SYMPLECTIC MANIFOLDS Contents 1. Symplectic Structures 1 2. the Cotangent Bundle 3 3. Hamiltonian Vector Fields 5 4. Hamiltonian
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Density of Thin Film Billiard Reflection Pseudogroup in Hamiltonian Symplectomorphism Pseudogroup Alexey Glutsyuk
Density of thin film billiard reflection pseudogroup in Hamiltonian symplectomorphism pseudogroup Alexey Glutsyuk To cite this version: Alexey Glutsyuk. Density of thin film billiard reflection pseudogroup in Hamiltonian symplectomor- phism pseudogroup. 2020. hal-03026432v2 HAL Id: hal-03026432 https://hal.archives-ouvertes.fr/hal-03026432v2 Preprint submitted on 6 Dec 2020 HAL is a multi-disciplinary open access L’archive ouverte pluridisciplinaire HAL, est archive for the deposit and dissemination of sci- destinée au dépôt et à la diffusion de documents entific research documents, whether they are pub- scientifiques de niveau recherche, publiés ou non, lished or not. The documents may come from émanant des établissements d’enseignement et de teaching and research institutions in France or recherche français ou étrangers, des laboratoires abroad, or from public or private research centers. publics ou privés. Density of thin film billiard reflection pseudogroup in Hamiltonian symplectomorphism pseudogroup Alexey Glutsyuk∗yzx December 3, 2020 Abstract Reflections from hypersurfaces act by symplectomorphisms on the space of oriented lines with respect to the canonical symplectic form. We consider an arbitrary C1-smooth hypersurface γ ⊂ Rn+1 that is either a global strictly convex closed hypersurface, or a germ of hy- persurface. We deal with the pseudogroup generated by compositional ratios of reflections from γ and of reflections from its small deforma- tions. In the case, when γ is a global convex hypersurface, we show that the latter pseudogroup is dense in the pseudogroup of Hamiltonian diffeomorphisms between subdomains of the phase cylinder: the space of oriented lines intersecting γ transversally. We prove an analogous local result in the case, when γ is a germ. -
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. -
Symplectic Structures on Gauge Theory?
Commun. Math. Phys. 193, 47 – 67 (1998) Communications in Mathematical Physics c Springer-Verlag 1998 Symplectic Structures on Gauge Theory? Nai-Chung Conan Leung School of Mathematics, 127 Vincent Hall, University of Minnesota, Minneapolis, MN 55455, USA Received: 27 February 1996 / Accepted: 7 July 1997 [ ] Abstract: We study certain natural differential forms ∗ and their equivariant ex- tensions on the space of connections. These forms are defined usingG the family local index theorem. When the base manifold is symplectic, they define a family of sym- plectic forms on the space of connections. We will explain their relationships with the Einstein metric and the stability of vector bundles. These forms also determine primary and secondary characteristic forms (and their higher level generalizations). Contents 1 Introduction ............................................... 47 2 Higher Index of the Universal Family ........................... 49 2.1 Dirac operators ............................................. 50 2.2 ∂-operator ................................................ 52 3 Moment Map and Stable Bundles .............................. 53 4 Space of Generalized Connections ............................. 58 5 Higer Level Chern–Simons Forms ............................. 60 [ ] 6 Equivariant Extensions of ∗ ................................ 63 1. Introduction In this paper, we study natural differential forms [2k] on the space of connections on a vector bundle E over X, A i [2k] FA (A)(B1,B2,...,B2 )= Tr[e 2π B1B2 B2 ] Aˆ(X). k ··· k sym ZX They are invariant closed differential forms on . They are introduced as the family local indexG for universal family of operators. A ? This research is supported by NSF grant number: DMS-9114456 48 N-C. C. Leung We use them to define higher Chern-Simons forms of E: [ ] ch(E; A0,...,A )=∗ L(A0,...,A ), l l ◦ l l and discuss their properties. -
Lecture 1: Basic Concepts, Problems, and Examples
LECTURE 1: BASIC CONCEPTS, PROBLEMS, AND EXAMPLES WEIMIN CHEN, UMASS, SPRING 07 In this lecture we give a general introduction to the basic concepts and some of the fundamental problems in symplectic geometry/topology, where along the way various examples are also given for the purpose of illustration. We will often give statements without proofs, which means that their proof is either beyond the scope of this course or will be treated more systematically in later lectures. 1. Symplectic Manifolds Throughout we will assume that M is a C1-smooth manifold without boundary (unless specific mention is made to the contrary). Very often, M will also be closed (i.e., compact). Definition 1.1. A symplectic structure on a smooth manifold M is a 2-form ! 2 Ω2(M), which is (1) nondegenerate, and (2) closed (i.e. d! = 0). (Recall that a 2-form ! 2 Ω2(M) is said to be nondegenerate if for every point p 2 M, !(u; v) = 0 for all u 2 TpM implies v 2 TpM equals 0.) The pair (M; !) is called a symplectic manifold. Before we discuss examples of symplectic manifolds, we shall first derive some im- mediate consequences of a symplectic structure. (1) The nondegeneracy condition on ! is equivalent to the condition that M has an even dimension 2n and the top wedge product !n ≡ ! ^ ! · · · ^ ! is nowhere vanishing on M, i.e., !n is a volume form. In particular, M must be orientable, and is canonically oriented by !n. The nondegeneracy condition is also equivalent to the condition that M is almost complex, i.e., there exists an endomorphism J of TM such that J 2 = −Id. -
Functor and Natural Operators on Symplectic Manifolds
Mathematical Methods and Techniques in Engineering and Environmental Science Functor and natural operators on symplectic manifolds CONSTANTIN PĂTRĂŞCOIU Faculty of Engineering and Management of Technological Systems, Drobeta Turnu Severin University of Craiova Drobeta Turnu Severin, Str. Călugăreni, No.1 ROMANIA [email protected] http://www.imst.ro Abstract. Symplectic manifolds arise naturally in abstract formulations of classical Mechanics because the phase–spaces in Hamiltonian mechanics is the cotangent bundle of configuration manifolds equipped with a symplectic structure. The natural functors and operators described in this paper can be helpful both for a unified description of specific proprieties of symplectic manifolds and for finding links to various fields of geometric objects with applications in Hamiltonian mechanics. Key-words: Functors, Natural operators, Symplectic manifolds, Complex structure, Hamiltonian. 1. Introduction. diffeomorphism f : M → N , a vector bundle morphism The geometrics objects (like as vectors, covectors, T*f : T*M→ T*N , which covers f , where −1 tensors, metrics, e.t.c.) on a smooth manifold M, x = x :)*)(()*( x * → xf )( * NTMTfTfT are the elements of the total spaces of a vector So, the cotangent functor, )(:* → VBmManT and bundles with base M. The fields of such geometrics objects are the section in corresponding vector more general ΛkT * : Man(m) → VB , are the bundles. bundle functors from Man(m) to VB, where Man(m) By example, the vectors on a smooth manifold M (subcategory of Man) is the category of m- are the elements of total space of vector bundles dimensional smooth manifolds, the morphisms of this (TM ,π M , M); the covectors on M are the elements category are local diffeomorphisms between of total space of vector bundles (T*M , pM , M). -
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 . -
Symplectic Topology Math 705 Notes by Patrick Lei, Spring 2020
Symplectic Topology Math 705 Notes by Patrick Lei, Spring 2020 Lectures by R. Inanç˙ Baykur University of Massachusetts Amherst Disclaimer These notes were taken during lecture using the vimtex package of the editor neovim. Any errors are mine and not the instructor’s. In addition, my notes are picture-free (but will include commutative diagrams) and are a mix of my mathematical style (omit lengthy computations, use category theory) and that of the instructor. If you find any errors, please contact me at [email protected]. Contents Contents • 2 1 January 21 • 5 1.1 Course Description • 5 1.2 Organization • 5 1.2.1 Notational conventions•5 1.3 Basic Notions • 5 1.4 Symplectic Linear Algebra • 6 2 January 23 • 8 2.1 More Basic Linear Algebra • 8 2.2 Compatible Complex Structures and Inner Products • 9 3 January 28 • 10 3.1 A Big Theorem • 10 3.2 More Compatibility • 11 4 January 30 • 13 4.1 Homework Exercises • 13 4.2 Subspaces of Symplectic Vector Spaces • 14 5 February 4 • 16 5.1 Linear Algebra, Conclusion • 16 5.2 Symplectic Vector Bundles • 16 6 February 6 • 18 6.1 Proof of Theorem 5.11 • 18 6.2 Vector Bundles, Continued • 18 6.3 Compatible Triples on Manifolds • 19 7 February 11 • 21 7.1 Obtaining Compatible Triples • 21 7.2 Complex Structures • 21 8 February 13 • 23 2 3 8.1 Kähler Forms Continued • 23 8.2 Some Algebraic Geometry • 24 8.3 Stein Manifolds • 25 9 February 20 • 26 9.1 Stein Manifolds Continued • 26 9.2 Topological Properties of Kähler Manifolds • 26 9.3 Complex and Symplectic Structures on 4-Manifolds • 27 10 February 25 -
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. -
Hamiltonian and Symplectic Symmetries: an Introduction
BULLETIN (New Series) OF THE AMERICAN MATHEMATICAL SOCIETY Volume 54, Number 3, July 2017, Pages 383–436 http://dx.doi.org/10.1090/bull/1572 Article electronically published on March 6, 2017 HAMILTONIAN AND SYMPLECTIC SYMMETRIES: AN INTRODUCTION ALVARO´ PELAYO In memory of Professor J.J. Duistermaat (1942–2010) Abstract. Classical mechanical systems are modeled by a symplectic mani- fold (M,ω), and their symmetries are encoded in the action of a Lie group G on M by diffeomorphisms which preserve ω. These actions, which are called sym- plectic, have been studied in the past forty years, following the works of Atiyah, Delzant, Duistermaat, Guillemin, Heckman, Kostant, Souriau, and Sternberg in the 1970s and 1980s on symplectic actions of compact Abelian Lie groups that are, in addition, of Hamiltonian type, i.e., they also satisfy Hamilton’s equations. Since then a number of connections with combinatorics, finite- dimensional integrable Hamiltonian systems, more general symplectic actions, and topology have flourished. In this paper we review classical and recent re- sults on Hamiltonian and non-Hamiltonian symplectic group actions roughly starting from the results of these authors. This paper also serves as a quick introduction to the basics of symplectic geometry. 1. Introduction Symplectic geometry is concerned with the study of a notion of signed area, rather than length, distance, or volume. It can be, as we will see, less intuitive than Euclidean or metric geometry and it is taking mathematicians many years to understand its intricacies (which is work in progress). The word “symplectic” goes back to the 1946 book [164] by Hermann Weyl (1885–1955) on classical groups. -
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.