Symplectic Geometry versus Riemannian Geometry
|
|
- Nigel Matthews
- 6 years ago
- Views:
Transcription
1 Symplectic Geometry versus Riemannian Geometry Inês Cruz, CMUP 2010/10/20 - seminar of PIUDM 1
2 necessarily very incomplete The aim of this talk is to give an overview of Symplectic Geometry (there is no intention of giving a list of recent results or open problems) 2
3 Notation: throughout the talk: - M - real, finite-dim al, differentiable manifold without boundary - C ( M ) = { f : M : f is smooth} - χ ( M ) = { X : M TM : X is a vector field} - Ω k M = { ω :TM TM } or Ω k ( M ) = ω : χ M ω p;v 1,,v k { χ ( M ) C ( M )} ω ( X 1,, X k ) C ( M ) given by: ω ( X 1,, X k )( p) = ω p; X 1p,, X kp 3
4 1. Symplectic Manifolds 1. Riemannian Manifolds Def: Symplectic manifold is a pair ( M,ω ), where: (a) ω Ω 2 ( M ) i.e., = ω ( X,Y ) ω Y, X ω ( fx + gy, Z ) = fω ( X, Z ) + gω ( Y,Z ) (b) ω is nondegenerate, i.e.: ω ( X,Y ) = 0, X χ ( M ) Y = 0 (c) ω is closed, i.e. dω = 0 We call ω a symplectic form. which manifolds qualify for being symplectic? Necessary conditions: (N1) dim M = 2n consider local coordinates ( x 1,, x m ) the matrix A with entries: and build Def: Riemannian manifold is a pair ( M,<,> ), where: (a) <,>: χ(m ) χ(m ) C M < Y, X > = < X,Y > < fx + gy, Z > = f < X, Z > +g < Y,Z > (b) <, > is positive definite. satisfies: Consequence: <, > is nondegenerate. which manifolds qualify for being Riemannian? all smooth manifolds! 4
5 then (a) and (b) imply: A T = A and det( A) 0 m = 2n (N2) M is oriented consider the nth exterior power: then (b) implies ω (n) = ω ω Ω 2n ( M ) ω (n) is a volume form on M. (N3) if M is compact then H 2 DR ( M, ) 0 ω = dα ω (n) = d α ω ω vol M a ij = ω, x i x i ω ( n) is called symplectic volume. dβ = β = 0 = ω (n) = M M M S 2n is not symplectic, for any n >1 (S 2 is symplectic) 5
6 2. Examples example 1: M = 2n with coords: { x 1,, x n, y 1,, y n } and symplectic structure given by: ω 0 = n i=1 dx i dy i = dx dy example 2: form: M = T * N ω = dλ with symplectic ( λ is the Liouville 1-form on M : =< α,dπ p,α > ) λ ( p,α ) X X The importance of example 1 will be clear soon. example 2 is behind the Hamiltonian formulation of Conservative Mechanics. 6
7 3. Special vector fields Nondegeneracy of ω implies that the following is an isomorphism: Def: Given vector field is: Ω 1 ( M ) I : χ M X i X ω = ω X, f C ( M ) X f = I 1 ( df ) its Hamiltonian = df (in other words: ω X f, ) Lemma X f is tangent to the level surface: ΣC = p M : f ( p) = C { } 3. Special vector fields Nondegeneracy of <, > implies that the following is an isomorphism: Def: Given vector field is: Ω 1 ( M ) I : χ M X < X, > f C ( M ) f = I 1 ( df ) its gradient (in other words: < f, >= df ( ) ) Lemma f is normal to the level surface: Σ C = p M : f ( p) = C { } (equivalently f is constant on the flow of X f ) note that: v T Σ C df v = 0 and: if v T Σ C then: < f,v >= df v = 0 = ω ( X f, X f ) = 0 df X f 7
8 f X f Other important vector fields: Def: A vector field X is said to be symplectic if I(X) is closed. In other words: di X ω = 0 Hamiltonian vector fields are symplectic, since ddf=0. Condition (c) implies that the flow of any symplectic vector field preserves ω : L X ω = di X ω + i X dω = 0 = 0 = 0 because of (c) 8
9 4. Poisson bracket One can use Hamiltonian vector fields to define an operation between smooth functions: Def: Poisson bracket on M is: {,} :C ( M ) C ( M ) C ( M ) ( f, g) ω X f, X g = df ( X g ) Condition (c) implies this bracket satisfies Jacobi s identity: f,{ g,h} { } { } { } + { g, h, f } + { h, f,g } = 0 which, together with obvious properties of {,}, implies that: ( C ( M ),{,}) is an infinite-dimensional Lie algebra. 9
10 5. Equivalence 5. Equivalence Def: Two symplectic manifolds ( M,ω ) M ',ω ' and are symplectomorphic if there exists a C 1 map: satisfying: i.e., ϕ : M M ' ϕ * ω ' = ω ( ) = ω p ( X,Y ) ω ' ϕ ( p) dϕ p ( X),dϕ p Y ϕ is called a symplectic map and necessarily dϕ p is injective, for all p so: dim M dim M ' A symplectomorphism is a symplectic diffeomorphism of M. Symplectomorphisms form an (infinite dimensional) subgroup of the group diff(m). Def: Two Riemannian manifolds ( M,<,> ) and ( M ',<,> ') are isometric if there exists a C 1 map: satisfying: < dϕ p ( X),dϕ p Y ϕ : M M ' > ' ϕ p = < X,Y > p ϕ is called an isometry and necessarily dϕ p is injective, for all p so: dim M dim M ' 10
11 Darboux-Weinstein theorem Let p be any point on a symplectic manifold of dimension 2n. Then there exist local coordinates ( x 1,, x n, y 1,, y n ) (in U) such that: ω U = n i=1 dx i dy i Therefore all symplectic manifolds are (locally) symplectomorphic to example 1. Consequence: there are no local invariants (apart from dimension) in Symplectic Geometry curvature is a local invariant in Riemannian Geometry 11
12 6. Global Invariants As seen before, a symplectic manifold carries a symplectic volume: If volumes: ϕ : M M ' ω (n) = ω ω is a symplectic map, then it preserves (symplectic) ϕ * ω ' (n) = ϕ * ( ω ' ω ') = ϕ * ω ' but the converse is not true for n > 1. ϕ * ( ω ') = ω ω = ω (n) example: take 2 ( M,ω ) = 4, dx i dy i i=1 ϕ x 1, x 2, y 1, y 2 and consider the map: = 1 2 x 1,2x 2, 1 2 y 1,2y 2 This map preserves the symplectic volume symplectic. 2dx 1 dx 2 dy 1 dy 2 but is not so what really characterizes symplectomorphisms? 12
13 6.1. Gromov s Nonsqueezing Theorem The key theorem for characterizing symplectomorphisms (using symplectic invariants) is: Nonsqueezing theorem - Gromov (1985) There is a symplectic embedding: ϕ : B 2n ( r),ω 0 if and only if r R. ( Z 2n ( R),ω 0 ) (where: B 2n Z 2n ( r) = x, y ( R) = x, y n 2n : x i 2 + y i 2 < r 2 i=1 { 2n : x y 2 1 < R 2 } open (symplectic) ball of radius r open symplectic cylinder of radius R ) a symplectic embedding is just a symplectic map which is also an embedding. It is denoted by: ϕ :( M,ω ) ( M ',ω '). U,ω U if M,ω is a symplectic manifold and U is open in M, then is also symplectic. 13
14 The theorem is not valid if symplectic cylinder is replaced by cylinder : C 2n ( R) = ( x, y) 2n : x x 2 2 < R 2 { } example: the following is a symplectic embedding from B 4 (2) into C 4 (1): = 1 2 x 1, 1 2 x 2,2y 1,2y 2 ϕ x 1, x 2, y 1, y 2 14
15 6.2. Symplectic Invariants - Capacities Using Gromov s nonsqueezing theorem, we will construct a symplectic invariant: Gromov s width. This is one of many symplectic invariants known as symplectic capacities. Let M(2n) denote the set of all symplectic manifolds of dimension 2n. Def Symplectic capacity is a map: satisfying all three properties: c : M ( 2n) (1) monotonicity - if there is a symplectic embedding: ( M 2,ω 2 ) ϕ : M 1,ω 1 then c( M 1,ω 1 ) c( M 2,ω 2 ) (2) conformality - c( M,λω ) = λ c( M,ω ) λ 0 (3) (strong) nontriviality - c B 2n ( 1),ω 0 = π = c( Z 2n ( 1),ω 0 ) 15
16 If n = 1 then (absolute value of) total volume of M is a symplectic capacity. If n > 1 then (absolute value of) total volume of M is not a symplectic capacity (nontriviality fails). Theorem Any symplectic capacity is a symplectic invariant, i.e., if there is a symplectomorphism: then c( M 1,ω 1 ) = c( M 2,ω 2 ). (proof: monotonicity in both ways) ϕ :( M 1,ω 1 ) ( M 2,ω 2 ) Lemma Any symplectic capacity satisfies: c( B 2n ( r),ω 0 ) = πr 2 = c( Z 2n ( r),ω 0 ). (proof: previous theorem+conformality+nontriviality) 16
17 Theorem The existence of a symplectic capacity is equivalent to Gromov s nonsqueezing theorem. suppose c exists and that: ( Z 2n ( R),ω 0 ) ϕ : B 2n ( r),ω 0 is a symplectic embedding. Then monotonicity+previous lemma imply: so r R. c( Z 2n ( R),ω 0 ) = π R 2 πr 2 = c B 2n ( r),ω 0 define the Gromov s width of a symplectic manifold: W G ( M,ω ) = sup{ πr 2 : ϕ :( B 2n ( r),ω 0 ) ( M,ω )} r >0 (the area of the disk of the bigger ball one can symplectically-embed on ( M,ω ) If Gromov s nonsqueezing theorem holds (it does!) then Gromov s width is a symplectic capacity. 17
18 Theorem Gromov s width W G is the smallest of all capacities: W G (proof: let c be any capacity and fix r such that there is an embedding: Then monotonicity of c implies: Since this holds for all r and proving the result). ( M,ω ) c( M,ω ), for any capacity c and any ( M,ω ). sup r >0 c( M,ω ) M,ω ϕ : B 2n ( r),ω 0 πr 2 = c B 2n ( r),ω 0 c M,ω is independent of r: { πr 2 : ϕ :( B 2n ( r),ω 0 ) ( M,ω )} c M,ω 18
19 6.3. Back to Symplectomorphisms We go back to the question: so what really characterizes symplectomorphisms? It turns out that symplectomorphisms of 2n are (almost) characterized by the property of preserving capacity of ellipsoids (1) : Theorem (Eliashberg, 1987) (Hofer, 1990) Let be a diffeomorphism and c a capacity. Then: ( 2n,ω 0 ) c( ϕ ( E,ω 0 )) = c E,ω 0 ϕ : 2n,ω 0 for any ellipsoid E 2n if and only if ϕ is symplectic or anti-symplectic (2). (1) ellipsoid is the image of a ball by a linear/affine diffeomorphism (2) meaning that ϕ * ω 0 = ω 0 19
Symplectic and Poisson Manifolds
Symplectic and Poisson Manifolds Harry Smith In this survey we look at the basic definitions relating to symplectic manifolds and Poisson manifolds and consider different examples of these. We go on to
More informationConvex Symplectic Manifolds
Convex Symplectic Manifolds Jie Min April 16, 2017 Abstract This note for my talk in student symplectic topology seminar in UMN is an gentle introduction to the concept of convex symplectic manifolds.
More informationTesi di Laurea Magistrale in Matematica presentata da. Claudia Dennetta. Symplectic Geometry. Il Relatore. Prof. Massimiliano Pontecorvo
UNIVERSITÀ DEGLI STUDI ROMA TRE FACOLTÀ DI SCIENZE M.F.N. Tesi di Laurea Magistrale in Matematica presentata da Claudia Dennetta Symplectic Geometry Relatore Prof. Massimiliano Pontecorvo Il Candidato
More informationSOME HIGHLIGHTS OF SYMPLECTIC GEOMETRY, SUMMER SCHOOL UTRECHT, GEOMETRY, AUGUST 16, 2017
SOME HIGHLIGHTS OF SYMPLECTIC GEOMETRY, SUMMER SCHOOL UTRECHT, GEOMETRY, AUGUST 16, 2017 FABIAN ZILTENER Overview Symplectic geometry originated from classical mechanics, where the canonical symplectic
More informationThe topology of symplectic four-manifolds
The topology of symplectic four-manifolds Michael Usher January 12, 2007 Definition A symplectic manifold is a pair (M, ω) where 1 M is a smooth manifold of some even dimension 2n. 2 ω Ω 2 (M) is a two-form
More informationLIE ALGEBROIDS AND POISSON GEOMETRY, OLIVETTI SEMINAR NOTES
LIE ALGEBROIDS AND POISSON GEOMETRY, OLIVETTI SEMINAR NOTES BENJAMIN HOFFMAN 1. Outline Lie algebroids are the infinitesimal counterpart of Lie groupoids, which generalize how we can talk about symmetries
More informationLecture I: Overview and motivation
Lecture I: Overview and motivation Jonathan Evans 23rd September 2010 Jonathan Evans () Lecture I: Overview and motivation 23rd September 2010 1 / 31 Difficulty of exercises is denoted by card suits in
More informationExercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1
Assistant: Saskia Voss Sheet 1 1. Conformal change of Riemannian metrics [3 points] Let (M, g) be a Riemannian manifold. A conformal change is a nonnegative function λ : M (0, ). Such a function defines
More informationPoisson geometry of b-manifolds. Eva Miranda
Poisson geometry of b-manifolds Eva Miranda UPC-Barcelona Rio de Janeiro, July 26, 2010 Eva Miranda (UPC) Poisson 2010 July 26, 2010 1 / 45 Outline 1 Motivation 2 Classification of b-poisson manifolds
More informationHamiltonian flows, cotangent lifts, and momentum maps
Hamiltonian flows, cotangent lifts, and momentum maps Jordan Bell jordan.bell@gmail.com Department of Mathematics, University of Toronto April 3, 2014 1 Symplectic manifolds Let (M, ω) and (N, η) be symplectic
More informationA little taste of symplectic geometry
A little taste of symplectic geometry Timothy Goldberg Thursday, October 4, 2007 Olivetti Club Talk Cornell University 1 2 What is symplectic geometry? Symplectic geometry is the study of the geometry
More informationBRST 2006 (jmf) 7. g X (M) X ξ X. X η = [ξ X,η]. (X θ)(η) := X θ(η) θ(x η) = ξ X θ(η) θ([ξ X,η]).
BRST 2006 (jmf) 7 Lecture 2: Symplectic reduction In this lecture we discuss group actions on symplectic manifolds and symplectic reduction. We start with some generalities about group actions on manifolds.
More informationEva Miranda. UPC-Barcelona. (joint with Victor Guillemin and Ana Rita Pires) Zaragoza, February
From b-poisson manifolds to symplectic mapping torus and back Eva Miranda UPC-Barcelona (joint with Victor Guillemin and Ana Rita Pires) Zaragoza, February 8 2011 Eva Miranda (UPC) Poisson Day February
More informationUNCERTAINTY PRINCIPLE, NON-SQUEEZING THEOREM AND THE SYMPLECTIC RIGIDITY LECTURE FOR THE 1995 DAEWOO WORKSHOP, CHUNGWON, KOREA
UNCERTAINTY PRINCIPLE, NON-SQUEEZING THEOREM AND THE SYMPLECTIC RIGIDITY LECTURE FOR THE 1995 DAEWOO WORKSHOP, CHUNGWON, KOREA Yong-Geun Oh, Department of Mathematics University of Wisconsin Madison, WI
More information1.2. Examples of symplectic manifolds 1. (R 2n, ω 0 = n
Introduction to the geometry of hamiltonian diffeomorphisms Augustin Banyaga The Pennsylvania State University, University Park, PA 16803, USA Lecture Notes of a mini-course delivered at the Seminaire
More informationCOMPUTING THE POISSON COHOMOLOGY OF A B-POISSON MANIFOLD
COMPUTING THE POISSON COHOMOLOGY OF A B-POISSON MANIFOLD MELINDA LANIUS 1. introduction Because Poisson cohomology is quite challenging to compute, there are only very select cases where the answer is
More informationDierential geometry for Physicists
Dierential geometry for Physicists (What we discussed in the course of lectures) Marián Fecko Comenius University, Bratislava Syllabus of lectures delivered at University of Regensburg in June and July
More informationEmbedded contact homology and its applications
Embedded contact homology and its applications Michael Hutchings Department of Mathematics University of California, Berkeley International Congress of Mathematicians, Hyderabad August 26, 2010 arxiv:1003.3209
More informationMany of the exercises are taken from the books referred at the end of the document.
Exercises in Geometry I University of Bonn, Winter semester 2014/15 Prof. Christian Blohmann Assistant: Néstor León Delgado The collection of exercises here presented corresponds to the exercises for the
More informationCALCULUS ON MANIFOLDS. 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M =
CALCULUS ON MANIFOLDS 1. Riemannian manifolds Recall that for any smooth manifold M, dim M = n, the union T M = a M T am, called the tangent bundle, is itself a smooth manifold, dim T M = 2n. Example 1.
More informationSYMPLECTIC CAPACITY AND CONVEXITY
SYMPLECTIC CAPACITY AND CONVEXITY MICHAEL BAILEY 1. Symplectic Capacities Gromov s nonsqueezing theorem shows that the radii of balls and cylinders are stored somehow as a symplectic invariant. In particular,
More informationA 2-COCYCLE ON A GROUP OF SYMPLECTOMORPHISMS
A 2-COCYCLE ON A GROUP OF SYMPLECTOMORPHISMS RAIS S. ISMAGILOV, MARK LOSIK, PETER W. MICHOR Abstract. For a symplectic manifold (M, ω) with exact symplectic form we construct a 2-cocycle on the group of
More informationGEOMETRIC QUANTIZATION
GEOMETRIC QUANTIZATION 1. The basic idea The setting of the Hamiltonian version of classical (Newtonian) mechanics is the phase space (position and momentum), which is a symplectic manifold. The typical
More informationReeb dynamics in dimension 3
Reeb dynamics in dimension 3 Dan Cristofaro-Gardiner Member, Institute for Advanced study Institute for Advanced Study September 24, 2013 Thank you! Thank you for inviting me to give this talk! Reeb vector
More informationSYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS. 1. Introduction
SYMPLECTIC MANIFOLDS, GEOMETRIC QUANTIZATION, AND UNITARY REPRESENTATIONS OF LIE GROUPS CRAIG JACKSON 1. Introduction Generally speaking, geometric quantization is a scheme for associating Hilbert spaces
More informationfy (X(g)) Y (f)x(g) gy (X(f)) Y (g)x(f)) = fx(y (g)) + gx(y (f)) fy (X(g)) gy (X(f))
1. Basic algebra of vector fields Let V be a finite dimensional vector space over R. Recall that V = {L : V R} is defined to be the set of all linear maps to R. V is isomorphic to V, but there is no canonical
More informationApplications of embedded contact homology
Applications of embedded contact homology Michael Hutchings Department of Mathematics University of California, Berkeley conference in honor of Michael Freedman June 8, 2011 Michael Hutchings (UC Berkeley)
More informationDifferential Geometry qualifying exam 562 January 2019 Show all your work for full credit
Differential Geometry qualifying exam 562 January 2019 Show all your work for full credit 1. (a) Show that the set M R 3 defined by the equation (1 z 2 )(x 2 + y 2 ) = 1 is a smooth submanifold of R 3.
More information(g 1, g 2 ) g 1 g 2. i : G G
1 Lie Groups Definition (4.1 1 A Lie Group G is a set that is a group a differential manifold with the property that and are smooth. µ : G G G (g 1, g 2 g 1 g 2 i : G G g g 1 Definition (4.1 2 A Lie Subgroup
More informationExercise 1 (Formula for connection 1-forms) Using the first structure equation, show that
1 Stokes s Theorem Let D R 2 be a connected compact smooth domain, so that D is a smooth embedded circle. Given a smooth function f : D R, define fdx dy fdxdy, D where the left-hand side is the integral
More informationLECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS. 1. Lie groups
LECTURE 16: LIE GROUPS AND THEIR LIE ALGEBRAS 1. Lie groups A Lie group is a special smooth manifold on which there is a group structure, and moreover, the two structures are compatible. Lie groups are
More information14 Higher order forms; divergence theorem
Tel Aviv University, 2013/14 Analysis-III,IV 221 14 Higher order forms; divergence theorem 14a Forms of order three................ 221 14b Divergence theorem in three dimensions.... 225 14c Order four,
More informationIntroduction to Index Theory. Elmar Schrohe Institut für Analysis
Introduction to Index Theory Elmar Schrohe Institut für Analysis Basics Background In analysis and pde, you want to solve equations. In good cases: Linearize, end up with Au = f, where A L(E, F ) is a
More informationPREQUANTIZATION OF SYMPLECTIC SUPERMANIFOLDS
Ninth International Conference on Geometry, Integrability and Quantization June 8 13, 2007, Varna, Bulgaria Ivaïlo M. Mladenov, Editor SOFTEX, Sofia 2008, pp 301 307 Geometry, Integrability and Quantization
More informationContact pairs (bicontact manifolds)
Contact pairs (bicontact manifolds) Gianluca Bande Università degli Studi di Cagliari XVII Geometrical Seminar, Zlatibor 6 September 2012 G. Bande (Università di Cagliari) Contact pairs (bicontact manifolds)
More informationMath 426H (Differential Geometry) Final Exam April 24, 2006.
Math 426H Differential Geometry Final Exam April 24, 6. 8 8 8 6 1. Let M be a surface and let : [0, 1] M be a smooth loop. Let φ be a 1-form on M. a Suppose φ is exact i.e. φ = df for some f : M R. Show
More informationVortex Equations on Riemannian Surfaces
Vortex Equations on Riemannian Surfaces Amanda Hood, Khoa Nguyen, Joseph Shao Advisor: Chris Woodward Vortex Equations on Riemannian Surfaces p.1/36 Introduction Yang Mills equations: originated from electromagnetism
More informationAn Introduction to Symplectic Geometry
An Introduction to Symplectic Geometry Alessandro Fasse Institute for Theoretical Physics University of Cologne These notes are a short sum up about two talks that I gave in August and September 2015 an
More informationMath 868 Final Exam. Part 1. Complete 5 of the following 7 sentences to make a precise definition (5 points each). Y (φ t ) Y lim
SOLUTIONS Dec 13, 218 Math 868 Final Exam In this exam, all manifolds, maps, vector fields, etc. are smooth. Part 1. Complete 5 of the following 7 sentences to make a precise definition (5 points each).
More informationThe Poisson Embedding Problem
The Poisson Embedding Problem Symposium in honor of Alan Weinstein Aaron McMillan Fraenkel Boston College July 2013 Notation / Terminology By a Poisson structure on a space X, we mean: A geometric space
More informationB 1 = {B(x, r) x = (x 1, x 2 ) H, 0 < r < x 2 }. (a) Show that B = B 1 B 2 is a basis for a topology on X.
Math 6342/7350: Topology and Geometry Sample Preliminary Exam Questions 1. For each of the following topological spaces X i, determine whether X i and X i X i are homeomorphic. (a) X 1 = [0, 1] (b) X 2
More informationBACKGROUND IN SYMPLECTIC GEOMETRY
BACKGROUND IN SYMPLECTIC GEOMETRY NILAY KUMAR Today I want to introduce some of the symplectic structure underlying classical mechanics. The key idea is actually quite old and in its various formulations
More informationDeformations of coisotropic submanifolds in symplectic geometry
Deformations of coisotropic submanifolds in symplectic geometry Marco Zambon IAP annual meeting 2015 Symplectic manifolds Definition Let M be a manifold. A symplectic form is a two-form ω Ω 2 (M) which
More informationChap. 1. Some Differential Geometric Tools
Chap. 1. Some Differential Geometric Tools 1. Manifold, Diffeomorphism 1.1. The Implicit Function Theorem ϕ : U R n R n p (0 p < n), of class C k (k 1) x 0 U such that ϕ(x 0 ) = 0 rank Dϕ(x) = n p x U
More informationLecture III: Neighbourhoods
Lecture III: Neighbourhoods Jonathan Evans 7th October 2010 Jonathan Evans () Lecture III: Neighbourhoods 7th October 2010 1 / 18 Jonathan Evans () Lecture III: Neighbourhoods 7th October 2010 2 / 18 In
More informationCalderón-Zygmund inequality on noncompact Riem. manifolds
The Calderón-Zygmund inequality on noncompact Riemannian manifolds Institut für Mathematik Humboldt-Universität zu Berlin Geometric Structures and Spectral Invariants Berlin, May 16, 2014 This talk is
More informationTwo simple ideas from calculus applied to Riemannian geometry
Calibrated Geometries and Special Holonomy p. 1/29 Two simple ideas from calculus applied to Riemannian geometry Spiro Karigiannis karigiannis@math.uwaterloo.ca Department of Pure Mathematics, University
More informationCOTANGENT MODELS FOR INTEGRABLE SYSTEMS
COTANGENT MODELS FOR INTEGRABLE SYSTEMS ANNA KIESENHOFER AND EVA MIRANDA Abstract. We associate cotangent models to a neighbourhood of a Liouville torus in symplectic and Poisson manifolds focusing on
More informationLECTURE 1: LINEAR SYMPLECTIC GEOMETRY
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 *********************************************************************************
More informationTwisted Poisson manifolds and their almost symplectically complete isotropic realizations
Twisted Poisson manifolds and their almost symplectically complete isotropic realizations Chi-Kwong Fok National Center for Theoretical Sciences Math Division National Tsing Hua University (Joint work
More informationA LITTLE TASTE OF SYMPLECTIC GEOMETRY: THE SCHUR-HORN THEOREM CONTENTS
A LITTLE TASTE OF SYMPLECTIC GEOMETRY: THE SCHUR-HORN THEOREM TIMOTHY E. GOLDBERG ABSTRACT. This is a handout for a talk given at Bard College on Tuesday, 1 May 2007 by the author. It gives careful versions
More informationEuler Characteristic of Two-Dimensional Manifolds
Euler Characteristic of Two-Dimensional Manifolds M. Hafiz Khusyairi August 2008 In this work we will discuss an important notion from topology, namely Euler Characteristic and we will discuss several
More informationCHARACTERISTIC CLASSES
1 CHARACTERISTIC CLASSES Andrew Ranicki Index theory seminar 14th February, 2011 2 The Index Theorem identifies Introduction analytic index = topological index for a differential operator on a compact
More informationNotes for Math 535 Differential Geometry Spring Francis Bonahon. Department of Mathematics, University of Southern California
Notes for Math 535 Differential Geometry Spring 2016 Francis Bonahon Department of Mathematics, University of Southern California Date of this version: April 27, 2016 c Francis Bonahon 2016 CHAPTER 1 A
More informationTHE NEWLANDER-NIRENBERG THEOREM. GL. The frame bundle F GL is given by x M Fx
THE NEWLANDER-NIRENBERG THEOREM BEN MCMILLAN Abstract. For any kind of geometry on smooth manifolds (Riemannian, Complex, foliation,...) it is of fundamental importance to be able to determine when two
More informationRelation of the covariant and Lie derivatives and its application to Hydrodynamics
Relation of the covariant and Lie derivatives and its application to Hydrodynamics Yang Cao TU Darmstadt March 12, 2010 The Euler hydrodynamic equation on manifolds Let M denote a compact oriented Riemannian
More information4-MANIFOLDS: CLASSIFICATION AND EXAMPLES. 1. Outline
4-MANIFOLDS: CLASSIFICATION AND EXAMPLES 1. Outline Throughout, 4-manifold will be used to mean closed, oriented, simply-connected 4-manifold. Hopefully I will remember to append smooth wherever necessary.
More informationMATRIX LIE GROUPS AND LIE GROUPS
MATRIX LIE GROUPS AND LIE GROUPS Steven Sy December 7, 2005 I MATRIX LIE GROUPS Definition: A matrix Lie group is a closed subgroup of Thus if is any sequence of matrices in, and for some, then either
More informationA MARSDEN WEINSTEIN REDUCTION THEOREM FOR PRESYMPLECTIC MANIFOLDS
A MARSDEN WEINSTEIN REDUCTION THEOREM FOR PRESYMPLECTIC MANIFOLDS FRANCESCO BOTTACIN Abstract. In this paper we prove an analogue of the Marsden Weinstein reduction theorem for presymplectic actions of
More informationEXERCISES IN POISSON GEOMETRY
EXERCISES IN POISSON GEOMETRY The suggested problems for the exercise sessions #1 and #2 are marked with an asterisk. The material from the last section will be discussed in lecture IV, but it s possible
More informationLie 2-algebras from 2-plectic geometry
Lie 2-algebras from 2-plectic geometry Chris Rogers joint with John Baez and Alex Hoffnung Department of Mathematics University of California, Riverside XVIIIth Oporto Meeting on Geometry, Topology and
More informationLie algebra cohomology
Lie algebra cohomology Relation to the de Rham cohomology of Lie groups Presented by: Gazmend Mavraj (Master Mathematics and Diploma Physics) Supervisor: J-Prof. Dr. Christoph Wockel (Section Algebra and
More informationInvariant Nonholonomic Riemannian Structures on Three-Dimensional Lie Groups
Invariant Nonholonomic Riemannian Structures on Three-Dimensional Lie Groups Dennis I. Barrett Geometry, Graphs and Control (GGC) Research Group Department of Mathematics, Rhodes University Grahamstown,
More informationLECTURE 9: MOVING FRAMES IN THE NONHOMOGENOUS CASE: FRAME BUNDLES. 1. Introduction
LECTURE 9: MOVING FRAMES IN THE NONHOMOGENOUS CASE: FRAME BUNDLES 1. Introduction Until now we have been considering homogenous spaces G/H where G is a Lie group and H is a closed subgroup. The natural
More informationLecture XI: The non-kähler world
Lecture XI: The non-kähler world Jonathan Evans 2nd December 2010 Jonathan Evans () Lecture XI: The non-kähler world 2nd December 2010 1 / 21 We ve spent most of the course so far discussing examples of
More informationMORSE HOMOLOGY. Contents. Manifolds are closed. Fields are Z/2.
MORSE HOMOLOGY STUDENT GEOMETRY AND TOPOLOGY SEMINAR FEBRUARY 26, 2015 MORGAN WEILER 1. Morse Functions 1 Morse Lemma 3 Existence 3 Genericness 4 Topology 4 2. The Morse Chain Complex 4 Generators 5 Differential
More informationNOTES ON DIFFERENTIAL FORMS. PART 1: FORMS ON R n
NOTES ON DIFFERENTIAL FORMS. PART 1: FORMS ON R n 1. What is a form? Since we re not following the development in Guillemin and Pollack, I d better write up an alternate approach. In this approach, we
More informationThe Geometry of Euler s equation. Introduction
The Geometry of Euler s equation Introduction Part 1 Mechanical systems with constraints, symmetries flexible joint fixed length In principle can be dealt with by applying F=ma, but this can become complicated
More informationAPPLICATIONS OF HOFER S GEOMETRY TO HAMILTONIAN DYNAMICS
APPLICATIONS OF HOFER S GEOMETRY TO HAMILTONIAN DYNAMICS FELIX SCHLENK Abstract. We prove that for every subset A of a tame symplectic manifold (W, ω) the π 1 -sensitive Hofer Zehnder capacity of A is
More informationMath 550 / David Dumas / Fall Problems
Math 550 / David Dumas / Fall 2014 Problems Please note: This list was last updated on November 30, 2014. Problems marked with * are challenge problems. Some problems are adapted from the course texts;
More informationLecture 5 - Hausdorff and Gromov-Hausdorff Distance
Lecture 5 - Hausdorff and Gromov-Hausdorff Distance August 1, 2011 1 Definition and Basic Properties Given a metric space X, the set of closed sets of X supports a metric, the Hausdorff metric. If A is
More informationLECTURE 8: THE MOMENT MAP
LECTURE 8: THE MOMENT MAP Contents 1. Properties of the moment map 1 2. Existence and Uniqueness of the moment map 4 3. Examples/Exercises of moment maps 7 4. Moment map in gauge theory 9 1. Properties
More informationRIEMANNIAN MANIFOLDS WITH INTEGRABLE GEODESIC FLOWS
RIEMANNIAN MANIFOLDS WITH INTEGRABLE GEODESIC FLOWS ANDREW MILLER 1. Introduction In this paper we will survey some recent results on the Hamiltonian dynamics of the geodesic flow of a Riemannian manifold.
More informationAn integral lift of contact homology
Columbia University University of Pennsylvannia, January 2017 Classical mechanics The phase space R 2n of a system consists of the position and momentum of a particle. Lagrange: The equations of motion
More informationCOMPUTABILITY AND THE GROWTH RATE OF SYMPLECTIC HOMOLOGY
COMPUTABILITY AND THE GROWTH RATE OF SYMPLECTIC HOMOLOGY MARK MCLEAN arxiv:1109.4466v1 [math.sg] 21 Sep 2011 Abstract. For each n greater than 7 we explicitly construct a sequence of Stein manifolds diffeomorphic
More informationTopics in Representation Theory: Lie Groups, Lie Algebras and the Exponential Map
Topics in Representation Theory: Lie Groups, Lie Algebras and the Exponential Map Most of the groups we will be considering this semester will be matrix groups, i.e. subgroups of G = Aut(V ), the group
More informationContact and Symplectic Geometry of Monge-Ampère Equations: Introduction and Examples
Contact and Symplectic Geometry of Monge-Ampère Equations: Introduction and Examples Vladimir Rubtsov, ITEP,Moscow and LAREMA, Université d Angers Workshop "Geometry and Fluids" Clay Mathematical Institute,
More informationL 2 Geometry of the Symplectomorphism Group
University of Notre Dame Workshop on Innite Dimensional Geometry, Vienna 2015 Outline 1 The Exponential Map on D s ω(m) 2 Existence of Multiplicity of Outline 1 The Exponential Map on D s ω(m) 2 Existence
More informationANSWERS TO VARIOUS HOMEWORK PROBLEMS IN MATH 3210, FALL 2015
ANSWERS TO VARIOUS HOMEWORK PROBLEMS IN MATH 3210, FALL 2015 HOMEWORK #11 (DUE FRIDAY DEC 4) Let M be a manifold. Define the Poincaré polynomial p M (t) := dim M i=0 (dim H i (M))t i and the Euler characteristic
More informationSection 2. Basic formulas and identities in Riemannian geometry
Section 2. Basic formulas and identities in Riemannian geometry Weimin Sheng and 1. Bianchi identities The first and second Bianchi identities are R ijkl + R iklj + R iljk = 0 R ijkl,m + R ijlm,k + R ijmk,l
More informationDirac structures. Henrique Bursztyn, IMPA. Geometry, mechanics and dynamics: the legacy of J. Marsden Fields Institute, July 2012
Dirac structures Henrique Bursztyn, IMPA Geometry, mechanics and dynamics: the legacy of J. Marsden Fields Institute, July 2012 Outline: 1. Mechanics and constraints (Dirac s theory) 2. Degenerate symplectic
More informationTHE FUNDAMENTAL GROUP OF MANIFOLDS OF POSITIVE ISOTROPIC CURVATURE AND SURFACE GROUPS
THE FUNDAMENTAL GROUP OF MANIFOLDS OF POSITIVE ISOTROPIC CURVATURE AND SURFACE GROUPS AILANA FRASER AND JON WOLFSON Abstract. In this paper we study the topology of compact manifolds of positive isotropic
More informationMath 147, Homework 5 Solutions Due: May 15, 2012
Math 147, Homework 5 Solutions Due: May 15, 2012 1 Let f : R 3 R 6 and φ : R 3 R 3 be the smooth maps defined by: f(x, y, z) = (x 2, y 2, z 2, xy, xz, yz) and φ(x, y, z) = ( x, y, z) (a) Show that f is
More information4 Divergence theorem and its consequences
Tel Aviv University, 205/6 Analysis-IV 65 4 Divergence theorem and its consequences 4a Divergence and flux................. 65 4b Piecewise smooth case............... 67 4c Divergence of gradient: Laplacian........
More informationLECTURE 4: SYMPLECTIC GROUP ACTIONS
LECTURE 4: SYMPLECTIC GROUP ACTIONS WEIMIN CHEN, UMASS, SPRING 07 1. Symplectic circle actions We set S 1 = R/2πZ throughout. Let (M, ω) be a symplectic manifold. A symplectic S 1 -action on (M, ω) is
More informationComplex manifolds, Kahler metrics, differential and harmonic forms
Complex manifolds, Kahler metrics, differential and harmonic forms Cattani June 16, 2010 1 Lecture 1 Definition 1.1 (Complex Manifold). A complex manifold is a manifold with coordinates holomorphic on
More informationLECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY
LECTURE 5: SOME BASIC CONSTRUCTIONS IN SYMPLECTIC TOPOLOGY WEIMIN CHEN, UMASS, SPRING 07 1. Blowing up and symplectic cutting In complex geometry the blowing-up operation amounts to replace a point in
More informationPractice Qualifying Exam Questions, Differentiable Manifolds, Fall, 2009.
Practice Qualifying Exam Questions, Differentiable Manifolds, Fall, 2009. Solutions (1) Let Γ be a discrete group acting on a manifold M. (a) Define what it means for Γ to act freely. Solution: Γ acts
More informationAN INTRODUCTION TO DIFFERENTIAL FORMS, STOKES THEOREM AND GAUSS-BONNET THEOREM
AN INTRODUCTION TO DIFFERENTIAL FORS, STOKES THEORE AND GAUSS-BONNET THEORE ANUBHAV NANAVATY Abstract. This paper serves as a brief introduction to differential geometry. It first discusses the language
More informationA BRIEF INTRODUCTION TO DIRAC MANIFOLDS
A BRIEF INTRODUCTION TO DIRAC MANIFOLDS HENRIQUE BURSZTYN Abstract. These notes are based on a series of lectures given at the school on Geometric and Topological Methods for Quantum Field Theory, in Villa
More informationIntroduction to Teichmüller Spaces
Introduction to Teichmüller Spaces Jing Tao Notes by Serena Yuan 1. Riemann Surfaces Definition 1.1. A conformal structure is an atlas on a manifold such that the differentials of the transition maps lie
More informationA NEW CONSTRUCTION OF POISSON MANIFOLDS
A NEW CONSTRUCTION OF POISSON MANIFOLDS A. IBORT, D. MARTÍNEZ TORRES Abstract A new technique to construct Poisson manifolds inspired both in surgery ideas used to define Poisson structures on 3-manifolds
More informationM4P52 Manifolds, 2016 Problem Sheet 1
Problem Sheet. Let X and Y be n-dimensional topological manifolds. Prove that the disjoint union X Y is an n-dimensional topological manifold. Is S S 2 a topological manifold? 2. Recall that that the discrete
More informationGroups up to quasi-isometry
OSU November 29, 2007 1 Introduction 2 3 Topological methods in group theory via the fundamental group. group theory topology group Γ, a topological space X with π 1 (X) = Γ. Γ acts on the universal cover
More informationGeneralized complex geometry
Generalized complex geometry Marco Gualtieri Abstract Generalized complex geometry encompasses complex and symplectic geometry as its extremal special cases. We explore the basic properties of this geometry,
More informationON NEARLY SEMIFREE CIRCLE ACTIONS
ON NEARLY SEMIFREE CIRCLE ACTIONS DUSA MCDUFF AND SUSAN TOLMAN Abstract. Recall that an effective circle action is semifree if the stabilizer subgroup of each point is connected. We show that if (M, ω)
More informationNicholas Proudfoot Department of Mathematics, University of Oregon, Eugene, OR 97403
Symplectic Geometry Nicholas Proudfoot Department of Mathematics, University of Oregon, Eugene, OR 97403 These notes are written for a ten week graduate class on symplectic geometry. Most of the material
More informationGlobal Lie-Tresse theorem
Global Lie-Tresse theorem Boris Kruglikov University of Tromsø, Norway (in part joint with: V.Lychagin, O. Morozov, E. Ferapontov) Group actions and classical Invariants The classical problem in invariant
More informationLECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES
LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES 1. Vector Bundles In general, smooth manifolds are very non-linear. However, there exist many smooth manifolds which admit very nice partial linear structures.
More informationInfinitesimal Einstein Deformations. Kähler Manifolds
on Nearly Kähler Manifolds (joint work with P.-A. Nagy and U. Semmelmann) Gemeinsame Jahrestagung DMV GDM Berlin, March 30, 2007 Nearly Kähler manifolds Definition and first properties Examples of NK manifolds
More information