Symplectic Vector Spaces, Lagrangian Subspaces, and Liouville's Theorem
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Symplectic Linear Algebra Let V Be a Real Vector Space. Definition 1.1. A
INTRODUCTION TO SYMPLECTIC TOPOLOGY C. VITERBO 1. Lecture 1: Symplectic linear algebra Let V be a real vector space. Definition 1.1. A symplectic form on V is a skew-symmetric bilinear nondegen- erate form: (1) ω(x, y)= ω(y, x) (= ω(x, x) = 0); (2) x, y such− that ω(x,⇒ y) = 0. ∀ ∃ % For a general 2-form ω on a vector space, V , we denote ker(ω) the subspace given by ker(ω)= v V w Vω(v, w)=0 { ∈ |∀ ∈ } The second condition implies that ker(ω) reduces to zero, so when ω is symplectic, there are no “preferred directions” in V . There are special types of subspaces in symplectic manifolds. For a vector sub- space F , we denote by F ω = v V w F,ω(v, w) = 0 { ∈ |∀ ∈ } From Grassmann’s formula it follows that dim(F ω)=codim(F ) = dim(V ) dim(F ) Also we have − Proposition 1.2. (F ω)ω = F (F + F )ω = F ω F ω 1 2 1 ∩ 2 Definition 1.3. A subspace F of V,ω is ω isotropic if F F ( ω F = 0); • coisotropic if F⊂ω F⇐⇒ | • Lagrangian if it is⊂ maximal isotropic. • Proposition 1.4. (1) Any symplectic vector space has even dimension (2) Any isotropic subspace is contained in a Lagrangian subspace and Lagrangians have dimension equal to half the dimension of the total space. (3) If (V1,ω1), (V2,ω2) are symplectic vector spaces with L1,L2 Lagrangian subspaces, and if dim(V1) = dim(V2), then there is a linear isomorphism ϕ : V V such that ϕ∗ω = ω and ϕ(L )=L . -
A Guide to Symplectic Geometry
OSU — SYMPLECTIC GEOMETRY CRASH COURSE IVO TEREK A GUIDE TO SYMPLECTIC GEOMETRY IVO TEREK* These are lecture notes for the SYMPLECTIC GEOMETRY CRASH COURSE held at The Ohio State University during the summer term of 2021, as our first attempt for a series of mini-courses run by graduate students for graduate students. Due to time and space constraints, many things will have to be omitted, but this should serve as a quick introduction to the subject, as courses on Symplectic Geometry are not currently offered at OSU. There will be many exercises scattered throughout these notes, most of them routine ones or just really remarks, not only useful to give the reader a working knowledge about the basic definitions and results, but also to serve as a self-study guide. And as far as references go, arXiv.org links as well as links for authors’ webpages were provided whenever possible. Columbus, May 2021 *[email protected] Page i OSU — SYMPLECTIC GEOMETRY CRASH COURSE IVO TEREK Contents 1 Symplectic Linear Algebra1 1.1 Symplectic spaces and their subspaces....................1 1.2 Symplectomorphisms..............................6 1.3 Local linear forms................................ 11 2 Symplectic Manifolds 13 2.1 Definitions and examples........................... 13 2.2 Symplectomorphisms (redux)......................... 17 2.3 Hamiltonian fields............................... 21 2.4 Submanifolds and local forms......................... 30 3 Hamiltonian Actions 39 3.1 Poisson Manifolds................................ 39 3.2 Group actions on manifolds.......................... 46 3.3 Moment maps and Noether’s Theorem................... 53 3.4 Marsden-Weinstein reduction......................... 63 Where to go from here? 74 References 78 Index 82 Page ii OSU — SYMPLECTIC GEOMETRY CRASH COURSE IVO TEREK 1 Symplectic Linear Algebra 1.1 Symplectic spaces and their subspaces There is nothing more natural than starting a text on Symplecic Geometry1 with the definition of a symplectic vector space. -
On the Complexification of the Classical Geometries And
On the complexication of the classical geometries and exceptional numb ers April Intro duction The classical groups On R Spn R and GLn C app ear as the isometry groups of sp ecial geome tries on a real vector space In fact the orthogonal group On R represents the linear isomorphisms of arealvector space V of dimension nleaving invariant a p ositive denite and symmetric bilinear form g on V The symplectic group Spn R represents the isometry group of a real vector space of dimension n leaving invariant a nondegenerate skewsymmetric bilinear form on V Finally GLn C represents the linear isomorphisms of a real vector space V of dimension nleaving invariant a complex structure on V ie an endomorphism J V V satisfying J These three geometries On R Spn R and GLn C in GLn Rintersect even pairwise in the unitary group Un C Considering now the relativeversions of these geometries on a real manifold of dimension n leads to the notions of a Riemannian manifold an almostsymplectic manifold and an almostcomplex manifold A symplectic manifold however is an almostsymplectic manifold X ie X is a manifold and is a nondegenerate form on X so that the form is closed Similarly an almostcomplex manifold X J is called complex if the torsion tensor N J vanishes These three geometries intersect in the notion of a Kahler manifold In view of the imp ortance of the complexication of the real Lie groupsfor instance in the structure theory and representation theory of semisimple real Lie groupswe consider here the question on the underlying geometrical -
Book: Lectures on Differential Geometry
Lectures on Differential geometry John W. Barrett 1 October 5, 2017 1Copyright c John W. Barrett 2006-2014 ii Contents Preface .................... vii 1 Differential forms 1 1.1 Differential forms in Rn ........... 1 1.2 Theexteriorderivative . 3 2 Integration 7 2.1 Integrationandorientation . 7 2.2 Pull-backs................... 9 2.3 Integrationonachain . 11 2.4 Changeofvariablestheorem. 11 3 Manifolds 15 3.1 Surfaces .................... 15 3.2 Topologicalmanifolds . 19 3.3 Smoothmanifolds . 22 iii iv CONTENTS 3.4 Smoothmapsofmanifolds. 23 4 Tangent vectors 27 4.1 Vectorsasderivatives . 27 4.2 Tangentvectorsonmanifolds . 30 4.3 Thetangentspace . 32 4.4 Push-forwards of tangent vectors . 33 5 Topology 37 5.1 Opensubsets ................. 37 5.2 Topologicalspaces . 40 5.3 Thedefinitionofamanifold . 42 6 Vector Fields 45 6.1 Vectorsfieldsasderivatives . 45 6.2 Velocityvectorfields . 47 6.3 Push-forwardsofvectorfields . 50 7 Examples of manifolds 55 7.1 Submanifolds . 55 7.2 Quotients ................... 59 7.2.1 Projectivespace . 62 7.3 Products.................... 65 8 Forms on manifolds 69 8.1 Thedefinition. 69 CONTENTS v 8.2 dθ ....................... 72 8.3 One-formsandtangentvectors . 73 8.4 Pairingwithvectorfields . 76 8.5 Closedandexactforms . 77 9 Lie Groups 81 9.1 Groups..................... 81 9.2 Liegroups................... 83 9.3 Homomorphisms . 86 9.4 Therotationgroup . 87 9.5 Complexmatrixgroups . 88 10 Tensors 93 10.1 Thecotangentspace . 93 10.2 Thetensorproduct. 95 10.3 Tensorfields. 97 10.3.1 Contraction . 98 10.3.2 Einstein summation convention . 100 10.3.3 Differential forms as tensor fields . 100 11 The metric 105 11.1 Thepull-backmetric . 107 11.2 Thesignature . 108 12 The Lie derivative 115 12.1 Commutator of vector fields . -
MAT 531 Geometry/Topology II Introduction to Smooth Manifolds
MAT 531 Geometry/Topology II Introduction to Smooth Manifolds Claude LeBrun Stony Brook University April 9, 2020 1 Dual of a vector space: 2 Dual of a vector space: Let V be a real, finite-dimensional vector space. 3 Dual of a vector space: Let V be a real, finite-dimensional vector space. Then the dual vector space of V is defined to be 4 Dual of a vector space: Let V be a real, finite-dimensional vector space. Then the dual vector space of V is defined to be ∗ V := fLinear maps V ! Rg: 5 Dual of a vector space: Let V be a real, finite-dimensional vector space. Then the dual vector space of V is defined to be ∗ V := fLinear maps V ! Rg: ∗ Proposition. V is finite-dimensional vector space, too, and 6 Dual of a vector space: Let V be a real, finite-dimensional vector space. Then the dual vector space of V is defined to be ∗ V := fLinear maps V ! Rg: ∗ Proposition. V is finite-dimensional vector space, too, and ∗ dimV = dimV: 7 Dual of a vector space: Let V be a real, finite-dimensional vector space. Then the dual vector space of V is defined to be ∗ V := fLinear maps V ! Rg: ∗ Proposition. V is finite-dimensional vector space, too, and ∗ dimV = dimV: ∗ ∼ In particular, V = V as vector spaces. 8 Dual of a vector space: Let V be a real, finite-dimensional vector space. Then the dual vector space of V is defined to be ∗ V := fLinear maps V ! Rg: ∗ Proposition. V is finite-dimensional vector space, too, and ∗ dimV = dimV: ∗ ∼ In particular, V = V as vector spaces. -
INTRODUCTION to ALGEBRAIC GEOMETRY 1. Preliminary Of
INTRODUCTION TO ALGEBRAIC GEOMETRY WEI-PING LI 1. Preliminary of Calculus on Manifolds 1.1. Tangent Vectors. What are tangent vectors we encounter in Calculus? 2 0 (1) Given a parametrised curve α(t) = x(t); y(t) in R , α (t) = x0(t); y0(t) is a tangent vector of the curve. (2) Given a surface given by a parameterisation x(u; v) = x(u; v); y(u; v); z(u; v); @x @x n = × is a normal vector of the surface. Any vector @u @v perpendicular to n is a tangent vector of the surface at the corresponding point. (3) Let v = (a; b; c) be a unit tangent vector of R3 at a point p 2 R3, f(x; y; z) be a differentiable function in an open neighbourhood of p, we can have the directional derivative of f in the direction v: @f @f @f D f = a (p) + b (p) + c (p): (1.1) v @x @y @z In fact, given any tangent vector v = (a; b; c), not necessarily a unit vector, we still can define an operator on the set of functions which are differentiable in open neighbourhood of p as in (1.1) Thus we can take the viewpoint that each tangent vector of R3 at p is an operator on the set of differential functions at p, i.e. @ @ @ v = (a; b; v) ! a + b + c j ; @x @y @z p or simply @ @ @ v = (a; b; c) ! a + b + c (1.2) @x @y @z 3 with the evaluation at p understood. -
Optimization Algorithms on Matrix Manifolds
00˙AMS September 23, 2007 © Copyright, Princeton University Press. No part of this book may be distributed, posted, or reproduced in any form by digital or mechanical means without prior written permission of the publisher. Index 0x, 55 of a topology, 192 C1, 196 bijection, 193 C∞, 19 blind source separation, 13 ∇2, 109 bracket (Lie), 97 F, 33, 37 BSS, 13 GL, 23 Grass(p, n), 32 Cauchy decrease, 142 JF , 71 Cauchy point, 142 On, 27 Cayley transform, 59 PU,V , 122 chain rule, 195 Px, 47 characteristic polynomial, 6 ⊥ chart Px , 47 Rn×p, 189 around a point, 20 n×p R∗ /GLp, 31 of a manifold, 20 n×p of a set, 18 R∗ , 23 Christoffel symbols, 94 Ssym, 26 − Sn 1, 27 closed set, 192 cocktail party problem, 13 S , 42 skew column space, 6 St(p, n), 26 commutator, 189 X, 37 compact, 27, 193 X(M), 94 complete, 56, 102 ∂ , 35 i conjugate directions, 180 p-plane, 31 connected, 21 S , 58 sym+ connection S (n), 58 upp+ affine, 94 ≃, 30 canonical, 94 skew, 48, 81 Levi-Civita, 97 span, 30 Riemannian, 97 sym, 48, 81 symmetric, 97 tr, 7 continuous, 194 vec, 23 continuously differentiable, 196 convergence, 63 acceleration, 102 cubic, 70 accumulation point, 64, 192 linear, 69 adjoint, 191 order of, 70 algebraic multiplicity, 6 quadratic, 70 arithmetic operation, 59 superlinear, 70 Armijo point, 62 convergent sequence, 192 asymptotically stable point, 67 convex set, 198 atlas, 19 coordinate domain, 20 compatible, 20 coordinate neighborhood, 20 complete, 19 coordinate representation, 24 maximal, 19 coordinate slice, 25 atlas topology, 20 coordinates, 18 cotangent bundle, 108 basis, 6 cotangent space, 108 For general queries, contact [email protected] 00˙AMS September 23, 2007 © Copyright, Princeton University Press. -
Symmetries of the Space of Linear Symplectic Connections
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 13 (2017), 002, 30 pages Symmetries of the Space of Linear Symplectic Connections Daniel J.F. FOX Departamento de Matem´aticas del Area´ Industrial, Escuela T´ecnica Superior de Ingenier´ıa y Dise~noIndustrial, Universidad Polit´ecnica de Madrid, Ronda de Valencia 3, 28012 Madrid, Spain E-mail: [email protected] Received June 30, 2016, in final form January 07, 2017; Published online January 10, 2017 https://doi.org/10.3842/SIGMA.2017.002 Abstract. There is constructed a family of Lie algebras that act in a Hamiltonian way on the symplectic affine space of linear symplectic connections on a symplectic manifold. The associated equivariant moment map is a formal sum of the Cahen{Gutt moment map, the Ricci tensor, and a translational term. The critical points of a functional constructed from it interpolate between the equations for preferred symplectic connections and the equations for critical symplectic connections. The commutative algebra of formal sums of symmetric tensors on a symplectic manifold carries a pair of compatible Poisson structures, one induced from the canonical Poisson bracket on the space of functions on the cotangent bundle poly- nomial in the fibers, and the other induced from the algebraic fiberwise Schouten bracket on the symmetric algebra of each fiber of the cotangent bundle. These structures are shown to be compatible, and the required Lie algebras are constructed as central extensions of their linear combinations restricted to formal sums of symmetric tensors whose first order term is a multiple of the differential of its zeroth order term. -
SYMPLECTIC GEOMETRY and MECHANICS a Useful Reference Is
GEOMETRIC QUANTIZATION I: SYMPLECTIC GEOMETRY AND MECHANICS A useful reference is Simms and Woodhouse, Lectures on Geometric Quantiza- tion, available online. 1. Symplectic Geometry. 1.1. Linear Algebra. A pre-symplectic form ω on a (real) vector space V is a skew bilinear form ω : V ⊗ V → R. For any W ⊂ V write W ⊥ = {v ∈ V | ω(v, w) = 0 ∀w ∈ W }. (V, ω) is symplectic if ω is non-degenerate: ker ω := V ⊥ = 0. Theorem 1.2. If (V, ω) is symplectic, there is a “canonical” basis P1,...,Pn,Q1,...,Qn such that ω(Pi,Pj) = 0 = ω(Qi,Qj) and ω(Pi,Qj) = δij. Pi and Qi are said to be conjugate. The theorem shows that all symplectic vector spaces of the same (always even) dimension are isomorphic. 1.3. Geometry. A symplectic manifold is one with a non-degenerate 2-form ω; this makes each tangent space TmM into a symplectic vector space with form ωm. We should also assume that ω is closed: dω = 0. We can also consider pre-symplectic manifolds, i.e. ones with a closed 2-form ω, which may be degenerate. In this case we also assume that dim ker ωm is constant; this means that the various ker ωm fit tog ether into a sub-bundle ker ω of TM. Example 1.1. If V is a symplectic vector space, then each TmV = V . Thus the symplectic form on V (as a vector space) determines a symplectic form on V (as a manifold). This is locally the only example: Theorem 1.4. -
M7210 Lecture 7. Vector Spaces Part 4, Dual Spaces Wednesday September 5, 2012 Assume V Is an N-Dimensional Vector Space Over a field F
M7210 Lecture 7. Vector Spaces Part 4, Dual Spaces Wednesday September 5, 2012 Assume V is an n-dimensional vector space over a field F. Definition. The dual space of V , denoted V ′ is the set of all linear maps from V to F. Comment. F is an ambiguous object. It may be regarded: (a) as a field, (b) vector space over F, (c) as a vector space equipped with basis, {1}. It is always important to bear in mind which of the three objects we are thinking about. In the definition of V ′, we use (c). The distinctions are often pushed into the background, but let us keep them in clear view for the remainder of this paragraph. Suppose V has basis E = {e1,e2,...,en}. ′ If w ∈ V , then for each ei, there is a unique scalar ω(ei) such that w(ei) = ω(ei)1. Then w = ω(e1) ω(e2) · · · ω(en) . (1) {1}E The columns (of height 1) on the right hand side are the coefficients required to write the images of the basis elements in E in terms of the basis {1}. (The definition of the matrix of a linear map that we gave in the last lecture demands that each entry in the matrix be an element of the scalar field, not of a vector space.) The notation w(ei) = ω(ei)1 is a reminder that w(ei) is an element of the vector space F, while ω(ei) is an element of the field F. But scalar multiplication of elements of the vector space F by elements of the field F is good old fashioned multiplication ∗ : F × F → F. -
1 Symplectic Vector Spaces
Symplectic Manifolds Tam´asHausel 1 Symplectic Vector Spaces Let us first fix a field k. Definition 1.1 (Symplectic form, symplectic vector space). Given a finite dimensional k-vector space W , a symplectic form on W is a bilinear, alternating non degenerate form on W , i.e. an element ! 2 ^2W ∗ such that if !(v; w) = 0 for all w 2 W then v = 0. We call the pair (W; !) a symplectic vector space. Example 1.2 (Standard example). Consider a finite dimensional k-vector space V and define W = V ⊕ V ∗. We can introduce a symplectic form on W : ! : W × W ! k ((v1; α1); (v2; α2)) 7! α1(v2) − α2(v1) Definition 1.3 (Symplectomorphism). Given two symplectic vector spaces (W1;!1) and (W2;!2), we say that a linear map f : W1 ! W2 is a symplectomorphism if it is an isomorphism of vector ∗ spaces and, furthermore, f !2 = !1. We denote by Sp(W ) the group of symplectomorphism W ! W . Let us consider a symplectic vector space (W; !) with even dimension 2n (we will see soon that it is always the case). Then ! ^ · · · ^ ! 2 ^2nW ∗ is a volume form, i.e. a non-degenerate top form. For f 2 Sp(W ) we must have !n = f ∗!n = det f · !n so that det f = 1 and, in particular, Sp(W ) ≤ SL(W ). We also note that, for n = 1, we have Sp(W ) =∼ SL(W ). Definition 1.4. Let U be a linear subspace of a symplectic vector space W . Then we define U ? = fv 2 W j !(v; w) = 0 8u 2 Ug: We say that U is isotropic if U ⊆ U ?, coisotropic if U ? ⊆ U and Lagrangian if U ? = U. -
Manifolds, Tangent Vectors and Covectors
Physics 250 Fall 2015 Notes 1 Manifolds, Tangent Vectors and Covectors 1. Introduction Most of the “spaces” used in physical applications are technically differentiable man- ifolds, and this will be true also for most of the spaces we use in the rest of this course. After a while we will drop the qualifier “differentiable” and it will be understood that all manifolds we refer to are differentiable. We will build up the definition in steps. A differentiable manifold is basically a topological manifold that has “coordinate sys- tems” imposed on it. Recall that a topological manifold is a topological space that is Hausdorff and locally homeomorphic to Rn. The number n is the dimension of the mani- fold. On a topological manifold, we can talk about the continuity of functions, for example, of functions such as f : M → R (a “scalar field”), but we cannot talk about the derivatives of such functions. To talk about derivatives, we need coordinates. 2. Charts and Coordinates Generally speaking it is impossible to cover a manifold with a single coordinate system, so we work in “patches,” technically charts. Given a topological manifold M of dimension m, a chart on M is a pair (U, φ), where U ⊂ M is an open set and φ : U → V ⊂ Rm is a homeomorphism. See Fig. 1. Since φ is a homeomorphism, V is also open (in Rm), and φ−1 : V → U exists. If p ∈ U is a point in the domain of φ, then φ(p) = (x1,...,xm) is the set of coordinates of p with respect to the given chart.