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NOTES on SYMPLECTIC GEOMETRY Contents 1. Week 1 1 NOTES ON SYMPLECTIC GEOMETRY NILAY KUMAR Contents 1. Week 11 1.1. The cotangent bundle1 1.2. Geodesic flow as Hamiltonian flow4 2. Week 27 2.1. Darboux's theorem7 3. Week 3 10 3.1. Submanifolds of symplectic manifolds 10 3.2. Contact manifolds 12 4. Week 4 15 4.1. Symplectic linear group and linear complex structures 15 4.2. Symplectic vector bundles 18 5. Week 5 21 5.1. Almost complex manifolds 21 5.2. K¨ahlermanifolds 24 6. Week 6 26 6.1. Poisson brackets 26 6.2. Group actions 28 6.3. Cohomological obstructions 30 7. Week 7 and winter break 33 7.1. Group actions on symplectic manifolds 33 7.2. Cohomological obstructions 35 7.3. Moment maps 38 7.4. Symplectic Reduction 42 7.5. Applications? 42 These notes were written for a reading course with Professor Eric Zaslow on the basics of symplectic geometry. They follow Mcduff/Salamon quite closely. These notes are rather rough, and in several places woefully incomplete: caveat lector.1 1. Week 1 1.1. The cotangent bundle. Date: Fall 2015. 1add references! 1 2 NILAY KUMAR Definition 1. Let X be a smooth n-manifold and π : M = T ∗X ! X be its cotangent bundle. We define the canonical one-form θ 2 Ω1(M) as follows. For any p = (x; ξ) 2 M, set θp(v) = ξ(dxπ(v)): The one-form θ is canonical (or tautological) in the sense that its value at a point is simply given by the covector determined by that point. More precisely, we have the following characterization. Proposition 2. The canonical one-form θ is the (unique) one-form such that for every λ 2 Ω1(X), λ∗θ = λ. Proof. We compute, for v 2 TpX, ∗ (λ θ)p(v) = θλ(p)(dpλ(v)) = λp(dp(π ◦ λ)(v)) = λp(v); where we have used the fact that λ is a section of π, i.e. π ◦ λ = idX . Uniqueness is easily checked. Definition 3. The canonical symplectic form ! 2 Ω2(M) is now defined to be the exterior derivative ! = −dθ; of the canonical one-form. To be symplectic, ! must be closed and nondegenerate. That it is closed is obvious. Proposition 4. The form ! 2 Ω2(M) is nondegenerate and thus defines a sym- plectic structure on M = T ∗X.2 Proof. For ! to be non-degenerate, it must be nondegenerate at each point p 2 M. 1 n Given coordinates p = (x; ξ) = (x ; : : : ; x ; ξ1; : : : ; ξn) in a neighborhood of p, we can compute @ @ @ θ vi + νi = ξ vi (x,ξ) @xi @ξi @xi i = ξiv and hence i θ = ξidx : Taking an exterior derivative, we find that ! = −dθ i = dx ^ dξi: Fix v 2 TpM and suppose that ιv!p = 0, i.e. !p(v; w) = 0 for all w 2 TpM. In coordinates, this implies that i i i i ιvj @ +νj @ (dx ^ dξi) = v dξi − ν dx @xj @ξj = 0; i i and hence that v = ν = 0, i.e. v = 0. We conclude that !p is nondegenerate at each p 2 M. 2Is there a coordinate invariant proof? NOTES ON SYMPLECTIC GEOMETRY 3 Remark 5. Note that a 2-form ! on a manifold M is nondegenerate if and only if n ! is nowhere vanishing. Fix p 2 M and consider the vector space (TpM; !p). If n !p is nondegenerate, we can find a symplectic basis for TpM, and so !p evaluated n on (u1; : : : ; un; v1; : : : ; vn) is nonzero, whence !p is not zero on V . On the other hand, suppose !p is degenerate, i.e. there is a v 6= 0 such that !p(v; w) = 0 for all w 2 V . Choosing a basis v1; : : : ; v2n for V such that v1 = v, we find that !p(v1; : : : ; v2n) = 0 and hence !p = 0 on V . We conclude that every symplectic manifold is orientable. ∼ ∗ It is easy to see that ! provides an isomorphism ι : TxX −! Tx X between tangent and cotangent spaces at each point x 2 X: since !x is nondegenerate, the linear map ι : v 7! !x(v; −) is injective and hence bijective. In fact, we can say more. Proposition 6. The metric ! induces an isomorphism of vector bundles ι : TX −!∼ T ∗X = M. Proof. Recall that an isomorphism in the category of smooth vector bundles is a smooth bijection3 ι such that the diagram TX ι T ∗X π1 π2 X ∗ commutes and for each x 2 X, the restriction ιx : TxX ! Tx X is linear. The map ι : TX ! T ∗X taking (x; v) 7! (x; !(v; −)) fits into the diagram above and is bijective and fiberwise linear. Moreover, ι is a smooth map, as is seen by its coordinate description computed above. Definition 7. A Hamiltonian is a smooth function H : M = T ∗X ! R. we define the Hamiltonian vector field vH associated to H to be the vector field on M satisfying ιvH ! = dH: The (local) flow F :(−"; ")×M ! M determined by vH is called the Hamiltonian flow.4 Note that an integral curve γvH :(−"; ") ! M of vH can be thought of as the trajectory of a physical state in phase space. Indeed, Hamilton's equations are given @xi @H = @t @ξi @ξ @H i = − ; @t @xi which is precisely the condition that γ0 (t) = (v ) . Moreover, H is constant vH H γ(t) along the Hamiltonian flow, as dH(vH ) = (ιvH !)(vH ) = !(vH ; vH ) = 0; 3Existence of a smooth inverse is automatic (reference?). 4Is this a global flow? Does it depend on X? 4 NILAY KUMAR i.e. vH is tangent to the level sets of H. In a physical system, where H is the energy functional on phase space, this phenomenon is the law of conservation of energy. ∗ 5 Proposition 8. The Hamiltonian flow is a symplectomorphism, i.e. Ft ! = !. Proof. We use the following trick: Z t d ∗ ∗ Ft ! dt = Ft ! − ! 0 dt since F0 = idM , and hence Ft is a symplectomorphism if and only if the integrand is zero. But d ∗ d ∗ ∗ d ∗ Ft ! = Ft+s! = Ft Fs ! dt ds s=0 ds s=0 ∗ = Ft LvH !; and Cartan's magic formula, LvH ! = dιvH ! + ιvH d!; tells us that LvH ! = 0 since ιvH ! = dH is closed, as is !. Corollary 9 (Liouville's Theorem). The volume form !n on M = T ∗X is preserved by the Hamiltonian flow. 1.2. Geodesic flow as Hamiltonian flow. We wish to discuss geodesics and ge- odesic flow. For this, we need the concept of connections and covariant derivatives.6 Definition 10. A connection on a vector bundle E ! X is an R-linear map r : Γ(X; E) ! Γ(X; E ⊗ T ∗X) such that the Leibniz rule r(fσ) = (rσ)f + σ ⊗ df; for all f 2 C1(X) and σ 2 Γ(X; E). Theorem 11. Given a Riemannian manifold (X; g), there exists a unique connec- tion on π : TX ! X, known as the Levi-Civita connection, satisfying (i) symmetry: rX Y − rY X − [X; Y ] = 0; for X; Y 2 Γ(X; T X); (ii) compatibility with g: Xg(Y; Z) − g(rX Y; Z) − g(Y; rX Z) = 0; for X; Y; Z 2 Γ(X; T X). Definition 12. Let v be a vector field on (X; g); we define the covariant deriv- ative of v along a smooth curve c : I ! X to be the vector field Dv = r v; dt dc=dt where r is the Levi-Civita connection. Explicitly, if we write v = vi@=@xi and c(t) = (c1(t); : : : ; cn(t)), i Dv X dv @ X dci @ = + viΓk : dt dt @xi dt ij @xk i ijk 5Is there a better proof? 6Reference do Carmo. NOTES ON SYMPLECTIC GEOMETRY 5 k Here Γij are the Christoffel symbols of r, determined by @ X k @ r i = Γ : @=@x @xj ij @xk ijk We say that c is geodesic at some t 2 I if D=dt(dc=dt) = 0 at t, and that c is geodesic if it is geodesic at all t 2 I. In coordinates, the condition for c to be geodesic is given by a system of second-order differential equations: d2ci X dcj dck + Γi = 0; dt2 jk dt dt jk for i = 1; : : : ; n: For the rest of the section, assume (X; g) is Riemannian and we fix the Hamil- tonian H : M = T ∗X ! R as 1 2 H(x; ξ) = ξx ; 2 g i.e. consisting of only a kinetic term. Here we are implicitly using the nondegeneracy −1 of g to associate ξx with its corresponding vector (or, equivalently, using g ). Proposition 13. The Hamiltonian flow on M = T ∗X is dual to the geodesic flow on TX. In other words, the integral curves of the Hamiltonian vector field vH associated to the Hamiltonian above project to geodesics of g on X.7 Proof. It suffices to show, in coordinates, that Hamilton's equations (i.e. the con- dition for being on the integral curve) yield the geodesic equations above after the necessary dualization. Note first that in coordinates the Hamiltonian becomes 1 H(x; ξ) = gijξ ξ : 2 i j For convenience we will denote the components of an integral curve as xi(t). Hamil- ton's equations yield i dx @ 1 jk = g ξjξk dt @ξi 2 1 1 = gjkδ ξ + gjkξ δ 2 ij k 2 j ik ij = g ξj dξ @ 1 i = − gjkξ ξ dt @xi 2 j k 1 @gjk = − ξ ξ : 2 @xi j k Differentiating the first equation with respect to t and using both of Hamilton's equations yields d2xi @gij dxk dξ = ξ + gim m dt2 @xk dt j dt @ 1 @ = gkl gij ξ ξ − gim gnr ξ ξ : @xk l j 2 @xm n r 7Is there a coordinate-free proof? See Paternain's book.
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