

at

is a real valued function on a manifold

This

7.

M

with critical

then the Hessian f**:

x T^

can be defined as follows. Given

- R

X1,X2 e TM^

choose a smooth map

(u1,u2) -*• a(u1,u2) defined on a neighborhood of values in M,

(0,0) in R 2, with

so that

“ (0 , 0 )

=

| 2 (0, 0)

P,

=

x,,

=

|g2(°,o)

x2

.

Then f**(X,,X2)

3 2f (a(u, ,u„)) =------2

1

This suggests defining E#*

as follows.

(0, 0)

Given vector fields W 1,W2

e TsU

choose a 2-parameter variation a: where U

is a neighborhood of

a(o,o,t)

(Compare §11 .)

=

7(t),

U x [0,1 ] -► M , (0,0) in R 2, so that

"3 1 ^

=

Then the Hessian

^1

(^),

( ° > 0 >t )

=

W2 ( t )

E**(W.j,W2) will be defined to be the

second partial derivative 8 2E ( 5 ( u 1 ,u 2))

SU1 Su2

where a(u1,u2) € ft denotes the path

(0, 0)

a(u.,,u2)(t)

second derivative will be written briefly as

=a(u1,u2,t) -(0,0)

The following theorem is needed to prove that

E**

. This

. is well defined.

§13.

THE HESSIAN OF

E

75

THEOREM 13. 1 (Second variation formula). Let a: U -*■ft be a 2-parameter variation of the geodesic 7 with variation vector fields wi

-

6 Tv 1 d2F -i

Then the second derivative

1 " 1'2 • (0,0)

of the energy

function is equal to v -2

DW, c < W 2(t),At ^ 1 > -

D 2W. < V8, - 4

J

77 z

+R(V,W1) V > dt

5

dt

0

where

V *^

denotes the velocity vector field and where DW.

DW.

DW.

At “err = ‘H T (t+) ■ *ur(t") DW.

denotes the jump in at one of its finitely many points of discontinuity in the open unit interval. PROOF:

According to 12.2 we have _

1 dE ^du2

-

J

“L ^

t dU > " J

du*

Y “

Zt ^/

8a

1

f 'J

D

da >.

dUg'dUj

At d^b >

1

/ \

D

0

da Dda^-,. du ' db db > dt

1

Let us evaluate this expression for

f ’ J

d

D D da x dt. du.dU d b >

0

(u1 ,u2) = (0,0).

Since

7 = a(o,o) is

an unbroken geodesic, we have

A da

D

A t HI

=

°»

da

HI HI

_ =

0 ’

so that the first and third terms are zero. Rearranging the second term, we obtain

1

n2t

(1 3 -2)

7 HG 1'Iu,2(0’0)

=

Y




* I