Cosmology in (2 + 1) -Dimensions, Cyclic Models, and Deformations of M2,1. (AM-121), Volume 121 9781400882410

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Annals of Mathematics Studies Number 121

Cosmology in (2+l)-Dimensions, Cyclic Models, and Deformations of M2,i by

Victor Guillemin

PRINCETON UNIVERSITY PRESS

PRINCETON, NEW JER SEY 1989

Copyright © 1989 by Princeton University Press ALL RIGHTS RESERVED

The Annals of Mathematics Studies are edited by William Browder, Robert P. Langlands, John Milnor and Elias M. Stein Corresponding editors: Stefan Hildebrandt, H. Blaine Lawson, Louis Nirenberg, and David Yogan

Clothbound editions of Princeton University Press books are printed on acid-free paper, and binding materials are chosen for strength and durability. Paperbacks, while satisfactory for personal collec­ tions, are not usually suitable for library rebinding

Printed in the United States of America by Princeton University Press, 41 William Street Princeton, New Jersey

L ib rary of C ongress C atalogin g-in -Pu blication D ata Guillemin, V., 1937Cosmology in (2-fl)-dimensions, cyclic models and deformations of M 2,l. (Annals of mathematics studies; no. 121) Bibliography: p. 1. Cosmology-Mathematical models. 2. Geometry, Differential. 3. Lorentz transformations. I. Title. II. Series. QB981.G875 1988

523.1’072’4

ISBN 0-691-08513-7 (alk. paper) ISBN 0-691-08514-5 (pbk.)

88-19517

CONTENTS

3

FOREWORD

PART I

A relativistic approach to Zoll phenomena phenomena 16

(§§1-3) PART II

The The general general theory theory of of Zollfrei Zollfrei deformations deformations 27

(§§4-8) PART III

Zollfrei deformations of M2 1 (§§9-13)

53

PART IV

The generalized x-ray transform

98

PART V

The Floquet theory (§§18-19)

(§§14-17)

189

223

BIBLIOGRAPHY

v

Cosmology in (2+1)-Dimensions, Cyclic Models, and Deformations of

Mg ^

FOREWORD

In this paper a "cyclic model" will mean a compact Lorentz manifold with the property that all its null-geodesics are periodic.

Such a model is cyclic in the sense that every

space-time event gets replicated infinitely often; it has an infinite number of antecedents with identical "pasts" and "futures".

We should warn the non-expert that this is not what

relativists usually mean by cyclicity. always used to equations.

In

This term is almost

describe periodic solutions of Einstein’ s (2+1)-dimensions this implies that the metric

involved is conformally flat; and, as we will see in §1 1 , this is practically

incompatible with cyclicity in our sense.

We will call a Lorentz metric all of whose null-geodesics are periodic a Zollfrei metric. term, see §1 .)

(For the etymology of this

Notice that the property of being Zollfrei is

conformally invariant.

This is because two Lorentz metrics

have the same null-geodesics if they differ by a conformality factor.

(Another way of stating this fact is that the trajec­

tories of light rays are independent of the metric structure of space-time but only depend on its causal structure: i.e., the specification of the future of every space-time event.)

3

FOREWORD

4

The Zollfrei problem is interesting even in dimension 2 ; in fact, as a warm-up for the problem in dimension 3 we will briefly describe what happens in dimension 2 :

Theorem.

Let

0

can

be the standard Zollfrei metric on

S^xS^; i.e., the metric, d0 ^ d0 2 > where

0^

and

are the

02

standard angle variables on the first and second factors. (X,#)

be any oriented Zollfrei two-fold.

covering map

1 : X -> S^xS^

such that

Let

Then there exists a can

and

g

are

conformally equivalent

Proof: First of all notice that every oriented compact Lorentzian two-fold has to be diffeomorphic to its Euler characteristic is zero.

1

S xS

Now suppose that

1

since X

is a

compact Lorentzian two-fold all of whose null-geodesics are periodic. Tp

The null-cone at

p e X

consists of two lines in

(See figure.) ?2

so the conformal geometry of

X

is completely described by a

pair of tranverse line element fields.

Let

vector fields defining these line element fields. tion the integral curves of

v^

and

^

are aH

and

v^

be

By assump­ closed.

5

FOREWORD

Choose an oriented curve, curve of

v^

7p

which intersects each integral

transversally.

[41], page 9.)

Let

the trajectory of

p v^

(This is always possible.

be the intersection number of through

x.

See 7^

with

It is clear that this

number is independent of

x.

trajectoryVLn exactly

points since the orientation numbers

p

Thus

7^

has to intersect each

at the points of intersect have to be all of the same sign. (See figure.)

Suppose in particular that next point at which The map

x

is on 7 ^.

Let

the trajectory through

f:

7 ^ -> 7 ^ which sends

morphism of

7 ^, and the points,

x

to

group,

7 ^.

Z ,

Thus on

7 ^.

is a diffeo2 T) 1

f (x),...,f^

are the distinct points where the trajectory through sects

be the

x intersects

f(x)

x, f(x),

f(x)

(x),

x inter­

f defines a free action of the finite cyclic 7 ^. In particular there exists a covering map

7l —

whose fibers are the

Z

p

orbits.

by associating to the point

x e X

S1

Now extend the

6, to Y1

all of

X

orbit in which the

FOREWORD

6

trajectory through curves of

^

x

intersects

integral curves of

^2

exp tE^ .

(This action is well-defined since when

exp

is the identity

2 j = t.)

Suppose now that we are given a function,

H, on

T*M

which is quadratic on each cotangent fiber and satisfies (7.6). Consider the equation (7.5).

A putative solution of this

equation is given by the variation of constants formula

p2 T r rt

(7.7)

JO U

(exp s Ep*Hds dt.

q

(It is easy to check that this is a solution of (7.5) if and only if G

on

H

satisfies (7.6).)

^ 2 x ; however, G

The formula,

can be

(7.7),

extended to all of E

requiring it to be homogeneous of degree one. extends

G

only defines by

Finally if one

arbitrarily to be a homogeneous function of degree

one on all of

T*M-0, and sets

(7.8)

then (7.4) will be identically satisfied on all of

T*M-0.

PART II

42

Let us now examine the problem of extending the infinitesmal deformation, H, (or

of

(dcr)

o

neighborhood”of 0 in the parameter space,

to the "n-th formal -e < s < e. By such

an extension we will mean the following three pieces of data:

1)

A canonical transformation,

♦ , of

T*M-0

onto itself.

2)

A smooth homogeneous function of degree zero,

F , on

T*M-0. 3)

A smooth function,

H , globally defined on s

T*M

and

quadratic on each cotangent fiber.

These data are required to depend smoothly on appropriate initial values (i.e. and

s, take on the

F q = 1, H q = H, (dHg/ds)Q = H

= identity) and, most important of all, to satisfy the

n-th order deformation equation:

(7.9)

rf*Hg = F gH 0 modulo 0(sn+1).

For instance, for n = l , (7.9) is equivalent to (7.4) if we set:

(7.10)

ij) = exp s Er , S \J

F

o

= 1 + sF ,

H

Suppose now that we have succeeded in extending (n-l)st formal neighborhood of

s = 0

o

= H + sfl .

H

to the

in deformation space,

i.e. suppose we have produced a canonical transformations,

¥ , o

43

ZOLLFREI DEFORMATIONS (§7)

and functions

F

s

and

H

s

with the properties described above

such that (7.9) is satisfied modulo

0(sn ).

(7.11)

+ 0(sn+1).

( « y * Hs = F SH + s \

Then we can write

Let us tryJ to find smooth functions,7 Gn 7.

n

T*M-0, such that, on each cotangent fiber,

Gn is homogeneous

of degree one,

Fn

and

homogeneous of degree zero and

Hn

Hn

on

a

homogeneous quadratic polynomial and such that (7.9) is satisfied with

¥,

F

and

H

replaced by

$ o (exp su E q ) , n

(7.12)

F* = F + snF

, n 5

H 1 = H + snH

n

.

It is easy to see that, because of (7.11), (7.9) reduces to

(7.13)

Let

7

be a null-bicharacteristic of

integrating (7.13) over integrability conditions

7

we get for

H

of period Rn

2x.

the n-th order

Then

44

PART II

(7.14)

Hn (7(t)dt - - Jo Rn (7 (t))dt. 0

Conversely suppose that we can find an holds for all

7.

Then we can define

Hn Gn

such that (7.14) as before (see

(7.7)) by the "variation of constants formula" on

^

(7.15)

'2tc 0

rt dt y

exP sHP * ( Hn + Rn)ds

and then, in the same way as before, extend it to all of T*M-0.

Finally letting

we get (7.13) to hold on all of

T*M-0.

The solvability of (7.14) can be formulated in terms of an integral operator which we will study in more detail in part 4.

Let

bundle of

S (T) M

be the symmetric tensor product of the tangent with itself and let

smooth global sections of

S^(T).

can be regarded as a function on each cotangent fiber.

C°°(S^(T))

be the space of

An element of T*M

C°°(S^(T))

which is quadratic on

By restricting this function to

we

get a C00 function on E 2 R . This procedure defines for us a map

(7.17)

C°°(S2 (T))

c“(s2x).

45

ZOLLFREI DEFORMATIONS (§7)

On the other hand, associated with the principal fibration, t

:

(see (5*1)) 9 Lhere is an operation of fiber

integration

(7.18)

C°°(S2x) -

C°°(P),

and, composing (7.17) and (7.18), we get a "generalized x-ray transform"

(7.19)

Rff:

C°°(S2 (T)) — * C°°(P).

Coming back to (7.14), as one varies

7 e P

the right

hand side of (7.14) varies smoothly, so we can regard the right hand side of (7.14) as defining a smooth function, C°°(P). section,

Solving (7.14) for all hn , of

9

S (T)

(7.20) v J

amounts to finding a smooth

7

such that

R h = (r ) S^(Vp/W) =

n ® \l one gets a map

PART III

58

Combining these two maps one can convert element of

^ (V^) ® (A^[V ]*)^)

one-dimensional space tion by varying

p

f(p)

(which is an

into f (p ) , an element of the

(V^/W) . If we globalize this construc­

in the set (9.6) we obtain a homogeneous

function of degree - 2 , f , on the two-dimensional vector space I1 (with origin deleted) (W^/W)-{0}. Notice, however, that W±/W comes equipped with a symplectic form, two dimensions,

and since we are in

u/y can be thought of as a volume form.

Therefore we can form the residue of

f

with respect to this

volume form to obtain a numerical quantity:

R estf^y)

(9.10)

=

R (f,/i)y .

(See the appendix at the end of this section.)

It is clear

from (9.9) that this quantity depends quadratically on Now let the point in space,

V, of

L RP

be the canonical line bundle of 3

RP

q

//. and

q

corresponding to the one-dimensional sub­

R^.

The fiber of

L

at

q

is

W; so the

quadratic form

// is an element of letting

q

1 R(f ,//)y, 2

// E W,

(L*)^* Globalizing this construction by 3 vary in RP , we obtain from f a global section,

ZOLLFREI DEFORMATIONS OF

M2 1 (§9)

59

9 (L*) . Summarizing we have exhibited the existence of

Rf, of

a transform

R: r ( S 4 ( V ) » (A2 [V*])2)

(9 .1 1 )

r(L * ) 2

which integrates data on the right hand side over the nullgeodesics of

^

1 ‘ ^ i s is, of course, just the x-ray

transform, (7.19), in disguise; the description we have just given of it displays clearly its

Sp( 2 ,R)-invariant character.

In part one we showed that an infinitesmal conformal deformation,

f, of

M2 ^

which corresponds to a deformation

of "Zollfrei" type has to satisfy the integrability condition: Rf = 0.

Ve also proved the converse (modulo some questions in

analysis which we've deferred to part four.)

It is clear from

the results in part one that the trivial deformations (the image of

in (9.2)) satisfy this condition.

k

else satisfies this condition?

However, what

We will give a definitive

answer to this question in the next section, but, for the moment, we will show that there are other elements in the kernel of

R

besides those in the image of

k

.

In keeping with the notation which we used in §2 we will denote by

the complexification of

C-linear extension of

v

to

, and let

which is homogeneous of degree

the vector field on

V

-n.

f Ve

associated with the

one-parameter group of homotheties

(9.13)

t— » e^ Identity.

Consider the

(n-l)-form

//f =

on

V- 0 .

(9.13), of field,

It is clearly invariant with respect to the action, (R.

Moreover, its interior product with the vector

5, is zero since

l

( E ) hf = t ( E )

l { E ) fu)

= 0.

PART III

62

Therefore, it is basic with respect to the fibration

V-0

that is, there exists an that

x*z/_p = /rp.

integral of see [12 ]).

Proj(V);

(n-l)-form,

One defines the residue, over

Proj(V).

on

Proj(V)

Res(f,w)

such to be the

(For a more detailed account,

ZOLLFREI DEFORMATIONS OF

§10.

63

M 2 1 (§10)

In this section we will show how to deduce almost all

the results which will be needed about the transform, R, from the fact that

R

is an

Sp( 2 ,IR)

invariant object.

Unfor­

tunately, in the course of the next few paragraphs we will need to cite a number of results in representation theory which are very technical (even to state). An extremely good reference for this material is Knapp:

Representation theory of semi­

simple Lie groups, an overview based on examples (Princeton U. Press, 1986). Let

G

be a connected semisimple Lie group and

parabolic subgroup of X, of by

P

E

G.

P

a

Given an irreducible representation,

on a finite dimensional vector space, we will denote the vector bundle over the coset

space,

M = G/P, in-

A

duced from sections

and by

Ind^

the representation of

of this vector bundle.

Let

is

G.

on

g be theLie algebra of

UQz)the universal enveloping algebra, and compact subgroup of

G

K a maximal

By definition a smooth section of

E

X

K-finite if there exists a finite-dimensional K-invariant

subspace of of all

containing it.

Let

space

K-finite sections of

generated to the

^(E^)

E . This space is a finitelyA K-U( 2 ) module, and is dense in T(E ) with respect

C00 topology. The first fact that we will need about these objects is

the following.

PART III

64

Theorem. T(E^)q finitely-generated

is an "Artinian" object in the category of K-U(£)

modules: i.e., there exists a

maximal chain

{ 0}

of

K—U(j2 )

cMjCHjC

submodules.

irreducible quotients,

•••

C

M r r(Ep c

0

This chain is not unique, but the | ^ , i = l,...,r+l

(and the

multiplicities with which they occur) are unique.

Proof: See Knapp, page 373, Corollary 10.39.

Thanks to some (very deep) recent developments in the theory of Verma modules, one can considerably strengthen this result: one can, in principle, write down formulas for the multiplicities of these irreducible quotients (see [44]).

The explicit de­

tails have only been carried out for a few groups.

Fortu­

nately, however, one of the groups for which this has been done is

SP(2,(R)

(loc. cit., pages 253— 255).

In fact, in some

unpublished work, Luis Casian has pushed these computations one step further.

For

Sp(2,DR)

he has actually computed the lat­

tice structure of the lattice of

K-U(^)

submodules of

T(E^) q . It turns out that even though an infinite number of representations are involved, a very small number of actual

ZOLLFREI DEFORMATIONS OF

lattice configurations occur.

*

M2 x (§10)

65

Ve will, henceforth, refer to

these as the Casian diagrams of the induced representations of S p (2,IR). We will now describe the Casian diagrams of the representations associated with Let Wq

Vq

R:

be a fixed Lagrangian subspace of

be a fixed one-dimensional subspace of

maximal parabolics,

and

stabilizers of

Vq

V q and

with them are transform,

M2 ^

R,

and

[R^ and let

V q . The two

0f Sp(2 ,CR)

are the

and the coset spaces associated 3

1RP . The domain of the integral

is,by (9.5), the space of sections of the

induced bundle,

E

, associated with the standard reprex\

sentation,

ofP^

on the space

S 4 (V0) ® (A2 [V0]*)2 .

The image of

R

is, by (9.9), contained in the space of

sections of the induced bundle,

E

, associated with the *2

standard representation,

*2 , of

P2

on the space,

9

(Wq) .

The Casian diagrams for these representations are:

*

Casian’ s results are not yet published; however, results similar to his (though not quite as definitive) can be found in [28] and [44].

66 1.

PART III

For

Xi

the diagram

Figure 1

The slashed arrow indicates that the module represented by the lower dot is of finite codimension in the module represented by the upper dot.)

2.

For

Figure 2

the diagram

A

ZOLLFREI DEFORMATIONS OF

(i.e., the module associated with

\ r ( s 2 ( v * ) ) -*-» r ( s 4 (v ) ® ( a 2 [ v ] * ) 2 ) r(s*(V) « a 2 [V]) M

r(s 2 (v) ® (a2 [V] )3 ) — . o .

By the Gasqui-Goldschmidt theorem, this complex computes the cohomology of

M2

vector fields.

with values in the sheaf of conformal

However, for

M2 ^

this sheaf is the constant

sheaf, sp(2 ,R ). Therefore, we can immediately write down its cohomology groups:

Proposition 11.3

The cohomology groups of (11.6) are just the

DeRham cohomology groups of H° = H 1 = sp(2,R)

and

M

tensored by

H 2 = H3 = 0.

sp(2,IR); i.e.,

ZOLLFREI DEFORMATIONS OF

73

M2 j (§11)

Now let us return to the Casian diagram associated with \y

Notice that the lattice depicted in this diagram is gen­

erated by the entries which we've labelled A, B, C, D and D; so we only have to identify these entries with subrepresentations of

indg^.

By experimenting with various possibilities it is

easy to convince oneself that there is only one way to choose A, B,

C, D and D so as to obtain the configuration infigure

one.

As we have pointed out

already, A has to be thepre-image

of the unique proper subrepresentation of and C, B has to be the kernel of

a

and

indg^C

As for B

the image of

k

.

(In view of Proposition 11.3, this accounts for the slashed arrow joining B to C.) ary data over

Finally D and D have to be the (bound­

of) holomorphic and

M2 r

anti-holomorphic sections of (9.5)

It is clear, in

fact, that these choicesare

consistent with figure one; and it is not hard to see that one can only obtain this figure by positioning A, B, C, D and D as we've indicated. Notice by the way that if we truncate Casian's lattice by throwing away everything below

A^g

we obtain the lattice for

which is consistent with Theorem 10.1 since sponds to the kernel of Abis

R.

A ^ g corre­

By throwing away everything above

we obtain the lattice structure of the kernel of

itself.

In particular we obtain

R

PART III

74

Proposition 11.4. of

The kernel of

R

is spanned by the image

and by the (distributional) boundary data of holomorphic

k

and anti-holomorphic sections of (9.5) over

Mg

Ve have already pointed out in section nine that the space of infinitesmal conformal deformations of

M2 ^

is not,

properly speaking, the space of sections of (9.5), but rather the quotient of this space by the image of

(11.7)

def(M2>1) = r(S 4 (V) «

k

\ i.e.,

A 2 [V*])2 ) A ( r ( S 2 (V*)).

Ve obtain the lattice structure of this space by throwing away everything in figure one below the position C.

In particular

we obtain:

Proposition: The space,

is the Grassmannian of two-

82

PART III

dimensional Lagrangian subspaces of

0 . This Grassmannian is a

four-dimensional homogeneous complex domain

(with SU( 2 ,2 )

its group of automorphisms); and itturns out that exactly the same relation to

^

namely its Shilov boundary is denote this domain by

p

it bears

thatdoes to

p

as

M2

^:

Henceforth, we will

Notice by the way that there is a

natural imbedding:

(12.5)

l:

Living on

^

M2 j

r

are the "free mass-zero spin-k/2 parti­

cles" of elementary particle physics, which are, by definition, certain irreducible representations of

SU( 2 ,2 ).

Ve will

briefly describe how these representations are defined: be the "tautology bundle" of as for

p

p

namely its fiber at

subspace of

represented by

definition that

V

bundle,

Let

(This is defined exactly p

p.)

is the two-dimensional It is clear from this

is a vector subbundle of the trivial

Let V be the quotient bundle,

out that for all integers,

C^/V.

k > 0, there is an

invariant first order differential operator,

It turns

SU(2,2)d

whose domain

is the space of holomorphic sections of the vector bundle

V

ZOLLFREI DEFORMATIONS OF

(12.6)

83

M2 1 (§12)

Sk (V) ® A2 [V],

and whose range is the space of holomorphic sections of the vector bundle

(12.7)

Sk_ 1 (V) ® (A2 [V] ) 2 » W*.

(For instance for

k = 1,

dk

is the usual Dirac operator,

is sometimes called the "spin— k/2M Dirac operator).

The "free

mass— zero spin— k /2 fields" are defined to be the holomorphic sections,

s, of (1 2 .6 ) which satisfy

(1 2 .8 )

dks = 0 .

(See [21 ], page 85, or [7].) Notice that except for

k = 1

the fiber dimension of

(12.6) is less than the fiber dimension of (12.7); so the equa­ tion (12.8) is over-determined.

Therefore, if

characteristic hypersurface in

^

s = s^

symplectic structure on to

on

M,

s^

has

We will describe this

constraint equation for the imbedding (12.5): there is an isomorphism of bundles

is a non­

and one wants to solve

(12.8) with pre-assigned initial data, to satisfy a "constraint equation."

M

V ^ W*

Over

given by the

so the bundle (12.7) is isomorphic

S^_^(V) ® (A^[V])^ ® V, which is also isomorphic to

84

PART III

(12.9)

Sk V )

® (A2 [V]) 3 ® V* .

By contracting the first and third terms in this tensor product one gets a morphism,

7 , of the bundle, (12.9), onto the bundle

(1 2 .10 )

Sk' 2 (V) ® (A2 [V])3 .

It is not hard to show that at every 7

is the image of the symbol map,

vector,

£, to

S^(V) ® A^[V] section of Mg ^

^

at

p.

p e

containing

[V]

Therefore, if

s

U, of

is a section of ^

and s is a

defined on some open subset,

U, and equal to

(1 2 .1 1 )

the kernel of

^(^)(^), for the conormal

defined on some subset,

S^(V) ®

^

s

on

U, of

U, the expression

7 (flk§|U)

is independent of the

choice of

Hence (1 2 .11 ) defines

a first-order differential operator,

(12.12)

s

fljj: T(Sk (V) ® A 2 [V]) —

(i.e., depends only on s).

T(Sk'2 (V) ® (A2 [V])3) ,

which is just the "tangential component" of ular, the constraint equation for

(12.13)

1

is

9 (r); so this question can be rephrased. onto the symplectic orthocomplement of

to

S

is

has a one-dimen­ map

bijectively onto the symplectic orthocomplement of W?"

p

Suppose that x and

are of corank one, and that the restriction of

injective and the restriction of

and

"Does p{r)?"

and p

(ker x)flS p(S)

in

/?(S) = map

ker x

This question,

however, is just a rephrasing of the conclusion of lemma 14.2. q .e .d

.

Remarks. 1.

The results we've just proved

have very simple singularities along s e S near

and

p

S, i.e. for every point,

there are simple local canonical s.

show that x

forms for x

and

p

The analytic details in the last two sections of part

PART IV

108

four could be enormously simplified if one could show that there is a "local symplectic canonical form" for the diagram (14.14) itself analogous to the canonical form for the "folding canonical relation" described in [20], XXI.

It turns out that

there is such a canonical form, but it requires an additional hypothesis on

a:

Replacing the image of

T

in

M

by M

itself we can, without loss of generality, assume that surjective.

Then

i

maps T-S

diffeomorphically onto

so there is a symplectic involution of sponding to

i = k+ 1 ,...,n

and

we obtain the canonical form described above.

Q.E.D.

THE GENERALIZED X-RAY TRANSFORM (§14)

113

Blowing-up is based on a global variant of this canonical form theorem.

Let

f : X -> Y

be a mapping which satisfies II

a)-c) and also satisfies the following global topological hypotheses: d)

f

is proper.

e)

The image,

W, of

dimension

(ii) f)

f: S -> V

(k-1 )

S

in

Y

is an imbedded

submanifold of

Y.

is a fiber mapping with connected

fibers.

(Apropos of condition e) the conditions II a)-c) imply that is an immersed dimension

(k-1 )

tion e) simply says that

V

condition f), since

f: S -» V

submanifold of

V

Y; so condi­

has no self— intersections.

As for

is a proper submersion, the Thom

isotopy theorem implies that it is a fibration.

Therefore f)

simply says that the typical fiber is connected.) Let

IPN be the projectivized normal bundle of

property II b ) , the differential of a lift of

(14.18)

f

to

f

along

S

V.

By

gives rise to

PN,

S

PN

and, from the canonical form theorem above, one can easily see that

/?£

is locally a diffeomorphism.

PART IV

114

However, since

f

image has to be all of covering. W

are

isproper, f}^ IPN, and ^

has to be

In fact even more is true:

IRPn_^ ,s, so if

n-k > 1 , ^

feomorphism or a double covering.

is also proper; so its a f inite-to-one

the fibers of

IPN

over

has to be either a difIn particular we have

proved:

Theorem. If

n-k > 1 , the fibers of

S

over

W

are either

(n-k)-dimensional spheres or (n-k)-dimensional real projective spaces.

Definition. If the first alternative occurs, blowing-up of occurs,

f

Y

along

f

is spherical

V, and if the second alternative

is a projective blowing-up of

Y

along

V.

There is a simple canonical form for each of these cases. Without loss of generality we can assume that

Y

is the total

space of a vector bundle E -» W and that

W

sits in

Y

the projectivization of triples, ment of

(p,o;,v), where IP(E )

and

sional subspace of which maps

(p,o;,v)

v E^ to

as the zero section.

Let

E,and let X

set of all

be the

p is a point of

W,

w

S = IP(E),

is an ele­

is a vector lying in the one-dimen­ corresponding to w. (p,v)

The map, 7 : X -* Y,

is a projective blowing-up of

THE GENERALIZED X-RAY TRANSFORM (§14)

Y

along

W; and it is easy to see that every projective

blowing-up of hood of

115

V.

Y

along

W

is isomorphic to

7

in a neighbor­

If we replace the projective-space bundle,

by the ray bundle,

IP(E),

S(E), we get the corresponding canonical

form for the spherical blowing-up.

PART IV

116

§15.

Before turning to the estimates, (8.12), we will describe

again some of the ingredients that go into the formulation of these estimates:

To begin with, a causal structure on a com­

pact three-manifold,

M, is a rule which assigns to each

m e M

a convex cone Cm c T . m m

One can always find a function, erty that, for all

H, on

T*M-0

with the prop­

m, the dual cone

C* c T* m m is just the set {t 6 T*, H(m,£) > 0} .

Of course,

H

is not unique; however, the solutions of the

Hamilton-Jacobi equations

for which

H(x,£) = 0

to parameterization).

are the same for all choices of A null-geodesic is a curve on

lifts to a null-solution curve of (15.1), and

M

H M

(up which

has the

"Wiederkehr property" if all its null— geodesics are periodic. (This means, by the conventions in force in this article, that

117

THE GENERALIZED X-RAY TRANSFORM (§15)

the lifted curve is periodic and that behavior of the type depicted in the figure below is ruled out.)

(spiraling of null-geodesics about an exceptional null-geodesic in

T*M.)

In particular, for such

M, the set of null-

geodesics is itself a compact three-manifold. will denote this manifold by For

m 6 M

let

(i.e. an element

r E

Z

m

As in §5, we

P.

be the set of null-rays in J is a ray,

T

m

{tv, t > 0 }, with

v e