SYMPLECTIC GEOMETRY: LECTURE 3

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1 SYMPLECTIC GEOMETRY: LECTURE 3 LIAT KESSLER 1. Local forms Vector fiels an the Lie erivative. A vector fiel on a manifol M is a smooth assignment of a vector tangent to M at each point. We think of M as embee in R N ; a vector fiel then is a smooth map v : M R N such that v(p) T p M for every p. More generally, a time-epenent vector fiel is a smooth family of vector fiels v t, t R. We think of a vector fiel v as v t such that v t = v for all t R. Imagine that we allow every point p M to flow along the vector fiel v for some time t, we thus obtain a map of M whose fixe points (if t is small enough) are exactly the zeros of v. The flow of v t is the map Φ: M R M, Φ t (p) := Φ(p, t), with Φ 0 = I M an each Φ t : M M is a iffeomorphism (such a map is calle an isotopy), such that t Φ t = v t Φ t. (Locally exists by Picar s theorem of the existence an uniqueness of solutions of first-orer equations with given initial conitions, globally exists when M is compact or v is of compact support.) Examples (raw): M = R, v = γ R, Φ t (m) = e γt m, M = S 1, v = iγ, Φ t (m) = e iγt m, M = S 2, v = iγ, Φ t (m) = e iγt m. Note that in the last example the flow is along the geoesic (locally minimizing length of constant velocity) curve starting from x whose tangent at x is v; this generalizes to (locally) efine a flow given a Riemann metric on a manifol, using the exponential map, in case v t = v is inepenent of t. The Lie erivative of a time epenent vector fiel v t is L v : Ω k (M) Ω k (M), L v α = t (Φ t) α t=0. 1

2 2 LIAT KESSLER (For the efinition of the Lie erivative it is enough that the isotopy exists locally, by Picar s theorem.) We will use the properties of the Lie erivative. Cartan s magic formula L v α = ι v α + ι v α, t Φ t α = Φ t L vt α. For a guie exercise towars the proof that the properties hol, see p. 42 in [2]. Using the chain rule an the linearity of the pullback, we euce a stronger version of the secon property Lemma. For a smooth family ω t, t R of k-forms we have ( (1.2) t Φ t ω t = Φ t L vt ω t + ) t ω t. Proof: [2, Proposition 6.4, pp ]. Moser s trick Lemma (Moser s trick). Let ω t, 0 t 1 be a smooth family of symplectic forms on a manifol M. Assume that (1) t ω t = α t for a smooth family of 1-forms on M, an (2) there is a vector fiel v t such that ι vt ω t = α t, an an isotopy φ t, t R such that t φ t = v t φ t an φ 0 = I. Then φ t ω t = ω 0. If a Lie group G acts on M, an the action is symplectic with respect to ω t an α t for every t, then v t (efine by ω t an α t ) is G-equivariant an so is its flow, φ t i.e., for every t we have φ t (g.x) = g.φ t (x) for all g G. Proof. ( t (φ t ω t ) = φ t L vt ω t + ) t ω t ) = φ t (ι vt ω t + ι vt ω t + α t = 0 The first equality is by (1.2), the secon equality is by Cartan s magic formula, an the thir is by the setting of v t an α t an the assumption that ω t are symplectic an in particular close. We first apply Moser s trick to get symplectomorphisms on a compact manifol M.

3 SYMPLECTIC GEOMETRY: LECTURE Corollary. Assume that M is compact an that [ω 0 ] = [ω 1 ] an the 2-form ω t = (1 t)ω 0 + tω 1 is symplectic for each t [0, 1]. Then there exists an isotopy φ: M R M such that φ t ω t = ω 0 for all t [0, 1]. In particular φ 1ω 1 = ω 0. If a compact Lie group G acts on M, an the action is symplectic with respect to ω t for every t, then φ t is G-equivariant for every t. Proof. Since [ω 0 ] = [ω 1 ] there exists a 1-form α such that ω 1 ω 0 = α. If a compact Lie group G acts on M, an the action is symplectic with respect to ω 0, ω 1 then for every a G we have τ a α = τ a α = τ a (ω 0 ω 1 ) = ω 1 ω 0. Averaging α w.r.t the G-action, we get the 1-form α(τ a u)τ a, G for u T M, where τ a is given by any measure on the group G, invariant by left translations, e.g., the Haar measure. By the above equation, G α(τ a u)τ a = ω 1 ω 0, so we can assume that α is G- invariant. We get that t ω t = t (1 t)ω 0 + tω 1 = ω 1 ω 0 = α. So it remains to fin v t such that ι vt ω t = α. By the non-egeneracy of ω t we can solve this pointwise, to obtain a unique (smooth) v t. The existence of the flow integrating v t is guarantee since M is compact. If ω t an α are G-invariant then so is v t an so is the flow φ: M R M integrating v t Note that if ω 0 an ω 1 are symplectic forms on M then ω t = (1 t)ω 0 +tω 1 are necessarily close but not necessarily non-egenerate. However (as we showe in PS2), if ω 0 an ω 1 are compatible with a fixe almost complex structure J on M, then ω t is non-egenerate If N is a compact complex manifol an h 0, h 1 Hermitian metrics on M, then for every t [0, 1], h t = (1 t)h 0 + th 1

4 4 LIAT KESSLER is again a Hermitian structure (CHECK). Therefore is symplectic for every t [0, 1]. ω t = Im h t = (1 t)ω 0 + tω 1 As a result of the last observation an Corollary 1.4, we get the following theorem Theorem (Banyaga). Let N be a compact complex manifol an h 0, h 1 Hermitian metrics on N (w.r.t to the given complex structure). Assume that [ω 0 ] = [ω 1 ]. then there exists a iffeomorphism φ of N (as a real manifol) such that φ ω 1 = ω 0. Darboux an Weinstein Theorems. By the Darboux theorem, the imension is the only local invariant of symplectic manifols, up to symplectomorphisms Theorem (Darboux Theorem). Let (M, ω) be a 2n-imensional symplectic manifol, an let p be any point in M. Then there is a coorinate chart (U, x 1,..., x n, y 1,..., y n ), with p U, such that on U, ω = n i=1 x i y i. Such a chart is calle a Darboux chart. Darboux first prove the theorem using inuction on imension. The proof given here is ue to Weinstein, an is vali in many infiniteimensional cases. Another recent proof is one of the suggeste topic for stuents presentations [1]. The proof applies Moser s trick to get a symplectomorphism, not on the manifol as in Corollary 1.4 but on a neighbourhoo of a point. We will also use Poincaré lemma Lemma (Poincaré Lemma). Let U R 2n be an open ball aroun 0. Then for every close form α Ω k (U) with k > 0 there is β Ω k 1 (U) such that α = β. Proof of the Darboux theorem. First, consier the exponential exp = exp p : T p M M of some Riemann metric on M. (The exponential map sens v T p M to γ v (1) where γ v is the unique geoesic (locally length-minimizing curve) satisfying γ v (0) = p an the initial tangent vector γ v(0) = v. Note that exp is locally efine since Picar s theorem of the existence an uniqueness of solutions of first-orer equations with given initial conitions hols locally.) The map exp is a iffeomorphism between a neighbourhoo V 0 of 0 in the vector space T p M an a neighbourhoo U p of p in the manifol M. Wlg, V 0 is an open ball.

5 SYMPLECTIC GEOMETRY: LECTURE 3 5 We get two symplectic forms on V 0 : ω 0 = ω p an ω 1 = exp ω. Then ω 0 an ω 1 agree at the origin. By Poincaré lemma, there is β Ω 1 (V 0 ) such that ω 1 ω 0 = β. Since ω 0 an ω 1 agree at 0, the form β 0 = 0. Let ω t = ω 0 + tβ. For 0 t 1, the 2-form ω t is close an agrees with ω 0 at 0 hence nonegenerate at 0. Non-egeneracy of a 2-form σ on a 2n-manifol means that σ n is nowhere vanishing, hence is an open conition. Therefore, shrinking V 0 if necessary, we may assume that ω t is non-egenerate on U for all t [0, 1]. The non-egeneracy of ω t implies, as before, that there is a unique vector fiel v t such that ι vt ω t = β. By shrinking V 0 we can assume that v t integrates to a flow φ t. Furthermore, since β 0 = 0 also v t 0 = 0 hence v t is small near 0, there is a neighbourhoo 0 V V 0 such that the flow φ t integrating v t in V oes not map V out of V 0 for t [0, 1]. Since v t 0 = 0 we have φ t (0) = 0. Now apply Moser s trick to show that φ t ω t = ω 0, hence φ 1(exp ω) = φ 1 ω 1 = ω 0 = ω p. The chart exp φ transforms coorinates of T p M with respect to an ω x -symplectic basis into local coorinates of M in U in which ω = n i=1 x i y i Theorem (Moser relative theorem). Let M be a manifol, ι: X M a compact submanifol of M, an ω 0, ω 1 symplectic forms on M. Assume that ω 0 X = ω 1 X. Then there exist neighbourhoos U 0, U 1 of X in M an a iffeomorphism Φ: U 0 U 1 such that ι = Ψ ι on X, an Φ ω 1 = ω 0 on U 0. If G is a compact Lie group acting on M with ω 0 an ω 1 both invariant uner the G-action, an if Y is invariant uner the action, then the map Φ can be chosen to commute with the action of G Notation. If σ is a ifferential form on M then σ X enotes the restriction of σ to (T M) X. Thus σ X can be evaluate on vectors that are not necessarily tangent to X. The proof is similar to the proof of Darboux theorem: the ientification of a neighbourhoo of a point p in M with a neighbourhoo of 0 in T p M will be replace with the tubular neighbourhoo theorem,

6 6 LIAT KESSLER using the exponential map, an Poincaré lemma will be generalize to the homotopy formula. See [2, 6.2, 6.3] Recall: The normal bunle of a k-imensional submanifol ι : X M of an n-imensional manifol M (with k < n) is NX = {(x, v) x X, v T x M/T x X}, where we ientify x with ι(x) an consier the tangent space to X as a subspace of the tangent space to M through the linear inclusion ι x : T x X T x M. The quotient N x X := T x M/T x X is an n k- im vector space. NX is a vector bunle over X with the projection π : N X, π(x, v) = x, hence a manifol of imension n. The zero section ι 0 : X NX; x (x, 0) embes X as a close submanifol of NX A neighbourhoo U 0 of the zero section X is calle convex if for every x X, the intersection U 0 N x X is convex. Note that on a convex neighbourhoo U of X in NX we can set φ 0 (x, v) = (x, 0), φ 1 (x, v) = (x, v), φ t (x, v) = (x, tv) for 0 t 1 an get a smooth retraction φ: U 0 [0, 1] U 0 of U 0 onto X (well efine since U 0 is convex an inclues the zero section). Let v t be the time-epenent vector fiel on U 0 that generate φ t, i.e., φ t t = v t φ t, v t (p) is the tangent vector to the curve φ(p, ) at t. For a form σ on U 0 we have the homotopy formula σ φ 0σ = where 1 0 t (φ t σ)t = 1 (1.14) Iβ = an 0 φ t L vt σ = (φ t (ι(v t )β)t φ t (ι vt σ+ι vt σ)t = Iσ+Iσ, φ t ι(v t )σ(u 1,..., u k ) = (ι vt(p)σ)(φ t u 1,..., φ t u k ). The first equality is by the funamental theorem of calculus an the fact that φ 1 is the ientity map, the secon equality is by the pullback property property of the Lie erivative state last week, the thir equality is by Cartan s magic formula, an the fourth is by the funamental theorem of calculus again. This is the homotopy formula in a convex neighbourhoo of X in NX. In particular, if σ is close an φ 0σ = 0 then σ = (Iσ). Moreover, since φ t (x, 0) = (x, 0) for every t, we get that v t X = 0, hence (Iσ) X = 0.

7 SYMPLECTIC GEOMETRY: LECTURE Homotopy operator. The operator (1.14) is a special case of a homotopy operator. In general, in the category of smooth manifols, let f 0, f 1 : M 1 M 2, such that there is a smooth homotopy f : [0, 1] M 1 M 2 between f 0 an f 1. Then there is a chain homotopy I k : Ω k (M 2 ) Ω k 1 (M 1 ) such that the homotopy formula (1.16) I + I = f 1 f 0 hols. I k 1 Ω k 1 (M 2 ) f 0 f 1 Ω k 1 (M 1 ) The operator I is efine by I(σ) = I k Ω k (M 2 ) f 0 f 1 Ω k (M 1 ) [0,1] f σ. I k+1 Ω k+1 (M 2 ) f 0 f 1 Ω k+1 (M 1 ) The homotopy formula (1.16) follows from the funamental theorem of calculus, as seen in the special case (1.14). If σ is close, we euce that f 1 σ f 0 σ = (Iσ). We conclue that if there is a homotopy between f 0 an f 1, then f 0 an f 1 inuce the same map on cohomology. In the spacial case that M 1 = {0} an M 2 = U R n is an open ball aroun 0 in R n, let ι: {0} U be the inclusion, an π : U {0} be the projection. We have a homotopy f t : U U, u tu, 0 t 1 such that f 0 = ι π an f 1 = I, i.e., a retraction of U onto {0}. Using the homotopy formula, we get that π ι = (ι π) an I inuce the same map on cohomology. Therefore ι : H k (U) H k ({0}) is an isomorphism with inverse π. Hence, for k > 0, every close k-form on U is exact. This proves Poincaré lemma 1.9 that we use in the proof of Darboux theorem To get a convex neighbourhoo of X in NX an a iffeomorphism from U 0 to a neighbourhoo U, we choose a Riemann metric g on M an use the exponential map. Note that the vector space N x X is ientifie with the orthogonal complement {v T x M g x (v, w) = 0 for any w T x X}. DRAW. Let NX ɛ = {(x, v) NX g x (v, v) < ɛ}. This is a convex neighbourhoo of X in N X (CHECK, use Cauchy-Schwarz inequality). Consier the exponential map exp: NX ɛ M that sens (x, v) to γ(1) where γ : [0, 1] M is the geoesic (locally minimizing length of constant velocity) curve starting from x whose tangent γ (0) = v at 0 is v. By Picar s uniqueness an existence theorem, if t X is a compact submanifol of M, then for ɛ small enough, the map

8 8 LIAT KESSLER exp is well efine. Then exp maps NX ɛ iffeomorphically to a neighbourhoo U ɛ of X in M, an is the ientity on the zero section X. (If X is not compact, replace ɛ by a continuous map X R + that tens to zero fast enough as x tens to infinity.) This is the tubular neighbourhoo theorem, [2, Theorem 6.5]. It allows us to apply the homotopy formula in a tubular neighbourhoo of X. Proof of Moser s relative theorem. On a tubular neighbourhoo U 0 of X, the 2-form ω 1 ω 0 is close an its restriction to X is zero. By the homotopy formula, there is a 1-form α on U 0 such that ω 1 ω 0 = α an α X = 0. Now, as in the proof of Darboux theorem set ω t = (1 t)ω 0 + tω 1 = ω 0 + tα. By shrinking U 0 we can assume that ω t is symplectic 0 t 1, so we can solve the Moser equation ι vt ω t = α with v t = 0 on X, an integrate v t to a flow ρ t. By Moser s trick, ρ t is an isotopy between ω 0 an ω 1, i.e., ρ t ω t = ω 0 for every 0 t 1. Now let U 1 = ρ 1 (U 0 ) an Φ = ρ If a compact Lie group acts on U an X is invariant uner the G-action, we can choose an invariant Riemann metric on U 0. Such a metric is obtaine from some Riemmann metric g by averaging with respect to the compact group G action to get g(u, v) = g (τ a u, τ a v)τ a, G for u, v T x M, where τ a is given by any measure on the group G, invariant by left translations, e.g., the Haar measure. Then the exponential map an hence the retraction φ t will commute with the action of G, an hence so will I, i.e., Iτ a β = τ a Iβ for every a G. This gives the equivariant tubular neighbourhoo an homotopy formula, an therefore an equivariant version of Moser s relative theorem Remark. Note that Darboux theorem has two parts: (1) Locally there is a change of coorinates so that the transforme symplectic form is constant. (2) There is a further change of coorinates yieling the canonical symplectic form x i y i. (This is the existence of a symplectic basis result in symplectic linear algebra we showe in the first lectures.) The equivariant version of the first part, the locally constant result still hols, moreover, the equivariant version of Moser relative theorem hols, as note before. However the secon part is not correct in the equivariant setting. This is relate to the representation of the group.

9 SYMPLECTIC GEOMETRY: LECTURE 3 9 For more etails see [3], which is one of the suggeste topics for a stuent presentation. There are groups with non-isomorphic invariant symplectic forms on any neighbourhoo of a fixe point Theorem (Weinstein s tubular neighbourhoo theorem). Let (M, ω) be a symplectic manifol an ι: L M a compact Lagrangian submanifol. Consier T L with the canonical symplectic form ω 0 an L as the zero section in T L, embee by ι 0 : L T L. Then there exist neigbourhoos U 0 of L in T L, U 1 of L in M an a iffeomorphism θ : U 0 U 1 such that θ ι 0 = ι an θ ω = ω Proposition. Let L be a Lagrangian sumanifol of a symplectic manifol (M, ω) Then the vector bunles NL an T L are canonically ientifie. Proof. For p L efine ω p : T p M/T p L (T p L), ω p ([v])( ) = ω p (v, ), where [v] is the equivalence class of v in T p M/T p L. Note that for u 1, u T p L, ω p (v + u 1, u) = ω p (v, u) since T p L is a Lagrangian, in particular isotropic, subspace of (T p M, ω p ). Hence ω p ([v]) is well efine. Moreover ω p is an isomorphism, since T p L is Lagrangian, i.e., T p L coincies with its symplectic orthogonal complement (T p L) ωp. Proof of Weinstein s tubular neighbourhoo theorem. By Proposition 1.21, T L = NL, so we can ientify the canonical form ω 0 with a symplectic form (to be enote again ω 0 ) on NL. By the tubular neighbourhoo theorem in 1.17, there is a neighbourhoo V of L in M, a convex neighbourhoo U of L in NL = T M L /T L, an a iffeomorphism exp: U V, etermine by a Riemannian metric on M, such that exp ι 0 = ι. Since L is Lagrangian both in T L (as the zero section) an in M, we have (1.22) exp ω T L = ω 0 T L = 0. However, to apply Moser s relative theorem we nee exp ω L = ω 0 L as 2-forms evaluate on vectors in T U = T (T L) that are not necessarily tangent to L. To get this, we can choose the Riemannian metric on M wisely: take it to be g(u, v) = ω(u, Jv) where J is an ω-compatible almost complex structure on M. Now, for p L, the space N p L is ientifie with the orthogonal complement of T p L w.r.t g, which equals JT p L (CHECK), hence T p V = T p L JT p L. We get, by choosing an orthonormal basis w.r.t the Hermitian form h(u, v) = ω p (u, Jv) +

10 10 LIAT KESSLER 1ωp (u, v) an ientifying it with the cotangent coorinates, that exp ω p = ω 0 p for every p L. Alternatively, we can use (1.22) to fin a linear isomorphism L p : T p U T p U such that L p TpL = I an L p (exp ω p ) = ω 0 p, that varies smoothly in p L. Then, by Whitney s extension theorem, there is an embeing h: U U, of some neighbourhoo U of L, such that h L = I an h p = L p for every p L, hence (h exp ω) p = (h p ) (exp ω p ) = L p (exp ω p ) = ω 0 p. For more etails, see [2, Proof of Theorem 8.4]. Now apply Moser s relative theorem to fin a neighbourhoo U 0 of L in T L an a iffeomorphism ψ : U 0 U 0 T L such that ψ exp ω = ω 0 an ψ ι 0 = ι 0. The iffeomorphism θ = exp ψ : U 0 exp(u 0 ) M satisfies the require properties θ ω = ω 0 an θ ι 0 = ι. References [1] H. Bursztyn, H. Lima, E. Meinrenken, Splitting theorems for Poisson an relate structures, to appear in J. Reine Angew. Math. arxiv: [2] Ana Cannas a Silva, Lectures on Symplectic Geometry, Lecture Notes in Mathematics, Springer-Verlag, [3] M. Dellnitz an I. Melbourne, The equivariant Darboux theorem, Exploiting Symmetry in Applie an Numerical Analysis (Eitors: E. Allgower et al): 1992 AMS-SIAM Summer Seminar Proceeings. Lectures in Appl. Math. 29 (1993), [4] V. Guillemin, S. Sternberg, Symplectic Techniques in Physics, Cambrige University Press [5] Eckhar Meinrenken, Symplectic Geometry - Lecture Notes, University of Toronto.