INTRODUCTION TO SYMPLECTIC TOPOLOGY Milena Pabiniak Friday October 20, 2006 GRADUATE STUDENT SEMINAR
A symplectic vector space is a pair ( V, ω ) con- sisting of finite dimensional real vector space V and a non-degenerate, skew symmetric bi- linear form ω : V × V → R , that is skew symmetry ∀ v,w ∈ V ω ( v, w ) = − ω ( w, v ) non-degeneracy � � ∀ w ∈ V ω ( v, w ) = 0 ⇒ v = 0 ∀ v ∈ V 1
Fact: The vector space V is necessary of even dimension. Linear map F : ( V 1 , ω 1 ) → ( V 2 , ω 2 ) is called symplectic if F ∗ ω 2 = ω 1 , where F ∗ ω 2 ( v, w ) = ω 2 ( Fv, Fw ) . 2
Example: V = R 2 n , ω ( x, y ) = x T J 0 y , where � 0 − I � J 0 = 0 I That is ω (( x 1 , ..., x 2 n ) T , ( y 1 , ..., y 2 n ) T ) = n � = ( y i x n + i − x i y n + i ) . i =1 3
Moreover, this is essentialy the only example of a symplectic vector space. Precisely: if ( V, ω ) is symplectic , then we can always find a cannonical basis e 1 , . . . , e n , f 1 , . . . , f n of V such that: ω ( e i , e j ) = ω ( f i , f j ) = 0 ω ( e i , f j ) = δ ij . Hence two symplectic vector spaces of the same dimension are isomorphic. 4
Let matrix A represtent linear map A : R 2 n → R 2 n . Map A is symplectic if and only if A T J 0 A = J 0 . Matrices satisfying condition above are called symplectic. Exercise: � A B � Ψ = C D A, B, C, D - real n × n matrices Prove that Ψ is symplectic iff � D T − B T Ψ − 1 = � − C T A T More explicitly it means A T C, B T D are sym- metric and A T D − C T B = I . 5
Let M be C ∞ smooth manifold, without bound- ary, compact. M is a symplectic manifold if there exist on M closed, non-degenerate 2-form ω (called sym- plectic structure). Diffeomorphism ψ : ( M 1 , ω 1 ) → ( M 2 , ω 2 ) is called symplectomorphism if ψ ∗ ω 2 = ω 1 . 6
Example: M = R 2 n with coordinates p 1 , . . . , p n , q 1 , . . . , q n , and n � ω 0 = dp i ∧ dq i i =1 Note that ω 0 (( x 1 , ..., x 2 n ) , ( y 1 , ..., y 2 n )) = � n i =1 ( x i y n + i − y i x n + i ) = − < x, J 0 y > . Fact: Diffeomorphism ψ : ( R 2 n , ω 0 ) → ( R 2 n , ω 0 ) is a symplectomorphism if and only if its Jacobi matrix dψ is a symplectic matrix. 7
Theorem 1 (Eliashberg) Group of symplecto- morphisms Symp ( M, ω ) = { g : M → M | g ∗ ω = ω } is C 0 -closed, that is if g i ∈ Symp ( M, ω ) and g i → g ∞ unifromly, then g ∞ ∈ Symp ( M, ω ) . Theorem 2 (Darboux) For any point y on a symplectic manifold ( M 2 n , ω ) of dimension 2 n , there exist an open neighborhood U of y and a differentiable map f : ( U, ω ) → ( R 2 n , ω 0 ) such that f ∗ ω 0 = ω | U . 8
Denote by B 2 n ( r ) the closed Euclidean ball in R 2 n with centre 0 and radius r and by Z 2 n ( r ) = B 2 ( r ) × R 2 n − 2 the symplectic cylinder. Theorem 3 (Gromov’s Nonsqueezing theorem) If there is a symplectic embedding B 2 n ( r ) ֒ → Z 2 n ( R ) then r ≤ R. For open subset U of a symplectic manifold ( M, ω ) define Gromov’s capacity c ( U ) = max { πr 2 | ∃ B 2 n ( r ) ֒ → U symplectic } . Theorem 4 Any diffeomorphism that preserves capacity i.e. c ( g ( U )) = c ( U ) for all open U is such that g ∗ ω = ω. 9
Example: S 4 does not admit a symplectic structure. Assume ω is a closed and non-degenerate 2- form on S 4 . As the second de Rham cohomol- ogy group of S 4 vanishes, ω has to be exact, that is there exist a 1-form α such that dα = ω . Then also the volume Ω = ω ∧ ω form is exact: d ( ω ∧ α ) = dω ∧ α + ω ∧ dα = ω ∧ ω = Ω . Thus by Stoke’s theorem we have � � S 4 Ω = ∂S 4 ω ∧ α = 0 , which is impossible for a volume form. So we see that on S 4 we cannot impose a symplectic form. 10
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