Symplectic Geometry versus Riemannian Geometry Inês Cruz, CMUP 2010/10/20 - seminar of PIUDM 1
necessarily very incomplete The aim of this talk is to give an overview of Symplectic Geometry (there is no intention of giving a list of recent results or open problems) 2
Notation: throughout the talk: - M - real, finite-dim al, differentiable manifold without boundary - C ( M ) = { f : M : f is smooth} - χ ( M ) = { X : M TM : X is a vector field} - Ω k M = { ω :TM TM } or Ω k ( M ) = ω : χ M ω p;v 1,,v k { χ ( M ) C ( M )} ω ( X 1,, X k ) C ( M ) given by: ω ( X 1,, X k )( p) = ω p; X 1p,, X kp 3
1. Symplectic Manifolds 1. Riemannian Manifolds Def: Symplectic manifold is a pair ( M,ω ), where: (a) ω Ω 2 ( M ) i.e., = ω ( X,Y ) ω Y, X ω ( fx + gy, Z ) = fω ( X, Z ) + gω ( Y,Z ) (b) ω is nondegenerate, i.e.: ω ( X,Y ) = 0, X χ ( M ) Y = 0 (c) ω is closed, i.e. dω = 0 We call ω a symplectic form. which manifolds qualify for being symplectic? Necessary conditions: (N1) dim M = 2n consider local coordinates ( x 1,, x m ) the matrix A with entries: and build Def: Riemannian manifold is a pair ( M,<,> ), where: (a) <,>: χ(m ) χ(m ) C M < Y, X > = < X,Y > < fx + gy, Z > = f < X, Z > +g < Y,Z > (b) <, > is positive definite. satisfies: Consequence: <, > is nondegenerate. which manifolds qualify for being Riemannian? all smooth manifolds! 4
then (a) and (b) imply: A T = A and det( A) 0 m = 2n (N2) M is oriented consider the nth exterior power: then (b) implies ω (n) = ω ω Ω 2n ( M ) ω (n) is a volume form on M. (N3) if M is compact then H 2 DR ( M, ) 0 ω = dα ω (n) = d α ω ω vol M a ij = ω, x i x i ω ( n) is called symplectic volume. dβ = β = 0 = ω (n) = M M M S 2n is not symplectic, for any n >1 (S 2 is symplectic) 5
2. Examples example 1: M = 2n with coords: { x 1,, x n, y 1,, y n } and symplectic structure given by: ω 0 = n i=1 dx i dy i = dx dy example 2: form: M = T * N ω = dλ with symplectic ( λ is the Liouville 1-form on M : =< α,dπ p,α > ) λ ( p,α ) X X The importance of example 1 will be clear soon. example 2 is behind the Hamiltonian formulation of Conservative Mechanics. 6
3. Special vector fields Nondegeneracy of ω implies that the following is an isomorphism: Def: Given vector field is: Ω 1 ( M ) I : χ M X i X ω = ω X, f C ( M ) X f = I 1 ( df ) its Hamiltonian = df (in other words: ω X f, ) Lemma X f is tangent to the level surface: ΣC = p M : f ( p) = C { } 3. Special vector fields Nondegeneracy of <, > implies that the following is an isomorphism: Def: Given vector field is: Ω 1 ( M ) I : χ M X < X, > f C ( M ) f = I 1 ( df ) its gradient (in other words: < f, >= df ( ) ) Lemma f is normal to the level surface: Σ C = p M : f ( p) = C { } (equivalently f is constant on the flow of X f ) note that: v T Σ C df v = 0 and: if v T Σ C then: < f,v >= df v = 0 = ω ( X f, X f ) = 0 df X f 7
f X f Other important vector fields: Def: A vector field X is said to be symplectic if I(X) is closed. In other words: di X ω = 0 Hamiltonian vector fields are symplectic, since ddf=0. Condition (c) implies that the flow of any symplectic vector field preserves ω : L X ω = di X ω + i X dω = 0 = 0 = 0 because of (c) 8
4. Poisson bracket One can use Hamiltonian vector fields to define an operation between smooth functions: Def: Poisson bracket on M is: {,} :C ( M ) C ( M ) C ( M ) ( f, g) ω X f, X g = df ( X g ) Condition (c) implies this bracket satisfies Jacobi s identity: f,{ g,h} { } { } { } + { g, h, f } + { h, f,g } = 0 which, together with obvious properties of {,}, implies that: ( C ( M ),{,}) is an infinite-dimensional Lie algebra. 9
5. Equivalence 5. Equivalence Def: Two symplectic manifolds ( M,ω ) M ',ω ' and are symplectomorphic if there exists a C 1 map: satisfying: i.e., ϕ : M M ' ϕ * ω ' = ω ( ) = ω p ( X,Y ) ω ' ϕ ( p) dϕ p ( X),dϕ p Y ϕ is called a symplectic map and necessarily dϕ p is injective, for all p so: dim M dim M ' A symplectomorphism is a symplectic diffeomorphism of M. Symplectomorphisms form an (infinite dimensional) subgroup of the group diff(m). Def: Two Riemannian manifolds ( M,<,> ) and ( M ',<,> ') are isometric if there exists a C 1 map: satisfying: < dϕ p ( X),dϕ p Y ϕ : M M ' > ' ϕ p = < X,Y > p ϕ is called an isometry and necessarily dϕ p is injective, for all p so: dim M dim M ' 10
Darboux-Weinstein theorem Let p be any point on a symplectic manifold of dimension 2n. Then there exist local coordinates ( x 1,, x n, y 1,, y n ) (in U) such that: ω U = n i=1 dx i dy i Therefore all symplectic manifolds are (locally) symplectomorphic to example 1. Consequence: there are no local invariants (apart from dimension) in Symplectic Geometry curvature is a local invariant in Riemannian Geometry 11
6. Global Invariants As seen before, a symplectic manifold carries a symplectic volume: If volumes: ϕ : M M ' ω (n) = ω ω is a symplectic map, then it preserves (symplectic) ϕ * ω ' (n) = ϕ * ( ω ' ω ') = ϕ * ω ' but the converse is not true for n > 1. ϕ * ( ω ') = ω ω = ω (n) example: take 2 ( M,ω ) = 4, dx i dy i i=1 ϕ x 1, x 2, y 1, y 2 and consider the map: = 1 2 x 1,2x 2, 1 2 y 1,2y 2 This map preserves the symplectic volume symplectic. 2dx 1 dx 2 dy 1 dy 2 but is not so what really characterizes symplectomorphisms? 12
6.1. Gromov s Nonsqueezing Theorem The key theorem for characterizing symplectomorphisms (using symplectic invariants) is: Nonsqueezing theorem - Gromov (1985) There is a symplectic embedding: ϕ : B 2n ( r),ω 0 if and only if r R. ( Z 2n ( R),ω 0 ) (where: B 2n Z 2n ( r) = x, y ( R) = x, y n 2n : x i 2 + y i 2 < r 2 i=1 { 2n : x 2 1 + y 2 1 < R 2 } open (symplectic) ball of radius r open symplectic cylinder of radius R ) a symplectic embedding is just a symplectic map which is also an embedding. It is denoted by: ϕ :( M,ω ) ( M ',ω '). U,ω U if M,ω is a symplectic manifold and U is open in M, then is also symplectic. 13
The theorem is not valid if symplectic cylinder is replaced by cylinder : C 2n ( R) = ( x, y) 2n : x 2 1 + x 2 2 < R 2 { } example: the following is a symplectic embedding from B 4 (2) into C 4 (1): = 1 2 x 1, 1 2 x 2,2y 1,2y 2 ϕ x 1, x 2, y 1, y 2 14
6.2. Symplectic Invariants - Capacities Using Gromov s nonsqueezing theorem, we will construct a symplectic invariant: Gromov s width. This is one of many symplectic invariants known as symplectic capacities. Let M(2n) denote the set of all symplectic manifolds of dimension 2n. Def Symplectic capacity is a map: satisfying all three properties: c : M ( 2n) + 0 + (1) monotonicity - if there is a symplectic embedding: ( M 2,ω 2 ) ϕ : M 1,ω 1 then c( M 1,ω 1 ) c( M 2,ω 2 ) (2) conformality - c( M,λω ) = λ c( M,ω ) λ 0 (3) (strong) nontriviality - c B 2n ( 1),ω 0 = π = c( Z 2n ( 1),ω 0 ) 15
If n = 1 then (absolute value of) total volume of M is a symplectic capacity. If n > 1 then (absolute value of) total volume of M is not a symplectic capacity (nontriviality fails). Theorem Any symplectic capacity is a symplectic invariant, i.e., if there is a symplectomorphism: then c( M 1,ω 1 ) = c( M 2,ω 2 ). (proof: monotonicity in both ways) ϕ :( M 1,ω 1 ) ( M 2,ω 2 ) Lemma Any symplectic capacity satisfies: c( B 2n ( r),ω 0 ) = πr 2 = c( Z 2n ( r),ω 0 ). (proof: previous theorem+conformality+nontriviality) 16
Theorem The existence of a symplectic capacity is equivalent to Gromov s nonsqueezing theorem. suppose c exists and that: ( Z 2n ( R),ω 0 ) ϕ : B 2n ( r),ω 0 is a symplectic embedding. Then monotonicity+previous lemma imply: so r R. c( Z 2n ( R),ω 0 ) = π R 2 πr 2 = c B 2n ( r),ω 0 define the Gromov s width of a symplectic manifold: W G ( M,ω ) = sup{ πr 2 : ϕ :( B 2n ( r),ω 0 ) ( M,ω )} r >0 (the area of the disk of the bigger ball one can symplectically-embed on ( M,ω ) If Gromov s nonsqueezing theorem holds (it does!) then Gromov s width is a symplectic capacity. 17
Theorem Gromov s width W G is the smallest of all capacities: W G (proof: let c be any capacity and fix r such that there is an embedding: Then monotonicity of c implies: Since this holds for all r and proving the result). ( M,ω ) c( M,ω ), for any capacity c and any ( M,ω ). sup r >0 c( M,ω ) M,ω ϕ : B 2n ( r),ω 0 πr 2 = c B 2n ( r),ω 0 c M,ω is independent of r: { πr 2 : ϕ :( B 2n ( r),ω 0 ) ( M,ω )} c M,ω 18
6.3. Back to Symplectomorphisms We go back to the question: so what really characterizes symplectomorphisms? It turns out that symplectomorphisms of 2n are (almost) characterized by the property of preserving capacity of ellipsoids (1) : Theorem (Eliashberg, 1987) (Hofer, 1990) Let be a diffeomorphism and c a capacity. Then: ( 2n,ω 0 ) c( ϕ ( E,ω 0 )) = c E,ω 0 ϕ : 2n,ω 0 for any ellipsoid E 2n if and only if ϕ is symplectic or anti-symplectic (2). (1) ellipsoid is the image of a ball by a linear/affine diffeomorphism (2) meaning that ϕ * ω 0 = ω 0 19