Symplectic and Contact Geometry with Complex Manifolds
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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. -
Gromov Receives 2009 Abel Prize
Gromov Receives 2009 Abel Prize . The Norwegian Academy of Science Medal (1997), and the Wolf Prize (1993). He is a and Letters has decided to award the foreign member of the U.S. National Academy of Abel Prize for 2009 to the Russian- Sciences and of the American Academy of Arts French mathematician Mikhail L. and Sciences, and a member of the Académie des Gromov for “his revolutionary con- Sciences of France. tributions to geometry”. The Abel Prize recognizes contributions of Citation http://www.abelprisen.no/en/ extraordinary depth and influence Geometry is one of the oldest fields of mathemat- to the mathematical sciences and ics; it has engaged the attention of great mathema- has been awarded annually since ticians through the centuries but has undergone Photo from from Photo 2003. It carries a cash award of revolutionary change during the last fifty years. Mikhail L. Gromov 6,000,000 Norwegian kroner (ap- Mikhail Gromov has led some of the most impor- proximately US$950,000). Gromov tant developments, producing profoundly original will receive the Abel Prize from His Majesty King general ideas, which have resulted in new perspec- Harald at an award ceremony in Oslo, Norway, on tives on geometry and other areas of mathematics. May 19, 2009. Riemannian geometry developed from the study Biographical Sketch of curved surfaces and their higher-dimensional analogues and has found applications, for in- Mikhail Leonidovich Gromov was born on Decem- stance, in the theory of general relativity. Gromov ber 23, 1943, in Boksitogorsk, USSR. He obtained played a decisive role in the creation of modern his master’s degree (1965) and his doctorate (1969) global Riemannian geometry. -
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. -
Lagrangian Intersections in Contact Geometry (First Draft)
Lagrangian intersections in contact geometry (First draft) Y. Eliashberg,∗ H. Hofer,y and D. Salamonz Stanford University∗ Ruhr-Universit¨at Bochumy Mathematics Institutez University of Warwick Coventry CV4 7AL Great Britain October 1994 1 Introduction Contact geometry Let M be a compact manifold of odd dimension 2n + 1 and ξ ⊂ T M be a contact structure. This means that ξ is an oriented field of hyperplanes and there exists a 1-form α on M such that ξ = ker α, α ^ dα^n 6= 0: The last condition means that the restriction of dα to the kernel of α is non- degenerate. Any such form is called a contact form for ξ. Any contact form determines a contact vector field (sometimes also called the Reeb vector field) Y : M ! T M via ι(Y )dα = 0; ι(Y )α = 1: A contactomorphism is a diffeomorphism : M ! M which preserves the contact structure ξ. This means that ∗α = ehα for some smooth function h : M ! . A contact isotopy is a smooth family of contactomorphisms s : M ! M such that ∗ hs s α = e α for all s. Any such isotopy with 0 = id and h0 = 0 is generated by Hamil- tonian functions Hs : M ! as follows. Each Hamiltonian function Hs determines a Hamiltonian vector field Xs = XHs = Xα,Hs : M ! T M via ι(Xs)α = −Hs; ι(Xs)dα = dHs − ι(Y )dHs · α, 1 and these vector fields determine s and hs via d d = X ◦ ; h = −ι(Y )dH : ds s s s ds s s A Legendrian submanifold of M is an n-dimensional integral submanifold L of the hyperplane field ξ, that is αjT L = 0. -
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 -
“Generalized Complex and Holomorphic Poisson Geometry”
“Generalized complex and holomorphic Poisson geometry” Marco Gualtieri (University of Toronto), Ruxandra Moraru (University of Waterloo), Nigel Hitchin (Oxford University), Jacques Hurtubise (McGill University), Henrique Bursztyn (IMPA), Gil Cavalcanti (Utrecht University) Sunday, 11-04-2010 to Friday, 16-04-2010 1 Overview of the Field Generalized complex geometry is a relatively new subject in differential geometry, originating in 2001 with the work of Hitchin on geometries defined by differential forms of mixed degree. It has the particularly inter- esting feature that it interpolates between two very classical areas in geometry: complex algebraic geometry on the one hand, and symplectic geometry on the other hand. As such, it has bearing on some of the most intriguing geometrical problems of the last few decades, namely the suggestion by physicists that a duality of quantum field theories leads to a ”mirror symmetry” between complex and symplectic geometry. Examples of generalized complex manifolds include complex and symplectic manifolds; these are at op- posite extremes of the spectrum of possibilities. Because of this fact, there are many connections between the subject and existing work on complex and symplectic geometry. More intriguing is the fact that complex and symplectic methods often apply, with subtle modifications, to the study of the intermediate cases. Un- like symplectic or complex geometry, the local behaviour of a generalized complex manifold is not uniform. Indeed, its local structure is characterized by a Poisson bracket, whose rank at any given point characterizes the local geometry. For this reason, the study of Poisson structures is central to the understanding of gen- eralized complex manifolds which are neither complex nor symplectic. -
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. -
Contact Geometry
Contact Geometry Hansj¨org Geiges Mathematisches Institut, Universit¨at zu K¨oln, Weyertal 86–90, 50931 K¨oln, Germany E-mail: [email protected] April 2004 Contents 1 Introduction 3 2 Contact manifolds 4 2.1 Contact manifolds and their submanifolds . 6 2.2 Gray stability and the Moser trick . 13 2.3 ContactHamiltonians . 16 2.4 Darboux’s theorem and neighbourhood theorems . 17 2.4.1 Darboux’stheorem. 17 2.4.2 Isotropic submanifolds . 19 2.4.3 Contact submanifolds . 24 2.4.4 Hypersurfaces.......................... 26 2.4.5 Applications .......................... 30 2.5 Isotopyextensiontheorems . 32 2.5.1 Isotropic submanifolds . 32 2.5.2 Contact submanifolds . 34 2.5.3 Surfaces in 3–manifolds . 36 2.6 Approximationtheorems. 37 2.6.1 Legendrianknots. 38 2.6.2 Transverseknots . 42 1 3 Contact structures on 3–manifolds 43 3.1 An invariant of transverse knots . 45 3.2 Martinet’sconstruction . 46 3.3 2–plane fields on 3–manifolds . 50 3.3.1 Hopf’s Umkehrhomomorphismus . 53 3.3.2 Representing homology classes by submanifolds . 54 3.3.3 Framedcobordisms. 56 3.3.4 Definition of the obstruction classes . 57 3.4 Let’sTwistAgain ........................... 59 3.5 Otherexistenceproofs . 64 3.5.1 Openbooks........................... 64 3.5.2 Branchedcovers ........................ 66 3.5.3 ...andmore ......................... 67 3.6 Tightandovertwisted . 67 3.7 Classificationresults . 73 4 A guide to the literature 75 4.1 Dimension3............................... 76 4.2 Higherdimensions ........................... 76 4.3 Symplecticfillings ........................... 77 4.4 DynamicsoftheReebvectorfield. 77 2 1 Introduction Over the past two decades, contact geometry has undergone a veritable meta- morphosis: once the ugly duckling known as ‘the odd-dimensional analogue of symplectic geometry’, it has now evolved into a proud field of study in its own right. -
Symplectic Geometry
Part III | Symplectic Geometry Based on lectures by A. R. Pires Notes taken by Dexter Chua Lent 2018 These notes are not endorsed by the lecturers, and I have modified them (often significantly) after lectures. They are nowhere near accurate representations of what was actually lectured, and in particular, all errors are almost surely mine. The first part of the course will be an overview of the basic structures of symplectic ge- ometry, including symplectic linear algebra, symplectic manifolds, symplectomorphisms, Darboux theorem, cotangent bundles, Lagrangian submanifolds, and Hamiltonian sys- tems. The course will then go further into two topics. The first one is moment maps and toric symplectic manifolds, and the second one is capacities and symplectic embedding problems. Pre-requisites Some familiarity with basic notions from Differential Geometry and Algebraic Topology will be assumed. The material covered in the respective Michaelmas Term courses would be more than enough background. 1 Contents III Symplectic Geometry Contents 1 Symplectic manifolds 3 1.1 Symplectic linear algebra . .3 1.2 Symplectic manifolds . .4 1.3 Symplectomorphisms and Lagrangians . .8 1.4 Periodic points of symplectomorphisms . 11 1.5 Lagrangian submanifolds and fixed points . 13 2 Complex structures 16 2.1 Almost complex structures . 16 2.2 Dolbeault theory . 18 2.3 K¨ahlermanifolds . 21 2.4 Hodge theory . 24 3 Hamiltonian vector fields 30 3.1 Hamiltonian vector fields . 30 3.2 Integrable systems . 32 3.3 Classical mechanics . 34 3.4 Hamiltonian actions . 36 3.5 Symplectic reduction . 39 3.6 The convexity theorem . 45 3.7 Toric manifolds . 51 4 Symplectic embeddings 56 Index 57 2 1 Symplectic manifolds III Symplectic Geometry 1 Symplectic manifolds 1.1 Symplectic linear algebra In symplectic geometry, we study symplectic manifolds. -
Symplectic and Contact Geometry and Hamiltonian Dynamics
Symplectic and Contact Geometry and Hamiltonian Dynamics Mikhail B. Sevryuk Abstract. This is an introduction to the contributions by the lecturers at the mini-symposium on symplectic and contact geometry. We present a very general and brief account of the prehistory of the eld and give references to some seminal papers and important survey works. Symplectic geometry is the geometry of a closed nondegenerate two-form on an even-dimensional manifold. Contact geometry is the geometry of a maximally nondegenerate eld of tangent hyperplanes on an odd-dimensional manifold. The symplectic structure is fundamental for Hamiltonian dynamics, and in this sense symplectic geometry (and its odd-dimensional counterpart, contact geometry) is as old as classical mechanics. However, the science the present mini-symposium is devoted to is usually believed to date from H. Poincare’s “last geometric theo- rem” [70] concerning xed points of area-preserving mappings of an annulus: Theorem 1. (Poincare-Birkho) An area-preserving dieomorphism of an annu- lus S1[0; 1] possesses at least two xed points provided that it rotates two boundary circles in opposite directions. This theorem proven by G. D. Birkho [16] was probably the rst statement describing the properties of symplectic manifolds and symplectomorphisms “in large”, thereby giving birth to symplectic topology. In the mid 1960s [2, 3] and later in the 1970s ([4, 5, 6] and [7, Appendix 9]), V. I. Arnol’d formulated his famous conjecture generalizing Poincare’s theorem to higher dimensions. This conjecture reads as follows: Conjecture 2. (Arnol’d) A ow map A of a (possibly nonautonomous) Hamilton- ian system of ordinary dierential equations on a closed symplectic manifold M possesses at least as many xed points as a smooth function on M must have critical points, both “algebraically” and “geometrically”. -
Symplectic Geometry Tara S
THE GRADUATE STUDENT SECTION WHAT IS... Symplectic Geometry Tara S. Holm Communicated by Cesar E. Silva In Euclidean geome- depending smoothly on the point 푝 ∈ 푀.A 2-form try in a vector space 휔 ∈ Ω2(푀) is symplectic if it is both closed (its exterior Symplectic over ℝ, lengths and derivative satisfies 푑휔 = 0) and nondegenerate (each structures are angles are the funda- function 휔푝 is nondegenerate). Nondegeneracy is equiva- mental measurements, lent to the statement that for each nonzero tangent vector floppier than and objects are rigid. 푣 ∈ 푇푝푀, there is a symplectic buddy: a vector 푤 ∈ 푇푝푀 In symplectic geome- so that 휔푝(푣, 푤) = 1.A symplectic manifold is a (real) holomorphic try, a two-dimensional manifold 푀 equipped with a symplectic form 휔. area measurement is Nondegeneracy has important consequences. Purely in functions or the key ingredient, and terms of linear algebra, at any point 푝 ∈ 푀 we may choose the complex numbers a basis of 푇푝푀 that is compatible with 휔푝, using a skew- metrics. are the natural scalars. symmetric analogue of the Gram-Schmidt procedure. We It turns out that sym- start by choosing any nonzero vector 푣1 and then finding a plectic structures are much floppier than holomorphic symplectic buddy 푤1. These must be linearly independent functions in complex geometry or metrics in Riemannian by skew-symmetry. We then peel off the two-dimensional geometry. subspace that 푣1 and 푤1 span and continue recursively, The word “symplectic” is a calque introduced by eventually arriving at a basis Hermann Weyl in his textbook on the classical groups.