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Symplectic Geometry SYMPLECTIC GEOMETRY Eckhard Meinrenken Lecture Notes UniversityofToronto These are lecture notes for two courses taughtattheUniversityofToronto in Spring and in Fall Our main sources have b een the b o oks Symplectic Techniques by GuilleminSternb erg and Intro duction to Symplectic Top ology by McDuSalamon and the pap er Stratied symplectic spaces and reduction Ann of Math by SjamaarLerman Contents Chapter Linear symplectic algebra Symplectic vector spaces Subspaces of a symplectic vector space Symplectic bases Compatible complex structures The group SpE of linear symplectomorphisms Polar decomp osition of symplectomorphisms Maslov indices and the Lagrangian Grassmannian The index of a Lagrangian triple Linear Reduction Chapter Review of Dierential Geometry Vector elds Dierential forms Chapter Foundations of symplectic geometry Denition of symplectic manifolds Examples Basic prop erties of symplectic manifolds Chapter Normal Form Theorems Mosers trick Homotopy op erators Darb ouxWeinstein theorems Chapter Lagrangian brations and actionangle variables Lagrangian brations Actionangle co ordinates Integrable systems The spherical p endulum Chapter Symplectic group actions and moment maps Background on Lie groups Generating vector elds for group actions Hamiltonian group actions Examples of Hamiltonian Gspaces CONTENTS Symplectic Reduction Normal forms and the DuistermaatHeckman theorem The symplectic slice theorem Chapter Hamiltonian torus actions The AtiyahGuilleminSternb erg convexitytheorem Some basic MorseBott theory Lo calization formulas Frankels theorem Delzant spaces CHAPTER Linear symplectic algebra Symplectic vector spaces Let E b e a nitedimensional real vector space and E its dual The space E can b e identied with the space of skewsymmetric bilinear forms E E R v w w v Definition The pair E is called a symplectic vector space if E is nondegenerate that is if the kernel ker fv E j v w for all w E g is trivial Two symplectic vector spaces E andE are called symplectomorphic if there is an isomorphism A E E with A The group of symplectomor phisms of E is denoted Sp E Since SpE is a closed subgroup of GlE it is by a standard theorem of Lie group theory a Lie subgroup n Example Let E R with basis fe e f f gThen n n e e f f e f i j i j i j ij denes a symplectic structure on E Examples of symplectomorphisms are Ae j f Af e or Ae e f Af f Also j j j j j j j j X X Ae B e Af B f j jk k j kj k k k for anyinvertible n nmatrix B is a symplectomorphism Example Let V bearealvector space of dimension nandV its dual space Then E V V has a natural symplectic structure v v v v If B V V is any isomorphism and B V V the dual map B B E E is a symplectomorphism Example Let E b e a complex vector space of complex dimension n with com plex p ositive denite inner pro duct Hermitian metric h E E C Then E viewed as a real vector space with bilinear form the imaginary part Imhisa symplectic vector space Every unitary map E E preserves h hence also and is therefore symplectic LINEAR SYMPLECTIC ALGEBRA Exercise Show that these three examples of symplectic vector spaces are in fact symplectomorphic Subspaces of a symplectic vector space Definition Let E b e a symplectic vector space For any subspace F E we dene the p erp endicular space F by F fv E v w for all w F g With our assumption that E is nite dimensional is nondegenerate if and only if the map E E h v wi v w is an isomorphism F is the preimage of the annihilator ann F E under From this it follows immediately that dim F dim E dim F and F F Definition A subspace F E of a symplectic vector space is called a isotropic if F F b coisotropic if F F c Lagrangian if F F d symplectic if F F fg The set of Lagrangian subspaces of E is called the Lagrangian Grassmannian and denoted Lag E Notice that F is isotropic if and only if F is coisotropic For example every dimensional subspace is isotropic and every co dimension subspace is coisotropic n Example In the ab ove example E R letL spanfg g g where for n all i g e or g f ThenL is a Lagrangian subspace i i i i Lemma For any symplectic vector space E there exists a Lagrangian sub space L Lag E Proof Let L b e an isotropic subspace of E which is maximal in the sense that it is not contained in any isotropic subspace of strictly larger dimension Then L is Lagrangian For if L L then cho osing any v L nL would pro duce a larger isotropic subspace L span v An immediate consequence is that any symplectic vector space E has even dimension For if L is a Lagrangian subspace dim E dimL dimL dim L Lemma can b e strengthened as follows Lemma Given any nite col lection of Lagrangian subspaces M M one r can nd a Lagrangian subspace L with L M fg for al l j j COMPATIBLE COMPLEX STRUCTURES Proof Let L b e an isotropic subspace with L M fg and not prop erly contained j in a larger isotropic subspace with this prop ertyWe claim that L is Lagrangian If not L is a coisotropic subspace prop erly containing LLet L L L b e the quotient map Cho ose any dimensional subspace F L L suchthatboth F is transversal to all M L This is p ossible since M L is isotropic and therefore has p ositive j j co dimension Then L F is an isotropic subspace with L L and L M fg j This contradiction shows L L Symplectic bases Theorem Every symplectic vector space E of dimension n is symplecto n morphic to R with the standard symplectic form from Example Proof Picktwo transversal Lagrangian subspaces L M Lag E The pairing L M R v w v w is nondegenerate In other words the comp osition E L M E where the last map is dual to the inclusion L E is an isomorphism Let e e n M By denition of the pairing b e a basis for L and f f the dual basis for L n is given in this basis by Definition Abasis fe e f f g of E for which has the stan n n dard form is called a symplectic basis Our pro of of Theorem has actually shown a little more Corollary Let E i be two symplectic vector spaces of equal di i i mension and L M Lag E such that L M fg Then there exists a symplecto i i i i i morphism A E E such that AL L and AM M Compare with isometries of inner pro duct spaces which are much more rigid In the following section wegive an alternative pro of of Theorem using complex structures Compatible complex structures Recall that a complex structure on a vector space V is an automorphism J V V such that J Id Definition A complex structure J on a symplectic vector space E is called compatible if g v w v J w denes a p ositive denite inner pro duct This means in particular that J is a symplec tomorphism J v w JvJw g Jvw g w J v w J v w v v w We denote by J E the space of compatible complex structures LINEAR SYMPLECTIC ALGEBRA We equip J E with the subset top ology induced from EndE Later we will see that it is in fact a smo oth submanifold Example In Example a compatible almost complex structure J is given by n n Je f Jf e This identies R Jwith C i i i i A compatible complex complex structure makes E into a Hermitian vector space complex inner pro duct space with Hermitian metric p hv w g v w v w That is h is complexlinear with resp ect to the second entry and complexantilinear with resp ect to the rst entry p p hv whJvw hv w hv J w and hv v for v We will show b elowthatJ E Assuming this for a moment let J JE and pick an orthonormal complex basis e e Letf Je n i i Then e e f f is a symplectic basis n n p e f Imhe Je Im he e e e Imhe e i j i j i j ij i j i j and similarly f f This is the promised alternative pro of of Theorem i j The next Theorem gives a convenient metho d for constructing compatible complex structures For anyvector space V let RiemV denote the convex op en subset of the space S V of symmetric bilinear forms consisting of p ositive denite inner pro ducts Theorem Let E be a symplectic vector space Thereisacanonical contin uous surjective map F RiemE JE The map G J E RiemE J g associating to each compatible complex struc ture the corresponding Riemannian structureisasection ie F GJ J Proof Given k RiemE letA GlE b e dened by k v w v Aw T Since is skewsymmetric A is skewadjoint with resp ect to k A A It follows T that in the p olar decomp osition A J jAj with jAj A A A J and jAj commute Therefore J Id The equation v J w v AjAj w k v jAj w k jAj v jAj w shows that g v w v J w denes a p ositive denite inner pro duct Wethus obtain acontinuous map F RiemE JE By construction it satises F G id in particular it is surjective Corollary The space J E is contractible In particular any two com patible complex structures can be deformed into each other THE GROUP Sp E OF LINEAR SYMPLECTOMORPHISMS Proof Let X RiemE and Y J E The space X is contractible since itisaconvex subset of a vector space Cho ose a contraction I X X where Id and is the map onto some p ointinX Then F Id Gisthe X required retraction of Y Given a Lagrangian subspace L of E any orthonormal basis e e of L is a an n orthornormal basis for E viewed as a complex Hermitian vector space The map taking this to an orthonormal basis e e of L Lag E is unitary Hence UE acts n transitively on the set of Lagrangian subspaces The stabilizer in UE ofL Lag E is canonically identied
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