LECTURE1:LINEARSYMPLECTICGEOMETRYCONTENTS31.Linear symplectic structure52.Distinguished subspaces73.Linear complex structure104.Thesymplecticgroup**************Information:CourseName:SymplecticGeometryInstructor: Zuoqin WangTime/Ro0m:Wed.2:00pm-6:00pm@1318Reference books:.Lectures on Symplectic Geometry by A. Canas de Silver?Symplectic Techniques in Physics by V.Guillemin and S.Sternberg. Lectures on Symplectic Manifolds by A. Weinstein.Introduction to Symplectic Topology by D.McDuff and D.Salamon.Foundations of MechanicsbyR.Abrahamand J.Marsden.GeometricQuantizationbyWoodhouseCourse webpage:http://staff.ustc.edu.cn/~wangzuoq/Symp15/SympGeom.html*******************************?Introduction:The word symplectic was invented by Hermann Weyl in 1939:he replaced theLatin roots in the word complex, com-plexus, by the corresponding Greek rootssym-plektikos.What is symplectic geometry?.Geometry=background space (smoothmanifold)+extra structure (tensor).-Riemannian geometry = smooth manifold + metric structure*metricstructure=positive-definitesymmetric2-tensor-Complexgeometry=smoothmanifold+complex structure.* complex structure = involutive endomorphism (1,1)-tensor)Symplecticgeometry=smoothmanifold+symplecticstructure
LECTURE 1: LINEAR SYMPLECTIC GEOMETRY Contents 1. Linear symplectic structure 3 2. Distinguished subspaces 5 3. Linear complex structure 7 4. The symplectic group 10 ********************************************************************************* Information: Course Name: Symplectic Geometry Instructor: Zuoqin Wang Time/Room: Wed. 2:00pm-6:00pm @ 1318 Reference books: • Lectures on Symplectic Geometry by A. Canas de Silver • Symplectic Techniques in Physics by V. Guillemin and S. Sternberg • Lectures on Symplectic Manifolds by A. Weinstein • Introduction to Symplectic Topology by D. McDuff and D. Salamon • Foundations of Mechanics by R. Abraham and J. Marsden • Geometric Quantization by Woodhouse Course webpage: http://staff.ustc.edu.cn/˜ wangzuoq/Symp15/SympGeom.html ********************************************************************************* Introduction: The word symplectic was invented by Hermann Weyl in 1939: he replaced the Latin roots in the word complex, com-plexus, by the corresponding Greek roots, sym-plektikos. What is symplectic geometry? • Geometry = background space (smooth manifold) + extra structure (tensor). – Riemannian geometry = smooth manifold + metric structure. ∗ metric structure = positive-definite symmetric 2-tensor – Complex geometry = smooth manifold + complex structure. ∗ complex structure = involutive endomorphism ((1,1)-tensor) – Symplectic geometry = smooth manifold + symplectic structure 1
2LECTURE1:LINEARSYMPLECTICGEOMETRY* Symplectic structure = closed non-degenerate 2-form:2-form=anti-symmetric 2-tensor-Contactgeometry=smooth manifold+contact structure*contactstructure=localcontact1-form. Symplectic geometry v.s. Riemannian geometryVery different (although the definitions look similar)* All smooth manifolds admit a Riemannian structure, but onlysome of them admit symplectic structures.* Riemannian geometry is very rigid (isometry group is small), whilesymplectic geometry is quite soft (the group of symplectomor-phisms is large)* Riemannian manifolds have rich local geometry (curvature etc).whilesymplecticmanifoldshavenolocalgeometry(Darbouxthe-orem)- Still closely related* Each cotangent bundle is a symplectic manifold.* Many Riemannian geometry objects have their symplectic inter-pretations, e.g. geodesics on Riemannian manifolds lifts to geo-desic flow on their cotangent bundles.Symplecticgeometryv.s.complexgeometry-Many similarities. For example, in complex geometry one combine pairsof real coordinates (r,y) into complex coordinates z= r + iy. In sym-plectic geometry one has Darboux coordinates that play a similar role.. Symplectic geometry v.s. contact geometrycontact geometry = the odd-dim analogue of symplectic geometry.Symplectic geometry v.s. analysis-Symplecticgeometryis a languagewhichcanfacilitatecommunicationbetween geometry and analysis (Alan Weinstein).(LAST SEMESTER) Quantization: one can construct analytic objects(e.g. FIOs) from symplectic ones (e.g. Lagrangians)..Symplectic geometry v.s.algebraThe orbit method (Kostant,Kirillov etc)in constructing Liegroup rep-resentations uses symplectic geometry in an essential way: coadjointorbits are naturally symplectic manifolds.-→ geometric quantization.Symplectic geometry v.s.physicsmathematics is created tosolvespecificproblems inphysicsandprovidesthe very language in which the laws of physics are formulated. (VictorGuillemin and Shlomo Sternberg)+general relativity* Riemannian geometry←*Symplecticgeometry←classical mechanics (and quantum me-chanics via quantization),geometrical optics etc.-Symplecticgeometryhas its origin in physics
2 LECTURE 1: LINEAR SYMPLECTIC GEOMETRY ∗ Symplectic structure = closed non-degenerate 2-form · 2-form = anti-symmetric 2-tensor – Contact geometry = smooth manifold + contact structure ∗ contact structure = local contact 1-form • Symplectic geometry v.s. Riemannian geometry – Very different (although the definitions look similar) ∗ All smooth manifolds admit a Riemannian structure, but only some of them admit symplectic structures. ∗ Riemannian geometry is very rigid (isometry group is small), while symplectic geometry is quite soft (the group of symplectomorphisms is large) ∗ Riemannian manifolds have rich local geometry (curvature etc), while symplectic manifolds have no local geometry (Darboux theorem) – Still closely related ∗ Each cotangent bundle is a symplectic manifold. ∗ Many Riemannian geometry objects have their symplectic interpretations, e.g. geodesics on Riemannian manifolds lifts to geodesic flow on their cotangent bundles • Symplectic geometry v.s. complex geometry – Many similarities. For example, in complex geometry one combine pairs of real coordinates (x, y) into complex coordinates z = x + iy. In symplectic geometry one has Darboux coordinates that play a similar role. • Symplectic geometry v.s. contact geometry – contact geometry = the odd-dim analogue of symplectic geometry • Symplectic geometry v.s. analysis – Symplectic geometry is a language which can facilitate communication between geometry and analysis (Alan Weinstein). – (LAST SEMESTER) Quantization: one can construct analytic objects (e.g. FIOs) from symplectic ones (e.g. Lagrangians). • Symplectic geometry v.s. algebra – The orbit method (Kostant, Kirillov etc) in constructing Lie group representations uses symplectic geometry in an essential way: coadjoint orbits are naturally symplectic manifolds. −→ geometric quantization • Symplectic geometry v.s. physics – mathematics is created to solve specific problems in physics and provides the very language in which the laws of physics are formulated. (Victor Guillemin and Shlomo Sternberg) ∗ Riemannian geometry ←→ general relativity ∗ Symplectic geometry ←→ classical mechanics (and quantum mechanics via quantization), geometrical optics etc. – Symplectic geometry has its origin in physics
3LECTURE1:LINEARSYMPLECTICGEOMETRY* Lagrange's work (1808)on celestial mechanics, Hamilton, Jacobi,Liouville,Poisson,Poincare.Arnold etc.-An old name of symplectic geometry: the theory of canonical transformationsIn this course, we plan to cover.Basic symplectic geometry-Linear symplectic geometry Symplectic manifolds- Local normal forms- Lagrangian submanifolds v.s. symplectomorphisms-Related geometric structures-Hamiltonian geometry.Symplectic group actions (= symmetry in classical mechanics)-Themoment mapSymplectic reduction-The convexity theorem-Toricmanifolds.Geometric quantization-Prequantization-Polarization一Geometric quantization***************************************************1.LINEAR SYMPLECTICSTRUCTURE Definitions and examples.Let V be a (finite dimensional) real vector space and 2: V × V-→R a bilinearmap. 2 is called anti-symmetric if for all u, v e V,(1)2(u, v) = -2(v, u).It is called non-degenerate if the associated map: V→V*, 2(u)(u) =2(u, v)(2)is bijective. Obviously the non-degeneracy is equivalent to the condition2(u,)=0,VE2u=0.Note that one can regard as a linear 2-form 2 E A2(V*) via2(u,0)=tutu2Definition 1.1. A symplectic vector space is a pair (V,2), where V is a real vectorspace, and 2 a non-degenerate anti-symmetric bilinear map. 2 is called a linearsymplectic structure or a linear symplectic form on V
LECTURE 1: LINEAR SYMPLECTIC GEOMETRY 3 ∗ Lagrange’s work (1808) on celestial mechanics, Hamilton, Jacobi, Liouville, Poisson, Poincare, Arnold etc. – An old name of symplectic geometry: the theory of canonical transformations In this course, we plan to cover • Basic symplectic geometry – Linear symplectic geometry – Symplectic manifolds – Local normal forms – Lagrangian submanifolds v.s. symplectomorphisms – Related geometric structures – Hamiltonian geometry • Symplectic group actions (= symmetry in classical mechanics) – The moment map – Symplectic reduction – The convexity theorem – Toric manifolds • Geometric quantization – Prequantization – Polarization – Geometric quantization ********************************************************************************* 1. Linear symplectic structure ¶ Definitions and examples. Let V be a (finite dimensional) real vector space and Ω : V × V → R a bilinear map. Ω is called anti-symmetric if for all u, v ∈ V , (1) Ω(u, v) = −Ω(v, u). It is called non-degenerate if the associated map (2) Ω : e V → V ∗ , Ω( e u)(v) = Ω(u, v) is bijective. Obviously the non-degeneracy is equivalent to the condition Ω(u, v) = 0, ∀v ∈ Ω =⇒ u = 0. Note that one can regard Ω as a linear 2-form Ω ∈ Λ 2 (V ∗ ) via Ω(u, v) = ιvιuΩ. Definition 1.1. A symplectic vector space is a pair (V, Ω), where V is a real vector space, and Ω a non-degenerate anti-symmetric bilinear map. Ω is called a linear symplectic structure or a linear symplectic form on V
4LECTURE1:LINEARSYMPLECTICGEOMETRYErample.Let V=R2n=Rn×Rn and define2o((r,s), (y,n)) := (r,n) -(s,y))then (V,2o)is a symplectic vector space.Let [ei,..,en,fi,..*,fn) bethe stan-dard basis of Rn × Rn, then 2 is determined by the relationsVi,j.2o(ei,e,)=o(fi,fi)=0,2o(ei,f)=dij,Denote by [ei,... ,en, ft, ... , f) the dual basis of (Rn)* × (R")*, then as a linear2-form onehas2o=e A ft.Erample. More generally, for any finitely dimensional vector space U, the vectorspace V = U @ U* admits a canonical symplectic structure2((u, α), (v, β)) = β(u) - α(v).Erample. For any nondegenerate skew-symmetric 2n × 2n matrix, the 2-form ZA onIR2ndefinedby2A(X,Y) = (X,AY)isasymplecticformonR2n. Linear Darboux theorem.Definition 1.2. Let (Vi,21) and (V2,2) be symplectic vector spaces. A linearmap F : Vi -→ V2 is called a linear symplectomorphism (or a linear canonical trans-formation)if it is a linear isomorphism and satisfies(3)F*22=21.Erample.AnylinearisomorphismL:U→U2liftstoalinearsymplectomorphismF : Ui④ U* → U2 由U2, F(u, α)) = (L(u), (L*)-1(α),It is not hard to check that F is a linear symplectomorphism.Theorem1.3(LinearDarboux theorem).For any linear symplectic wector space(V,2), there erists a basis [ei,..-,en, fi,.., fn] of V so that(4)Vi,j.2(ei,ej)=2(fi, fi)= 0,2(es,fi)=dij,The basis is called a Darboux basis of (V,2)Remark. The theorem is equivalent to saying that given any symplectic vector space(V,), there exists a dual basis (ei, ..,en, fi,..-, fn] of V* so that as a linear 2-form,(5)n-eifti=1
4 LECTURE 1: LINEAR SYMPLECTIC GEOMETRY Example. Let V = R 2n = R n × R n and define Ω0((x, ξ),(y, η)) := hx, ηi − hξ, yi, then (V, Ω0) is a symplectic vector space. Let {e1, · · · , en, f1, · · · , fn} be the standard basis of R n × R n , then Ω is determined by the relations Ω0(ei , ej ) = Ω0(fi , fj ) = 0, Ω0(ei , fj ) = δij , ∀i, j. Denote by {e ∗ 1 , · · · , e∗ n , f ∗ 1 , · · · , f ∗ n} the dual basis of (R n ) ∗ × (R n ) ∗ , then as a linear 2-form one has Ω0 = Xn i=1 e ∗ i ∧ f ∗ i . Example. More generally, for any finitely dimensional vector space U, the vector space V = U ⊕ U ∗ admits a canonical symplectic structure Ω((u, α),(v, β)) = β(u) − α(v). Example. For any nondegenerate skew-symmetric 2n×2n matrix, the 2-form ΩA on R 2n defined by ΩA(X, Y ) = hX, AY i is a symplectic form on R 2n . ¶ Linear Darboux theorem. Definition 1.2. Let (V1, Ω1) and (V2, Ω2) be symplectic vector spaces. A linear map F : V1 → V2 is called a linear symplectomorphism (or a linear canonical transformation) if it is a linear isomorphism and satisfies (3) F ∗Ω2 = Ω1. Example. Any linear isomorphism L : U1 → U2 lifts to a linear symplectomorphism F : U1 ⊕ U ∗ 1 → U2 ⊕ U ∗ 2 , F((u, α)) = (L(u),(L ∗ ) −1 (α)). It is not hard to check that F is a linear symplectomorphism. Theorem 1.3 (Linear Darboux theorem). For any linear symplectic vector space (V, Ω), there exists a basis {e1, · · · , en, f1, · · · , fn} of V so that (4) Ω(ei , ej ) = Ω(fi , fj ) = 0, Ω(ei , fj ) = δij , ∀i, j. The basis is called a Darboux basis of (V, Ω). Remark. The theorem is equivalent to saying that given any symplectic vector space (V, Ω), there exists a dual basis {e ∗ 1 , · · · , e∗ n , f ∗ 1 , · · · , f ∗ n} of V ∗ so that as a linear 2- form, (5) Ω = Xn i=1 e ∗ i ∧ f ∗ i
LECTURE1:LINEARSYMPLECTICGEOMETRY5This is also equivalent to saying that there exists a linear symplectomorphismF : (V,2) → (R2n,20).Inparticular,. Any symplectic vector space is even-dimensional. Any even dimensional vector space admits a linear symplectic form..Up to linear symplectomorphisms, there is a unique linear symplecticformon each even dimensional vector space.Proof of the linear Darbouc theorem. Apply the Gram-Schmidt process with respect口to the linear symplectic form 2. Details left as an exercise. Symplectic volume form.Since a linear symplectic form is a linear 2-form, a natural question is: which2-form in A?(V*) is a linear symplectic form on V?Proposition 1.4.Let V be a 2n dimensional vector space.A linear 2-form EA2(V*) is a linear symplectic form on V if and only if as a 2n-form,(6)2" =...20EA2n(V*).[We will call a symplectic volume form or a Liouville volume form on V.]Proof. If is symplectic, then according to the linear Darboux theorem, one canchoose a dual basis of V* so that 2 is given by (5). It follows"=nleift..enfn0.Conversely, if is degenerate, then there exists u e V so that (u, u) = o forall u e V.Extend u into a basis {ui,...,u2n) of Vwith ui = u.Then sincedim A2n(V) = 1, ui A.... Au2n is a basis of A2n(V). But (2n(ui A ..- A u2n) = 0. So口2n = 0.2. DISTINGUISHED SUBSPACESSymplectic ortho-complement.Nowwe turn to study interesting vector subspaces of a symplectic vector space(V,2). A vector subspace W of V is called a symplectic subspace if 2/wxw is a linearsymplectic form on W. Symplectic subspaces are of course important.However,insymplectic vector spaces there are many other types of vector subspaces that areeven more important.Definition 2.1. The symplectic ortho-complement of a vector subspace W V is(7)w? = (e V I 2(u, w) = o for all w E W).Erample.If (V,2)=(R2n,2o)and W= span[ei,e2,fi,f3],thenW?= spanfe2, f3,es,...,en, fa,..., fn]
LECTURE 1: LINEAR SYMPLECTIC GEOMETRY 5 This is also equivalent to saying that there exists a linear symplectomorphism F : (V, Ω) → (R 2n , Ω0). In particular, • Any symplectic vector space is even-dimensional. • Any even dimensional vector space admits a linear symplectic form. • Up to linear symplectomorphisms, there is a unique linear symplectic form on each even dimensional vector space. Proof of the linear Darboux theorem. Apply the Gram-Schmidt process with respect to the linear symplectic form Ω. Details left as an exercise. ¶ Symplectic volume form. Since a linear symplectic form is a linear 2-form, a natural question is: which 2-form in Λ2 (V ∗ ) is a linear symplectic form on V ? Proposition 1.4. Let V be a 2n dimensional vector space. A linear 2-form Ω ∈ Λ 2 (V ∗ ) is a linear symplectic form on V if and only if as a 2n-form, (6) Ωn = Ω ∧ · · · ∧ Ω 6= 0 ∈ Λ 2n (V ∗ ). [We will call Ωn n! a symplectic volume form or a Liouville volume form on V .] Proof. If Ω is symplectic, then according to the linear Darboux theorem, one can choose a dual basis of V ∗ so that Ω is given by (5). It follows Ω n = n!e ∗ 1 ∧ f ∗ 1 ∧ · · · ∧ e ∗ n ∧ f ∗ n 6= 0. Conversely, if Ω is degenerate, then there exists u ∈ V so that Ω(u, v) = 0 for all v ∈ V . Extend u into a basis {u1, · · · , u2n} of V with u1 = u. Then since dim Λ2n (V ) = 1, u1 ∧ · · · ∧ u2n is a basis of Λ2n (V ). But Ωn (u1 ∧ · · · ∧ u2n) = 0. So Ω n = 0. 2. Distinguished subspaces ¶ Symplectic ortho-complement. Now we turn to study interesting vector subspaces of a symplectic vector space (V, Ω). A vector subspace W of V is called a symplectic subspace if Ω|W×W is a linear symplectic form on W. Symplectic subspaces are of course important. However, in symplectic vector spaces there are many other types of vector subspaces that are even more important. Definition 2.1. The symplectic ortho-complement of a vector subspace W ⊂ V is (7) WΩ = {v ∈ V | Ω(v, w) = 0 for all w ∈ W}. Example. If (V, Ω) = (R 2n , Ω0) and W = span{e1, e2, f1, f3}, then WΩ = span{e2, f3, e4, · · · , en, f4, · · · , fn}.