1986_arnold first steps in symplectic topology

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First steps in symplectic topology

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1986 Russ. Math. Surv. 41 1

(http://iopscience.iop.org/0036-0279/41/6/R01)

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Uspekhi Mat. Nauk 41:6 (1986), 3- 18 Russian Math. Surveys 41 :6 (198 6), 1-21


(1)

V . I .

Arnol d

CONTENTS

Introduction 1

1. Is there such a thing as symp lectic topology? 2

2. Generalizations of the geom etric theorem of Poincare 4

3.

Hyperbolic Morse theory 5

4. Intersections of Lagrangian manifolds 7

5.

Legendrian submanifolds of contac t man ifolds 9

6. Lagrangian and Legendrian kn ots 11

7.

Two theorems of Given tal' on Lagrangian embeddings 13

8. Odd-dimensional analogues 15

9. Optical Lagrangian manifolds 16

References 19

Introduction

By symplectic topology I mean the discipline having the same relation to

ordinary topology as the th eor y of Ham iltonian dyn am ical system s has to

the general theory of dyn am ical systems. The co rrespond ence here is similar

to that between real and complex geometry.

A complex linear space can be considered as an even-dimensional real

space, furnished with additional structure (the operation of multiplication

by /'). How ever, com plexification of a the ory does no t boil dow n t o

reducing the pile of spaces and th e additio n of a new op era tio n: all th e

concepts take on new meanings. Fo r examp le, complex subspaces or

operators are not the same as subspaces or operators in the underlying real

space. Th us complex geom etry is an analogue of real geo m etry, bu t no t a

particular case of it.

In precisely the same way, symplectic geometry can be considered, of

course, as ordinary geom etry in the presence of additional structu re. But it

(1 )

The papers of V.I. Arnol'd, A.N. Dranishnikov, E.V. Shchepin, and V.V. Fedorchuk

brought together in this issue are reports of plenary sessions held on 27-29 May 1986 at

the Aleksandrov Colloquium (jointly w ith a session of the Moscow M athematical

Society and the All-Moscow topological seminar in the name of P.S. Aleksandrov (the

topological association)), dedicated to the 90th anniversary of Aleksandrov's birth

{Editor's note).


3/22

2

V.I. Arnol'd

is possible to adopt another point of view in which symplectic geometry can

be considered rather as an analogue of ordinary geometry in its own right.

For example, the symplectic group may be considered not as a subgroup of

a group of matrices of even order, but as the simple Lie group C

k

, having

equal rights with the group of non-singular matrices

A

k

,

not least in having a

distinctive system of roots and so on.

Questions of symplectic topology, which we shall be speaking about later,

can be considered as questions of ordinary topology in the presence of

add itiona l structu re. But of much gre ater interest to me is no t the use of

ordinary topology in the study of objects of symplectic geometry, but

divining sym plectic results by me ans of sym ple ctizatio n .

Symplectization transforms not only the initial objects (manifolds,

m aps , . . .) , bu t also the whole theo ry . F or exam ple, the concep ts of

boundary and homology theory in symplectic topology are quite different

from the ordinary ones. The dimension of a symp lectic bo un da ry should

not be one, but two units less than the dimension of the original manifold

(lowering dimension in symplectic geometry is always accomplished in two

stages, one of which is section and the other projection).

I do not intend here to formalize these nebulous ideas^

1

^, but pass to

specific conjectures which they give rise to (omitting rather lengthy

intermediate considerations).

Some of the conjectures of this type published in the years 1965-1976

([9]-[12]) have recently been proved by Conley, Zehnder, Sicorav,

G rom ov, and oth ers, and powerful new techniq ues have been developed. It

seems to me that now is the time to return to other conjectures of this type

and even perhaps to look at the whole programme of symplectization.

Odd-dimensional variants (related to contact topology) are also considered

below.

The author thanks A.V. Alekseev, M.L. Byaly, Yu.V. Chekanov,

Ya.M. Eliashberg, D.B. Fuks, V.L. Ginzburg, A.V. Givental ' , V.P. Kolokol'tsov,

V.V. Kozlov, V.P. Maslov, S.P. Novikov, J. Nye, L.V. Polterovich,

E.V. Shchepin, A.I. Shnirel 'man, and V.A. Vasil 'ev for numerous useful

discussions.

1

Is there such a thing as symplectic topology?

A

symplectic structure

on a ma nifold is a closed non-d egene rate 2-form.

Th e simplest example of a sym plectic manifold is the plane; the (o riented)

element of area provides the symplectic structure.

A symplectic diffeomorphism (o rsymplectomorphism) is a diffeom orphism

preserving the symp lectic structu re. It is clear that this condition pu ts a

regards symplectic boundaries see the theory of Lagrangian cobordism in [1] - [ 7 ] ;

the complexification of the concept of boundary is a branching divisor, Z

2

is replaced by

Z,

Stiefel-Whitney classes by Chern classes, and so on (see [8]).


4/22

*

First steps in sym plectic topology

topological limitation on the diffeomo rphism. F or exam ple, a region of

finite volume cannot be transform ed by a sym plectic diffeomorphism in to

one lying strictly inside it, since a sym plec tic diffeom orphism sends a ny

region into one of the same volume.

A richer example is the geom etrical th eo re m of Poincare (proved by

Birkhoff) according to which an area-preserving diffeomorphism of an

annulus that moves the two bounding circles in opposite directions has no

fewer than two fixed points. I m ention also the theorem of Nikishin [ 1 3 ] :

a diffeomorphism of a two-dimensional sphere preserving oriented area has

no fewer than two fixed points.

Thus the topological properties of general diffeomorphisms and volume

(area) preserving diffeom orphism s are different. Bu t d o sym plec tic

diffeomorphisms have specific properties distinguishing them from those

preserving volume? This qu est ion has been form alized b y Eliashberg in the

following way.

Can every volume-preserving diffeomorphism of a symplectic man ifold of

dimension greater than two be approxima ted topologically (C) by a

symplectic diffeomorphism

1

.

If this were so, then the stable topological properties of volume-preserving

diffeomorphisms would be the same as for symplectic ones.

Recently Grom ov has proved the existence theorem of symplectic

topology , also formulated by Eliashberg.

Theorem [ 1 4 ] . If the limit of a uniformly (C) converging sequence of

symplectomorphisms is a diffeomorphism, then it is sym plectic.

A typical question, showing how symplectic geometry differs from the

geometry of volume-preserving diffeomorphisms, is the following problem,

also investigated by Eliashberg and Gro m ov : can a sym plectic cam el go

through the eye of a needle?

What is meant by this is the following: can one by a symplectic isotopy

move a ball, lying in the left half-space of a four-dimensional standard

symplectic subspace, into the right half-space through an arbitrarily sm all

hole in the plane separating the tw o subspacesl

In the class of volume-preserving diffeomorphisms such an isotopy clearly

exists. Grom ov has proved, however, that there is no symplectic isoto py : a

symplectic camel is prevented from going thr ou gh a small gap by sym ple ctic

ribs a special non-linear analogue of the inequalities of Rayleigh-Fisher-

Courant.

Of the unsolved problems of the geom etry of symp lectic diffeomo rphisms

I mention also the following: is the diameter of the group of symplectic

diffeomorphisms of a ball with left-invariant standard metric bounded

Shnirel 'man has proved the boundedness of the diameter of the group of

volume-preserving diffeomorphisms of the three-dimensional ball [45].

Somewhat esoteric symplectic diffeomorphisms are those to which one is led

from positively twisting Hamiltonians (see [31], [33]).


5/22

4 V.I.Arnol d

2 .

Generalizations

of th

geometr ic theorem

ofPoincare

Consider a symplectic (area preserving) diffeomorphism of a two

dimensional torus to

itself,

homotopic to the identity transformation. Such

a diffeomorphism is said to be

homologousto the identity

(or

preserving the

centreof

gravity) if it isgivenby a map of the plane, covering the torus,

> + /(x), for which the mean value of the periodic vector function / is

equal to zero.

A diffeomorphism homologous to the identity can be joined to the

identity by a one parameter family of symplectic diffeomorphisms in such a

way that the derivative with respect to the parameter is a Hamiltonian vector

field

with singlevalued Hamiltonian, and, conversely, all symplectomorphisms

joining such a map to the identity are homologous to the identity.

Diffeomorphisms homologous to the identity form a group (the commutator

of the connected component of unity in the group of all symplectic

diffeomorphisms see [15]).

Poincare's geometric theorem (on the diffeomorphisms of an annulus)

generalizes in the following way (the conjecture of [9], [10], proved in

Theorem.

A

symplectomorphism

of a

torus homologous

to the

identity

has

nofewer than

four fixed

points

{taking

multiplicities

into

account) and no

fewer thanthree geometrically

distinct fixed

points.

Poincare's theorem on transformations of an annulus can be obtained

from this asfollows: a torus can be glued together from two symmetrically

placed annuli joined by intermediate annuli which are then moved along

themselves in opposite directions. By regulating the width of the intermediate

annuli one can arrange that the resulting diffeomorphism of the torus

preserves the centre of

gravity.

Half of its

fixed

points (that is, no

fewer

t han two of thefixed points) are to be found in the original annulus.

The following multidimensional generalization of Poincare's theorem was

formulated in [11], [12]:

Conjecture.A symplectomorphism of acompactmanifold,

homologous

to

the identity

transformation^,

has at leastas many

fixed

pointsas asmooth

function on the

manifold

hascritical points.

(One has in mind in both cases either the number of geometrically

distinct points or the algebraic sum of the multiplicities.)

conjectured direct generalization of the theorem on the annulus was stated in [10]

( t h e condition of moving the boundaries in opposite directions is transformed into the

linking of the sphere S*

1

the boundary of a disc in the fibre of

T*M

n

with its image

in the universal covering of the sphere bundle T I M ) . This, it seems, has not been

proved.

^Joined

by a one parameter family of symplectomorphisms with singlevalued (but time

dependent)

Hflmiltonians.


6/22


5

This conjecture has been proved in the following cases:

1) for two-dimensional surfaces

( [ 1 7 ] ,

[ 1 8 ] ) ;

2) for the 2-dimensional torus with the standard symplectic structure

( [ 1 9 ] ,

[2 0 ] , [ 2 1 ] ) ;

3) for CP with standard structure [ 2 2 ] ;

4) for many Kahler manifolds of negative curvature

( [ 1 8 ] ,

[ 2 3 ] ) ;

5) for diffeomorphisms C-close to the identity [24].

Estimates, established by the conjecture, have been obtained for

diffeomorphisms enclose to the identity transformation (obtained by

integrating the Ham iltonian field over a small time interval). In th e sim plest

cases the minimal numbers of critical points are the following:

Manifold

Circle

Two-dimensional torus

Surface of genusg

2rt-dimensional torus

CP

2

4

2 ? + 2

2

2 n

n + 1

2

3

3

2 n + l

For symplectic maps not homologous to the identity (but lying in the

same connected component of the group of symplectomorphisms) the

number of fixed points, clearly, is bounded below by the number of

critical points of a closed

1-form

on the manifold (the inequalities of Morse

theory are replaced by the Novikov inequalities [25], [26]).

3

Hyperbolic Morse theory

The new technique, on which the proofs of the stated results are based, is

a fresh version of variational Morse theory for functionals, unbounded on

either side. The basis of the ordina ry Morse the ory of positive fun ction als is

the reduction of the problem to the finite-dimensional one by taking

account of the fact that the functional increases rapidly in the directions of

the high harm onics (of the Fo urier series): it appro aches infinity in

these direction s like a positive-definite q ua dratic form. Fixing the finite-

dimensional inform ation (th e values of th e coefficients of th e lower

harmonics ), we stratify the functional space into subspaces of finite

codimension. The restriction of a functional to such a subspace has a

unique minimum point, smoothly dependent on the point of the (finite-

dimensional) basis. The further choice of actual extrem als from the

manifold of conditional extremals found is already a finite-dimensional

problem, solved by finite-dimensional Morse thoery, that is, by studying

functions on a finite-dimensional m anifold. This manifold of co nditio nal

extremals can be obtained by gradient descent along the subspaces of finite

codimension described above.


7/22

6 V.I.ArnoVd

Thus at the

base

of variational Morse theory there

lies

the structural

stability of linearly attract ing equilibrium positions. The basic idea of the

new methods consists in the fact that instead of attracting equilibrium

positions one may use hyperbolic ones, which are also structurally stable

(theorems

of Grobman H artm an, Anosov).

Hyperbolicity here has to be taken in the sense of the theory of dynamical

systems

and not in the sense of the theory of partial differential equations:

t h e corresponding quadratic form has an infinite number of squares, as many

with plus as with minus.

I n applying these methods it is necessary that the functional under study

has the hyperbolicity property (on submanifolds of finite codimension

obtained by fixing the coefficients of the lower harmon ics ). An example

is the action functional

\p dq~ dt.

Consider for simplicity the case of

o n e degree of freedom, when the functional is determined on curves

[0, 1] vR

2

= {(/>,

q)}.

On loops of curves, traversed in the positive

direction, the integral of

pdq

is positive, but in the negative direction it is

negative. If =

p + iq,

t h e n e

iht

is traversed in the one direction for

positive

and in the other for negative

k.

I n th is way high frequency hyperbolicity arises: the harmonics for large

\k\

form spaces where the action functional grows or shrinks quadratically

(depending on the

sign

of

k).

The integral of

Hdt,

playing the role of

perturbation, does not invalidate this.

Suppose th at is a periodic function of

p, q,

and

t,

and consider our

functional on closed curves. After reducing the problem to the finite

dimensional case one obtains a function on th e space of the finite dimensional

vector bundle over the manifold under study, tending to infinity on each

fibre like

a non degenerate quadratic form (of signature zero) . An estimate

of the number of critical points of such a function is obtainable by means

of a generalization of Morse th eory, the so called theory of th e Conley

index [27].

As a result we obtain a lower bound for the number of critical points

of the action, that is, for the number of closed trajectories of Hamilton's

equations, and that means also for the number of fixed points of the

symplectomorphism brought about by the solutions of these equations over

a

period. Finally, any symplectomorphism homologous to the identity can

be obtained in this way from a

single valued

Hamiltonian, periodic in

t i m e .

By this method Conley and Zehnder [19] have also proved a

generalization of the geometric theorem of Poincare (for a multidimensional

torus with standard symplectic structure).

M o r e recently Chaperon [21] has developed a symplectic version of

Morse's theory of broken geodesies, based on a technically more convenient

finite dimensional approximation (broken

geodesies

in place of Fourier series).


8/22

First steps

insymplectic topology

7

In the last few years many papers have appeared that use or develop this

m e t h o d ,

for e xa m p le [ 1 8 ] [ 2 4 ] , [ 1 6 ] .

I n Sicorav's pap er [28] th e co n di t ion of preserving th e cen t re of

gravity

is thrown awayin this case the number of

fixed

p oin t s is bou n ded be low

by the number of cri t ical points of a closed

1form

(a minimum for a l l

forms has been eva lua ted by N ovikov [ 2 5 ] , bu t for to r i th ese bo u n d s are

em pty) . I t st ill r ema ins un kn own , however , "whethera symplectomorphism,

homologous

to the identity, of afourdimensional

torus with arbitrary

symplectic structure

can have three

different fixed points.

4. Intersec tions of Lagrangian m anifold s

O ne of the genera l pr in ciples of sym plect ic geom etry h as been s t r ikingly

form ulated by Weinstein [29 ] in th e fol lowing way: everyth ing in th e

world is a Lagrangian m an ifold .

ALagrangian submanifold of a sym plect ic m an ifold is a subm anifold o f

th e grea test possible dimension on which the symplect ic s t ruc ture rest r ic t s to

zero ( th is grea test d imension is equal to ha l f the dimension of the ambient

manifold) .

Sym plectom orph ism s can be con sidered as Lagrangian m anifolds . In fac t ,

th e

gr a ph o f a sym p le c t om or ph ism (

1;

o ^ )

*(M

2

,

2

) is a Lagrangian

subm anifold of the p ro d u ct M

1

2

, furnished wi th th e sym plect ic s t ruc tu re

1

7 2

2

(where

a n d

2

a re the canonica l p ro jec t ions of the produc t

on

th e first an d secon d fac to rs) . C on verse ly, a di ffeom orph ism with a

Lagrangian graph is a symplectomorphism.

A Lagrangian submanifold of a product tha t i s not the graph of a

di ffeom orphism det erm ines asymplectic correspondence

a

manyva lued

generaliza tion of a sym plec tom orp h ism (o r a can on ica l t ra n sfor m a t io n of

classical mechanics).

T h e fixed

poin t s of a symplec tomorph ism a re the poin t s of in t e rsec t ion of

it s graph wi th the diagon al. T o est ima te the n um ber of fixed poin t s of a

symplec tomorph ism i t i s enough the re fore to e s t ima te the number of poin t s

of in tersec t ion of two Lagrangian manifolds, the graph and the diagonal .

Th e co n di t ion of preserving th e cent re of

gravity

i s then t ransformed in to

th e

con di t ion of H am ilton ian h om ologou sness: Lagrangian subm an ifolds a re

Hamiltonian

homologous

if on e of them can be ob ta ined from th e o th e r by

m eans of a phase t ransform a t ion of a H am i lton ian vec tor

field

with single

valued ( t im e de pe n d e n t ) H a m i lt on i a n .

In this way one obta ins ye t another conjec tured genera l iza t ion of the

Poincare conjec ture :

T he num ber of

points

of

intersection

of a

symplectic correspondence with

one that is H am iltonian

homologous

to it is not less than the minima l

number of

critical points

of afunction on it (counting both sets of

points

either

algebraically with

multiplicities, or geometrically).


9/22

8 V.I.Arnold

The

simplest cases of this conjecture were discussed back in 1965 [9].

Consider

as a symplectic manifold the space of the cotangent bundleT*Bof

a

manifold (that is, the phase space with configuration spaceB), furnished

with the standard symplectic structure

dpdq ,

and as a Lagrangian

submanifold L the zero section. For example, if is a circle, thenT*Bis a

cylinder andL is the equator.

Thenon compactness of the phase space compels one to make precise the

definition of Hamiltonian homologousness: a Lagrangian submanifold is

assumed to be compact and a Hamiltonian

field

to be of compact support

(equal to zero outside some compact set).

h or m

[30]. The number of

points

of

intersection

of the

zero section

of

the spaceof a

cotangentbundlewith

amanifoldHamiltonianhomologousto

it is not

less

than the sum of theBetti

numbers

of the manifold {taking

multiplicities into

account) and is

greater

than the

cohomological length

A curve on the cylinder that is Hamiltonian homologous to the central

circle is the embedding in the cylinder of a closed curve going around the

cylinder once in such a way that the oriented area between it and the

equator

is equal to zero. Clearly such a curve has no

fewer than

two points

of intersection with the equator (just this consideration was the starting

poin t

in Poincare's attempts to prove his geometrical theorem and in

Birkhoff sproof).

A neighbourhood of a Lagrangian manifold in a symplectic one isalways

symplectomorphic to a neighbourhood of the zero section of the space of

a

cotangent bundle [19].

Therefore from the bound on the number of points of intersection of

th e

zero section of a cotangent bundle with a Lagrangian submanifold

Hamiltonian

homologous to it there

follows

the same bound for the

number of points of intersection of any Lagrangian manifold with any one

Hamiltonian

homologous to it (in a sufficiently small neighbourhood). In

particular, one obtains a bound on the number of fixed points of a

symplectomorphism joined to the identity diffeomorphism in such a way

tha t

the derivative has singlevalued Hamiltonian, while the trajectories of

th e

points are sufficiently small.

If the Lagrangian manifold is an dimensional torus, then the number of

points

of intersection is not

less

than

2 (taking multiplicities into account),

including nofewer than +1 geometrically distinct points.

Such bounds cannot be obtained for Lagrangian immersions: Fig. 1

shows an immersion inT*S

X

of a circle, Hamiltonian homologous to the zero

section in the sense that their difference is the boundary of a chain of area

zero,

but nevertheless not intersecting the zero section (this effect was already

discussed in [9]).


10/22

First stepsin

symplectic

topology

A condit ion under which Lagrangian immersions have

n o

fewer p o i n t s

of

intersection

t h a n

embeddings hasrecen t ly been po in ted out by

Yu.V. Chekanov. I tsfo r m u l a tio n d e m a n d s th esimplest con ceptsofc o n t a c t

geometry.

s

Fig.1.

An

exact Lagrangian immersion,

n o tintersecting th e

null section.

5 . Legendrian subm anifoldsofcontact m anifo lds

Acontact manifold

is an

odd d imen sional m anifo ld ,

possessinga

m axim al ly

n on de ge n e r a t e field ofhyperp lanes . Atypical exam ple is th e th ree

dimensional sphere

S

3

,

which wesha ll co n side r as th eb o u n d a r y of aballin

C

2

. The tan gent co m plex lines form on

S

3

afield of two d imens ion al p lanes

(one

can

define th em also

as th e

n o r m a ls

to the

fibres

of

th e H opf fibrat ion

S

3

> S

2

). In

th e same

way

on e defines the s tan dard con tac t s t ruc t ure

on

any oddd imens ional sphere

of

h igher d im ens ion .

All

c on tac t m an i fo lds

o f

th e same dimension areloc a lly c on tac tom or ph ic .

T h e s tandard example is the manifold of 1 jetso ffun ct ions. T h e 1 jet

( the par t

of

th e T aylor series

of

degree

1) of a

fun c t ion

of

variables

is

given

by th e

cho ice

of2n+1

n u m b er s

=(

..., ) th e source

p o i n t ) ,

the value of t h e fu n c t io n ) , a n d =(p

lt

..., p

n

)

( th e pa rtial derivatives).

T h e n a tu r a l c on tac t s t r uc tu r e isd e t e r m i n e d by th ec on d i t iondy = pdx.

T h e space of 1 jetsof fun c t ion s on am an ifold isfibred over t h e space of it s

cotangent bundle (bym e a n s of th e m ap forgett ing th e value oft h e fu n c t io n ,

J \B ,

R)

* T * B ) .

A submanifold

of a

contac t manifo ld

is

said

to be

Legendrian

if it is

integral and has thegreate st possible dim en sion (nin a manifold of

dimension

2n+

1) . T h e

1graph

of afun c t ion m ay

serve

as ane xam ple

(y=f(x),p

=bflbx .

T h e

embedding

of a

Legendr ian manifold

in th e

space

of 1jetsof

funct ions

on it is

said

to be a

quasi function

if it is

r egu la rly h om ot op ic

to

t h e

1graph

of a

fun ct ion

in th e

space

of

Le ge n dr ian e m be d d in gs

( 1 )

.

With th e

C' topology:

embeddings aresaidto be close

to

each other

ifthey

areclose

not

only

in

value

bu t

also

in

their

derivatives.


11/22

10

V.I. Arnold

T h e

natural projection J\B, R)

T*B (forgetting the value of a function)

locally diffeomorphically maps Legendrian manifolds to immersed Lagrangian

manifolds. Each germ of a Lagrangian manifold in T*B is obtainable by this

c o n s t r u c t i o n from local quasi functions (because = \pdx can be recovered

from th e Lagrangian man ifold). This quasi function is determined uniquely

u p

to an additive constant.

A point of a quasi function is said to be

critical

if its projection on the

space of the cotangent bundle belongs to its zero section.

h or m

(Chekanov). A quasi function has at

least

as many

critical points

as the sum of the Betti numbers of the

manifold

(taking

multiplicities

into

account), and

among

them the number of geometrically different critical

points isgreater than the cohomological

length.

Example. Fig. 2 shows two Lagrangian immersions of a circle in T*S

1

.

They are both projections of Legendrian embeddings of a circle in J^iS

1

, R)

t h a t

is, p dx=0). But these embeddings are not regularly homotopic in

t h e

class of Legendrian embeddings.

Fig.

2. A quasi function and a non quasi function

I n

fact, to the Lagrangian curve on the left there corresponds a quasi

function (a Legendrian curve is one regularly hom otopic to a 1 graph in the

class of Legendrian embeddings). F or th is Lagrangian curve can be deformed

i n t o a section in the class of Lagrangian immersions with pdx = 0,

without creating any Legendrian self intersections.

T o

the Lagrangian curve on the right there corresponds a Legendrian

embedding of a circle t h a t is not a quasi function: it is not regularly

h o m o t o p i c

to a 1graph in the class of Legendrian embeddings (though it is

regularly homotopic to a 1graph in the class of all embeddings). For if it

were regularly homotopic to a

1 graph

in the class of Legendrian embeddings

in

brief,

Legendrian homotopic), t h e n its Lagrangian curve would intersect

t h e zero section in two points (by Chekanov's t h e o r e m ) , and this is not so.

Remark. The number of points of intersection in the theorems of

Laudenbach Sicorav and Chekanov can be bounded below by the


12/22

First steps in symplectic topology 11

stable minimal nu m be r of critical p oi n ts of a function on the m anifo ld

und er study . This nu m be r m ay be defined as follows. Con sider a ve cto r

bun dle over this manifold with fibre of dim ens ion 27V and fun ctio ns on it

coinciding outside a tubular neighbourhood of the zero section with a

quadratic form of signature NN on each fibre. Th e minimal nu m be r of

critical points of such functions (for all fibrations) then serves as a lower

bound for the number of points of intersection.

It appears not to be known whether this number coincides with the

minimal num ber of critical poin ts of function s on the man ifold. In

examplesfor surfaces, tori, projective spaces, and so onthe two numbers

usually coincide and both are equal to the sum of the Betti numbers (if one

counts points with multiplicities) or both are equal to the Lyusternik-

Shnirel'man category (if one counts geometrically different critical points).

It is not known also whe ther it is possible to bound the number of

points of intersection below by the minimal number of critical points

of functions on the original manifold in the theorems of Chekanov and

Lau denbac h-Sicorav, th at is, wh ether it is true th at the nu m ber of critical

points of a quasi-function is no smaller than that of a function.

The generalization of Chekanov's theorem for closed

1-forms

(with

Novikov's theory [25] replacing Morse theory) remains an open question.

6

Lagrangian and Legendrian knots

By aLagrangian {Legendrian) knot I mean a connected co m pon ent of the

space of Lagrangian (Legendrian) embeddings in a fixed symplectic (contact)

manifold

(1 )

.

Example 1. Consider Legen drian kn ot s in the sphere S

3

with its standard

contact structure. An embedding S

1

- S

3

determines a knot in the usual

sense. Fo r each kn ot in the usual sense one can choose an isotop ic

Legendrian embedding (this follows easily from the classical theorems of

Caratheodory or Rashevsky and Cho w). However, these em bedding s,

determining the same ordinary knot, may determine different Legendrian

knots.

One may convince oneself tha t this is so, for exam ple, with th e help

of the Maslov ind ex , determined here by the following con struction

(see [31]).

Considering S

3

as the boundary of a ball in C

2

, let us furnish each point

of the embedded curve with the outward unit normal vector and the unit

tangent vector. The con dition th at th e curve is Legendrian mea ns precisely

that these unit vectors areHermitian ortho gon al. The frame dete rm ines a

may take as the basis of the diffeomorphism either isotopies of the ambient

manifold or of the complement of the submanifold, preserving the corresponding

structure (one can also demand single-valuedness of the Hamiltonian isotopy).


13/22

2

V.I. Arnol d

map of the embedded circle to the unitary group. Together with the

determinant it gives a map of oriented circles:

7,

,

Idet

S

The degree of this map is said to be the index.

Theorem

(Alekseev)

(1)

.

The index is the only invariant of Legendrian

immersions of a circle in S

3

: all Legendrian immersions of the same index

are regularly homotopic in the class of Legendrian immersions and all values

of the index are realized on Legendrian embeddings from a C-neighbourhood

of any map S

1

-*S

3

.

In particular, there are infinitely many Legendrian non-homotopic

Legendrian embeddings, unknotted in the ordinary sense: they differ by

values of the index. Legendrian knots, unknotted in the ordinary sense, are

said to bepurely Legendrian.

Example 2. In the three-dimensional space /

1

(R, R) = { x, y, p)} with

contact structure

dy pdx

consider a Legendrian curve R -

J

1

,

coinciding

outside the unit ball with centre at the origin with the embedding of the

jc-axis.

Any knot in R

3

can be realized by such a curve. Even if this curve is

unknotted in the ordinary sense, it may be purely Legendrian knotted, as

Fig. 2 shows. This example shows, evidently, that there are more different

Legendrian knots in

S

3

than simply ordinary knots, furnished with the

Maslov index.

A multidimensional analogue of the preceding construction is determined

by the Gaussian map of a Legendrian submanifold L

1

S

2n

~

x

in the

Lagrangian

G rassm an n manifold

= U(n)/ O(n). F or = 2this manifold

is the space of the non trivial sphere bundle over the circle [33] (both

proofs of triviality in [34] are in error) .

F o r

Lagrangian embeddings R

2

* R

4

(coinciding with embeddings of the

plane

0 outside some sphere in the standard four dimensional symplectic

space)

not a single

k n o t t e d

example is known.

Bo th

the following questions

he re are o p e n .

1)Can any knot in the ordinary sensebe realized by a Lagrangian one (I t

is not known even whether the re is a Lagrangian embeddingk no t t ed in the

ord i na ry

sense, t h a t is, not

h o m o t o p ic

to an embedding of the plane in the

class of n ot necessarily Lagrangian embeddings.)

is is a particular case of Alekseev's general theory of immersions in a space with a

nonholonomicdistribution, generalizing Smale's theory of immersions [32].


14/22

First stepsin

symplectic topology

13

2)

Are

there purely Lagrangian

knots,

t h a t is, Lagrangian embeddings

h o m o t o p i c to the plane in the class of all (not necessarily Lagrangian)

embeddings, but non homotopic in the class of Lagrangian embeddings?

Remark

1. By a homotopy of a Lagrangian embedding in general position

in the class of embeddings we obtain a series of metamorphoses of caustics

of embeddings. Locally each typical caustic or typical metamorphosis of a

caustic on the plane is realizable as a caustic or metamorphosis of a

Lagrangian surface. Therefore the question reduces to the study of

obstructions to the global realization of caustics and their metamorphoses as

lines of critical values of an embedded Lagrangian surface. F or immersion s

the re is no obstruction (Alekseev).

Remark 2.

A Lagrangian kn ot R > R

2

determines an element of th e

h o m o t o p y group 7 ( ) of the Lagrangian G rassmann manifold. It is not

known whether this element can be non trivial or whether there are

characterist ic classes coming from the cohomology groups of the Lagrangian

G r assm a nn ia n ,

enabling one to distinguish purely Lagrangian knots.

Two theorems of Givental on Lagrangian embeddings

Consider

an embedding R

T R

n

,

obtained from the zero section by a

symplectic isotopy with compact carrier. The projection of a Lagrangian

submanifold

o n t o

the base of the fibration is said to be aLagrangian

projection.

Consider a regular value of a Lagrangian project ion . All its

inverse images are regular poin ts. The

Maslov

index of a

regular

point

is

defined as the Maslov index of a curve coming from infinity to this point on

o ur Lagrangian manifold (the index of intersection with the manifold of

singularities, see [31] for more details). On Fig. 3 the value of the Maslov

index has been written against each point.

Fig. 3. The indices of poin ts for a Lagrangian embedding and for an immersion

Theorem

(Givental') Overeach

regular

value there

is an

inverse image point

of index

zero.

This theorem was formulated as a conjecture and has been proved in the

one d imens iona l

case by Kolokol'tsov. F or immersions the theorem is not

t rue (Fig. 3, on the right).

A second theorem is obtained by trying to symplectify the fact t h a t an

embedding

M

n c

N

n

induces an inclusion of homology groups.


15/22

14

V.I.

Arnold

A Lagrangian submanifold of the space of a cotangent bundle is said to be

exact

if it is the projection of a Legendrian submanifold of the space of

1 jets

of functionsin o th er words, if th e generat ing function

dq is

single valued.

Theorem. An exact

Lagrangian embedding

M

n


16/22

First stepsin

symplectictopology 15

F r o m the conjecture itfollows th at a symplectic embedding

(M

compact, dim

=

dim

N

=

n)

induces an isomorphism of the

dimensional homology groups. F or if all the trajectories from the po in t in

go to infinity, th en they form an dimensional non compact cycle,

intersecting the fundamental cycle with index 1.

Corollary

1

No compact manifold admits an exact

Lagrangian embedding

in T*R" with the standard symplectic structure.

This conjecture was discussed long ago (see [35]) and has recently been

proved in another way by G romov [14 ] .

Corollary

2. The

image

of the

zero

section

in T*M under the

action

of a

symplectomorphism homologous

to the

identity

intersects the zero

section.

Otherwise there would be an exact Lagrangian embedding of two non

intersecting copies of in T*M, reducing the fundamental class of the

difference to 0.

8 .

Odd d im ensional analogues

Theorems on fixed points of symplectomorphisms and on intersections of

Lagrangian manifolds can be reformulated as assertions about closed phase

curves and about Legendrian chains in special contact spaces. By carrying

these ideas over to more general contact manifolds we obtain many new

conjectures.

Suppose that a contact structure is given as a field of zeros of a

1 form.

All such

contact

\ forms are distinguished by a non invertible nowhere zero

factor. The choice of a contact form a determines a field of characteristic

directions (the kernels of da), everywhere transversal to th e contact plane

a

= 0

If the integral curves of the field of kernels form a fibration,

t h e n

its base

is a symplectic manifold and the preceding theorem is applicable. However,

by changing the factor in the form a the field changes and the fibration

disappears. Symplectic results, formulated without reference to th e

fibration, also serve as contact conjectures.

Example. Consider a closed Legendrian curve (let us say inS

3

with its

standard contact structu re). For the standard

1form

the characteristics are

fibres of the Hopf fibration S

3

S

2

. The projection of th e Legendrian

curve on

S

2

is a curve immersed in the two dimensional sphere boundin g a

chain of area 0 (mod 4 ).

A curve of area 0 has a poin t of self intersection. We obtain the following

conjecture.

Conjecture.

Every closed Legendrian

curve

in S

3

with standard contact

structure has for any choice of contact

l form

a characteristic

chord

(it

intersects

twice

some integral curve

of the

field

of

kernels

of the

form

da).


17/22

16 V.I.

Arnol d

By the same considerations for each immersion of a Legendrian manifold

(let

us say in

S*"*

1

or in R

2

*

1

with the standard contact structure) for any

choice

of contact

1 form,

for any C close Legendrian immersion one must

find an interval of characteristics, starting at an embedded Legendrian

manifold and ending at another (and, moreover, the number of such

intervals must be bounded below by inequalities of Morse type).

I nthis direction all that has been proved is the following generalization of

Poincare's

geometric theorem.

Consider :

2 + 1

^

2

, a fibration with fibre an oriented circle. Let

be a closed non degenerate 2 form on the fibration space. We shall say that

preserves

the

centre

of

gravity

if its

cohomology class

[ ]

*{ )

is

induced

from the base (belongs top*H*(B)) ,that is, if the integral of over

vertical cycles is equal to zero.

Theorem(Ginzburg). / /thecharacteristic field (thefieldof kernels) of a

form preservingthe centreof gravityisenclose to thefieldof thedirections

of the

fibres

of the

fibration,

then the number of closedcharacteristicsclose

to thefibres

once

traversedis not lessthan the sum of the Betti numbersof

the base(if the characteristics are

non degenerate

or

counted

with

multiplicities),andamongthem the number ofgeometricallydistinct

closed

characteristicsis not lessthan the categoryof thebasefor = 1; for > 1

the number of characteristics is not lessthan v(p).

Remark, ^ closeness may here be replaced by C closeness (as in the

theorem onfixed points of symplectomorphisms, by contactization of which

th epresent result has been obtained) and perhaps also by a still weaker

condition.

If one does not ask for preservation of the centre of

gravity,

then the

inequalities of Morse type must be replaced by inequalities of Novikov

type [25].

Example. Consider the motion of a charged point on a Riemann surface of

genusg in a transverse magnetic field not reducible to zero.

F r o m the theorem it

follows

that if the initialspeedv

0

issufficiently

small,then

there

exist (taking

multiplicities

intoaccount) nofewer than

2g+2

closed

trajectories(curvesof

prescribed

geodesic curvature

k(x )

= H(x)/v

o

y

HereE

3

* is a manifold of energy level, while is the sum of the

restriction

of the symplectic structure and the 2form HdSinduced fromM.

In this situation there are supposedly 2g+2 closed curves on any energy

level

(see [25], [26], [36]).

9 . Opt ical

Lagrangian

manifolds

A Lagrangian submanifold of the space of a cotangent bundle is said to be

optical if it lies in a hypersurface transversal to the fibres, the intersections


18/22

First stepsinsymplectictopology 17

of which with the fibres are quadratically convex at every point (with

respect to the natural affine structure of the fibres).

Example.

The solution of the eikonal equation

(dS/dq)

2

=

1 determines the

optical

Lagrangian submanifold

= dS/dq

of the hypersurface

p

2

=

1.

Every stable Lagrangian singularity is locally realizable as the singularity

of the projection of an optical Lagrangian manifold (even the manifold of

th e preceding example) on to the base [37 ]. However, the global property

of being put in a hypersurface of phase space, convex with respect t o

m o m e n t a , imposes restr ictions on the co existence of singularities. F or the

same reason not all local metamorphoses of caustics are realized in the class

of optical Lagrangian singularities (singularities born under metamorphosis

may be shown to be incompatible for optical caustics).

T h e first examples of this type were discovered by Nye [38], who

unsuccessfully tried to realize with the aid of a laser the bifurcation of the

birth of a flying saucer and t h e n proved that it could not be realized in

optics. The general theory, presented below, is due to Chekanov [39] .

Fig. 4. Aflying saucera nonoptical caustic.

Consider

the manifold of critical points of the projection of a Lagrangian

manifold on to the base of the cotangent bundle. For Lagrangian manifolds in

general position (it is un importan t whether they are optical or not) this

critical submanifold of the Lagrangian manifold is a hypersurface in it (it is

defined in it by a single equation ). On this hypersurface there are singular

points forming in it a set of codimension 2 (similar to the vertex on the

surface of a

cone) .

We suppose now that the manifold is optical and in

general position.

Definition.

The characteristic direction of a hypersurface in a symplectic

space is the skew orthogonal complemen t to its tangent hyperplane (t hat is,

th e direction of a Hamiltonian field whose Hamiltonian is constant on the

hypersurface).

Theorem (Chekanov).

The

characteristic direction

of a convex fibred

hypersurface

containing

an

optical Lagrangian manifold

is not

tangent

to the

set of

critical

points

of its

projection

(even at the

singular points

of this set).

Corollary.

On the

non singular

part of the

critical variety

there is

defined

a

smooth

field

of

directions, coinciding with

the

field

of kernels of a

Lagrangian

map at

points

where

the kernel is one dimensional and

touches

the

critical

variety.


19/22

18 V.I.

Arnold

This field is cut out by the two dimensional plane spanned by the kernel

a n d the characteristic direct ion of th e convex fibred hypersurface.

Corollary. The Euler characteristic of asmooth compact optical critical

manifold

is equal to

zero.

I n particular, it follows from this that the optical realization of the

metamorphosis of the birth of aflying saucer is impossible (Fig. 4; pan cake

in Zel'dovich's terminology [40] , [41] , lip in Thorn 's):

(

2

)

0.

I n

[39] Chekanov proved the formula +

2(#D^+ #D\A) =

0, expressing

t h e Euler characteristic of a compact two dimensional critical manifold in

terms of the number of critical points of type Z)

4

(in the optical case one

has to distinguish not two, as usual, but three versions of D

4

, corresponding

t o

the three types of umbilical

p o i n t ) .

I n

a space of any dimension no metam orphoses of the type lip or

pancake (of any signature) are realized as optical. In the three dimensional

case the metamorphosis ofD~^ (the birth of two pyramids) is not realized;

all the remaining metamorphoses

(A

4

, A

5

, D%, D

s

)

are realizable optically

an d have been observed in experiments.

I n

the optical situation one can repeat the constructions of

Vasil'ev

complexes, algebraizing the adjacency relations of singularities, and the

c o m p u t a t i o n of the Lagrangian and Legendrian cobordism rings (see [1] [7 ]) .

T h e theory of Lagrangian and Legendrian knots (see above in 6) gives rise

in the opt ical case to a special form of the theory of optical knots. An

optical Lagrangian manifold is invariant relative to the corresponding

H a m i l t o n i a n flow (for which the level surface of the Hamilton function is a

convex fibred hypersurface). Therefore globally optical manifolds, as a rule,

are rigid (like Kolmogorov tori) on a

fixed level

surface of the Hamilton

function. Therefore in defining homotopies of optical manifolds it is natural

t o

admit deformations of the ambient hypersurface (preserving its convex

sets).

Some Lagrangian knots do not a priori admit optical realizations. Some

Lagrangian regularly hom otopic optical manifolds may be optically non

h o m o t o p i c (untying a Lagrangian knot may a priori demand that one goes

outside the class of Lagrangian manifolds, embedded in convex hypersurfaces).

I

note an interesting property, recently proved for Lagrangian manifolds

of the hypersurface of cotangent unit vectors of a Riemannian torus:

Theorem

(Byalyi and Polterovich [42]). A

Lagrangian

section

belonging

to

the stated hypersurface is

filled

up with lifts in the phase

space

of geodesies,

minimal between any of their points on the covering

plane

of the torus;

conversely, an invariant torus

whose

geodesies

possess

this

property

of

minimality is necessarily a

section.

Apparently an analogous theorem is true not only for geodesic

flows,

but

generally also for optical Lagrangian manifolds.


20/22

First steps in symplectic topology 19

The proof would be rather simple if it were possible to join the torus

under study with a section of a continuous family of optical Lagrangian tori

(deforming along with the torus and containing its convex hypersurface: the

minimality is an obstruction to the birth of caustics.

Exactly this analysis of the dependence of the character of the extremum

of variational principles for Kolmogorov tori on their situation in the space

of the cotangent bundle has led the author and Shchepin to problems on

Lagrangian and Legendrian knots.

Remark. A Lagrangian embedding T

2

c - T*T*cannot realize more than a

single generator of the two-dimensional hom ology group ofphasespace.

In fact, otherwise, going over to the #-fold cover, we would obtain q

mutually non-intersecting Lagrangian embeddings. Each of these is obtained

from another symplectomorphism with single-valued Hamiltonian (induced

by a shift of the base). According to Sicorav's theorem ( 4) the manifolds

must intersect each other.

More generally, a two-dimensional submanifold in T'T

1

may cover the

base with arbitrary m ultiplicity. Therefore, Lagrangianness sets a global

bound on an embedding. Multidimensional generalizations are obvious.

It is instructive that such natural problems and theorems of symplectic

topology as the problem of Lagrangian knots and Chekanov's theorem on

the topological properties of the critical sets of generalized solutions of the

Hamilton-Jacobi equations with Hamiltonians convex with respect to

momenta, were discovered only as a result of experiments in laser optics

[38] and the analysis of the variational principles of Percival, Aubri, and

others, connected with the theory of corrosion

( [43] ,

[44]).

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First stepsinsymplectic topology 21

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Translated

by I.R. Porteous Moscow State University

Received by the Editors 4 September 1986

1986_arnold first steps in symplectic topology

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