Exercises in Geometry II University of Bonn, Summer semester 2015 Professor: Prof. Christian Blohmann Assistant: Saskia Voss Sheet 1


 Logan Rose
 2 years ago
 Views:
Transcription
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 a new Riemannian manifold (M, λ 2 g). Show that any Riemannian manifold (M, g) admits a conformal change λ, such that (M, λ 2 g) is complete. 2. Tautological bundle of RP 1 [3 points] Let E := {(p, v) RP 1 R 2 v p}. a) Show that E is a smooth line bundle. b) Is E trivial? Justify your answer. 3. Integration of Vectorfields [3 points] Calculate the flow of each vector field and sketch the flowlines. a) V = x 2 x + xy y b) W = (x + y) + (x y) x y c) X = (x y) x + x y 4. Metric connections on vector bundles [3 points] Let π : E M be a vector bundle equipped with a fiberwise metric.,.. Show that E admits a linear metric connection. Recall: A linear connection is called metric if the covariant derivative satisfies v e, e = v e, e + e, v e for all v X (M) and e, e Γ (M, E).
2 Sheet 2 1. Exact forms on S 1 [3 points] Show that a oneform ω Ω 1 (S 1 ) on S 1 is exact if and only if S 1 ω = 0 holds. 2. Harmonic functions [3 points] On an oriented Riemannian manifold (M, g) the Laplacian is a linear operator : C (M) C (M) defined by u = div(grad u). Here div denotes the divergence where the index µ is ommitted, since we fix µ to be the volume form such that orthonormal bases get 1 in coordinates. A function u C (M) is called harmonic if u = 0. a) Assume M to be compact. Show Green s identities: u v µ = grad u, grad v g µ u n(v) µ (1) M M M (u v v u) µ = (v n(u) u n(v)) µ (2) M M Here n denotes the outward unit normal vector field along M and µ the volume form of M with respect to the induced metric on M. b) Show that the only harmonic functions on a closed Riemannian manifold M are the constants. c) Assume M to be a compact and connected Riemannian manifold with nonempty boundary. Let u, v be harmonic functions on M. Show that if the restrictions of u and v to M agrees then u v. 3. Divergence and Laplacian in coordinates [3 points] Let (M, g) be an ndimensional oriented Riemannian manifold and (x 1,... x n ) smooth local coordinates. a) Show that in these coordinated the divergence is given by ( ) 1 ( ) div X i = X i det(g) x i det(g) x i for any vector field X = X i x i. b) Show that the Laplacian can be written as for any smooth function u. 1 u = det(g) x i ( g ij det(g) u x j c) Conclude that on R n with the Euclidean metric and standard coordinates we obtain ( ) n X i n 2 u div X i =, u = xi x i ( x i ). 2 i=1 i=1 ),
3 4. Stoke s Theorem [3 points] Let T 2 := S 1 S 1 R 4 be the 2torus defined by w 2 + x 2 = y 2 + z 2 = 1. Compute T 2 xyz dw dy.
4 Sheet 3 1. First and second fundamental form [3 points] Let M i R n+1 be an embedded hypersurface. Denote by G : M S n the Gauß map. Show that i(m) takes values in a sphere of radius 1 if the first and the second fundamental form coincide. 2. Cartan s structure equation [3 points] Let be the LeviCivita connection on a Riemannian manifold (M, g). Denote by {e i } a local orthonormal frame on some open subset U M and let {θ i } be the dual coframe, i.e. θ i (e j ) = δ ij. The connection 1forms ω j i are defined by for all v T M. v e i = ω j i (v)e j a) Prove Cartan s first structure equation: b) Define the curvature 2forms Ω j i by dθ j = θ i ω j i. Ω j i (u, v)e j := R(u, v)e i for all vector fields u, v X (M), where R(u, v) = u v v u [u,v] is the curvature operator. Show Cartan s second structure equation: Ω j i = dωj i + ωk i ω j k. 3. GaußBonnet for closed embedded surfaces [3 points] Let M i R 3 be a closed embedded surface with the induced metric. a) Show that M can t have K 0 everywhere. b) Show that M can t have K 0 everywhere unless χ(m) > Immersed curves in R 2 [3 points] Let γ : (a, b) R 2 be an immersed curve with γ(t) = 1 for all t (a, b). a) Calculate the components of the second fundamental form. b) Show that γ is uniquely determined by its first and second fundamental form up to rotation and translation. Hint: Identify R 2 with C.
5 Sheet 4 1. Index of a Killing vector field [3 points] A vector field X X (M) on a Riemannian manifold (M, g) is called a Killing vector field if L X g = 0, i.e. if its integrating flow consists of local isometries. Show that the index of any isolated zero of a Killing field X on a Riemannian surface is Geodesic polygons [3 points] Let (M, g) be a Riemannian surface. A curved polygon in M whose sides are geodsic segments is called a geodesic polygon. a) Show that there are no geodesic polygons with exactly 0, 1 or 2 vertices if g has everywhere nonpositive Gaussian curvature. b) Give examples of geodesic polygons with 0, 1 and 2 vertices on surfaces for which the curvature hypothesis of a) is not satisfied. 3. Metrics on S 2 [3 points] Let g be a metric on S 2 with Gaussian curvature K 0. Using the GaußBonnet theorem, prove the inequality where g denotes the standard metric on S 2. vol(s 2, g) vol(s 2, g) = 4π, 4. Geodesics in different metrics [3 points] Let γ be a geodesic in a Riemannian manifold (M, g). Let g be another Riemannian metric on M such that g ( γ, γ) = g( γ, γ) holds and g (X, γ) = 0 if and only if g(x, γ) = 0. Show that γ is also a geodesic with respect to g.
6 Sheet 5 1. Conjugate points [3 points] Let (M, g) be a complete Riemannian manifold. Let SM := {v T M v = 1} denote the sphere bundle. For all v SM we define con(v) (0, ] to be the first t > 0 such that γ v (t) a conjugate point to γ(0). Here γ v denotes the geodesic with γ v (0) = v. Show that con( γ v (con(v))) = con(v) holds for all v SM. 2. More about the 2torus T 2 [3 points] a) Show that the standard 2torus T 2 R 3 with the induced metric is flat, i.e. its curvature tensor R vanishes everywhere. b) Calculate the geodesics on T 2. c) Show that T 2 has no conjugate points. 3. Jacobi fields on manifolds with nonpositive sectional curvature [3 points] Let (M, g) be a Riemannian manifold with nonpositive sectional curvature. a) Let c : [a, b] M be a differentiable curve and J be a Jacobi field along it. Define f(t) = J(t) 2. Show that f is a convex function, i.e. f (t) 0 for all t. b) Conclude from a) that M has no conjugate points. 4. Killing and Jacobi fields [3 points] Let X X (M) be a Killing field on a Riemannian manifold (M, g). Show that for every geodesic γ : [a, b] M the vector field X γ is a Jacobi field along γ.
7 Sheet 6 1. Jacobi fields [3 points] Let γ be a geodesic in a Riemannian manifold (M, g) and J 1, J 2 Jacobi fields along γ. Show that is constant. 2. FermiWalker transport [6 points] g( J 1, J 2 ) g(j 1, J 2 ) Let γ : [a, b] M be a curve into a Riemannian manifold, such that γ(t) never vanishes and let T := the unit tangent of γ. We say that V is a FermiWalker field along γ if V satisfies γ γ V = g(v, T ) T g(v, T )T = ( T T )(V ). a) Show that for given V (t 0 ) there is a unique FermiWalker field V along γ whose value at t 0 is the given value V (t 0 ). b) Show that T is a FermiWalker field along γ. c) Show that g(v, W ) is constant along γ if V and W are FermiWalker fields along γ. d) Show that FermiWalker fields along geodesics are parallel. 3. Taylor series of Riemannian metrics [3 points] Let (M, g) be a Riemannian manifold and p a point in M. Show that the second order Taylor series of g is g ij (x) = δ ij 1 3 n R iklj x k x l + O( x 3 ), k,l=1 in Riemannian normal coordinates (x 1,..., x n ) centered in p. Hint: Consider a radial geodesic γ(t) = (tv 1,..., tv n ) and a Jacobi field J(t) = tw i i along γ. Compute the first four tderivatives of J(t) 2 at t = 0 in two different ways using the Jacobiequation.
8 Sheet 7 1. Transversality [3 points] Two smooth maps f : M N and g : P N are called transverse, denoted by f g, if whenever f(x) = g(y) = z, the image of T x f and T y g span T z N. Show that f g if and only if f g : M P N N is transverse to the diagonal : N N N, n (n, n). 2. Quotients [3 points] Let M be a differentiable manifold, τ : M M an involution without fixed points, i.e. τ τ = id, τ(x) x for all x M. We call points x and y equivalent if y = τ(x). Show that the space M/τ of equivalence classes possesses a unique differentiable structure for which the projection M M/τ is a local diffeomorphism. For further thought [0 points]: What goes wrong with the differentiable structure of the quotient when τ has fixed points? 3. Covering maps [3 points] Let M be a connected smooth manifold, and let π : M M be a topological covering map. Show that there is only one smooth structure on M such that π is a smooth covering map. 4. Rank of maps of vectorbundles [3 points] Let f : E F a map of vector bundles over the same base manifold M. The rank of f is given by the function rank f : M N 0, m dim ( f(e m ) ), where E m is the fiber over m. Show that rank f is lower semicontinuous.
9 Sheet 8 1. Group actions [3 points] Define an action of Z on R 2 by n (x, y) = (x + n, ( 1) n y). a) Show that this action is smooth free and proper. Let E := R 2 /Z denote the orbit space. b) Show that the projection π 1 : R 2 R onto the first coordinate descends to a smooth map π : E S 1. c) Conclude that E is a nontrivial smooth rank1 vector bundle over S 1 with projection π. 2. Quotients and proper Group actions [3 points] Let G be a Lie Group acting smoothly and freely on a smooth manifold M. Assume further that the orbit space M/G has a smooth manifold structure such that the quotient map π : M M/G is a smooth submersion. Show that G acts properly. 3. Group actions on discrete spaces [6 points] Let G be a connected Lie Group. a) Suppose that G acts smoothly on a discrete space K. Show that the action is trivial. b) Show that any discrete normal subgroup of G is contained in the center of G. c) Show that π 1 (G, e) is abelian. Hint: Consider the covering group on the universal covering G of G.
10 Sheet 9 1. Trivial principal bundle [3 points] Let G be a Lie group and P π M be a Gprincipal bundle. Show that the following are equivalent: i) P is trivializable. ii) P π M has a section. 2. Lie subgroups and principal bundles [3 points] a) Let P M be an Hprincipal bundle and H G a Lie subgroup. Show that the associated bundle P H G M is naturally a Gprincipal bundle. b) A reduction of a G bundle P M to an H bundle is a pair consisting of an H bundle P M and an isomorphism of G bundles P H G P. Show that a Gprincipal bundle reduces to the subgroup H = {1} if and only if the G bundle P is trivial. 3. Invariant metric on homogenous space [3 points] Let G be a Lie group and M a homogenous Gspace. Suppose further that there is an m M such that the stabilizer {g G g m = m} is compact. a) Show that there exists a Ginvariant Riemannian metric on M, i.e. a Riemannian metric for which G acts by isometries. b) Is this also true for noncompact stabilizer? Justify your answer. 4. Vertical tangent bundle of a principal bundle [3 points] Let P π M be a Gprincipal bundle for a Lie group G. Let E := Ker(T π), where T π : T P T M is the tangent map. Show that the vector bundle E is isomorphic to the associated bundle P Ad g, where g is the Lie algebra of G and Ad the adjoint representation of G.
11 Sheet Frame bundle [3 points] The frame bundle of a manifold M is the bundle with fiber over m M given by the set of all bases of T m M. (A basis of T m M is also called a frame.) Show that the frame bundle is a prinicpal GL(n, R)bundle. 2. Equivariant cohomology [3 points] Let G be a compact connected Lie group and P a manifold with a free Gaction. Let ρ : g X (M) be the structure homomorphism of the associated infinitesimal action. A differential form ϕ Ω(P ) is called (i) horizontal if i ρ(a) ϕ = 0, (ii) invariant if L ρ(a) ϕ = 0, for all a g. It is called basic if it is both horizontal and invariant. Show that ϕ Ω(P ) is the pullback of a form on P/G if and only if ϕ is basic. 3. Lagrangian subspaces [3 points] Let (V 1, ω 1 ) and (V 2, ω 2 ) be symplectic vector spaces. Show that a linear map f : V 1 V 2 is a morphism of symplectic vector spaces if and only if its graph, Graph(f) := {(v 1, v 2 ) V 1 V 2 v 2 = f(v 1 )}, is a lagrangian subspace of (V 1, ω 1 ) (V 2, ω 1 ). 4. Linear symplectic reduction [3 points] Let (V, ω) be a symplectic vector space and W V a coisotropic subspace. Show that there is a unique symplectic form ω on W/W so that i ω = π ω, where i : W V is the inclusion and π : W W/W the quotient map.
12 Sheet Darboux theorem in 2 dimensions [4 points] (a) Show that any nonvanishing 1form α on a 2dimensional manifold can be written locally as α = fdg for some functions f and g. Show that this is no longer true if the manifold is of dimension higher than 2. (b) Use (a) to prove the Darboux theorem in 2 dimensions: Show that every symplectic form ω on a 2dimensional manifold can be written locally as ω = dq dp. 2. Symplectomorphisms of the cotangent bundle [4 points] Show that the group of fiberpreserving symplectomorphisms of the cotangent bundle T Q (with the canonical symplectic form) is isomorphic to the semidirect product of the group D = Diff(Q) of diffeomorphisms of Q and the additive group Z = Ω 1 cl (Q) of closed 1forms on Q, where D acts on Z by pullback. 3. Hamiltonian functions on a presymplectic manifold [4 points] Let ω Ω 2 (M) be a closed 2form (which is also called a presymplectic form). A smooth function f C (M) is called hamiltonian if there is a vector field v X (M) such that df = i v ω. The Poisson bracket of two hamiltonian functions f, g is defined by {f, g} := ω(v, w), where v and w are vector fields such that i v ω = df and i w ω = dg. Show that this is a welldefined operation that endows the space of hamiltonian functions with the structure of a Lie algebra.
13 Sheet Lagrangian subspaces [5 points] a) Show that the image of a 1form µ Ω 1 (Q) viewed as map µ : Q T Q is a lagrangian submanifold of T Q if and only if it is closed, dµ = 0. b) Let S Q be a submanifold of Q. Recall that the conormalbundle of S is defined as N S := {α T Q α(v) = 0 for all v T S}. Show that N S T Q is a langrangian submanifold. 2. Coisotropic subspaces [4 points] Let M be a symplectic manifold. a) Show that X {f,g} = [X f, X g ] for all smooth functions f, g. Here X f denotes the Hamiltonian vector field of f. b) Let S be a submanifold of M and define I S := {f C (M) f S = 0} to be the ideal of smooth functions vanishing on S. Show that S is coisotropic if and only if I S is closed under the Poisson bracket. 3. Classification of complex structures [3 points] Let (V,.,. ) be a four dimensional euclidean vector space. An almost complex structure J GL(V ) is called compatible if Jv, Jw = v, w for all v, w V. Show that the set of compatible almost complex structures on V is a submanifold of GL(V ) that is diffeomorphic to two copies of S 2.
Many 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 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 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 twoform ω Ω 2 (M) which
More information1. Geometry of the unit tangent bundle
1 1. Geometry of the unit tangent bundle The main reference for this section is [8]. In the following, we consider (M, g) an ndimensional smooth manifold endowed with a Riemannian metric g. 1.1. Notations
More informationLECTURE 3 MATH 261A. Office hours are now settled to be after class on Thursdays from 12 : 30 2 in Evans 815, or still by appointment.
LECTURE 3 MATH 261A LECTURES BY: PROFESSOR DAVID NADLER PROFESSOR NOTES BY: JACKSON VAN DYKE Office hours are now settled to be after class on Thursdays from 12 : 30 2 in Evans 815, or still by appointment.
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 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 informationSYMPLECTIC GEOMETRY: LECTURE 5
SYMPLECTIC GEOMETRY: LECTURE 5 LIAT KESSLER Let (M, ω) be a connected compact symplectic manifold, T a torus, T M M a Hamiltonian action of T on M, and Φ: M t the assoaciated moment map. Theorem 0.1 (The
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 informationDIFFERENTIAL GEOMETRY HW 12
DIFFERENTIAL GEOMETRY HW 1 CLAY SHONKWILER 3 Find the Lie algebra so(n) of the special orthogonal group SO(n), and the explicit formula for the Lie bracket there. Proof. Since SO(n) is a subgroup of GL(n),
More informationSubgroups of Lie groups. Definition 0.7. A Lie subgroup of a Lie group G is a subgroup which is also a submanifold.
Recollections from finite group theory. The notion of a group acting on a set is extremely useful. Indeed, the whole of group theory arose through this route. As an example of the abstract power of this
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 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 information1 v >, which will be Ginvariant by construction.
1. Riemannian symmetric spaces Definition 1.1. A (globally, Riemannian) symmetric space is a Riemannian manifold (X, g) such that for all x X, there exists an isometry s x Iso(X, g) such that s x (x) =
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 informationARITHMETICITY OF TOTALLY GEODESIC LIE FOLIATIONS WITH LOCALLY SYMMETRIC LEAVES
ASIAN J. MATH. c 2008 International Press Vol. 12, No. 3, pp. 289 298, September 2008 002 ARITHMETICITY OF TOTALLY GEODESIC LIE FOLIATIONS WITH LOCALLY SYMMETRIC LEAVES RAUL QUIROGABARRANCO Abstract.
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 informationSmooth Dynamics 2. Problem Set Nr. 1. Instructor: Submitted by: Prof. Wilkinson Clark Butler. University of Chicago Winter 2013
Smooth Dynamics 2 Problem Set Nr. 1 University of Chicago Winter 2013 Instructor: Submitted by: Prof. Wilkinson Clark Butler Problem 1 Let M be a Riemannian manifold with metric, and LeviCivita connection.
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 informationA brief introduction to SemiRiemannian geometry and general relativity. Hans Ringström
A brief introduction to SemiRiemannian geometry and general relativity Hans Ringström May 5, 2015 2 Contents 1 Scalar product spaces 1 1.1 Scalar products...................................... 1 1.2 Orthonormal
More informationLECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE
LECTURE: KOBORDISMENTHEORIE, WINTER TERM 2011/12; SUMMARY AND LITERATURE JOHANNES EBERT 1.1. October 11th. 1. Recapitulation from differential topology Definition 1.1. Let M m, N n, be two smooth manifolds
More informationLECTURE 14: LIE GROUP ACTIONS
LECTURE 14: LIE GROUP ACTIONS 1. Smooth actions Let M be a smooth manifold, Diff(M) the group of diffeomorphisms on M. Definition 1.1. An action of a Lie group G on M is a homomorphism of groups τ : G
More informationTHE NEWLANDERNIRENBERG THEOREM. GL. The frame bundle F GL is given by x M Fx
THE NEWLANDERNIRENBERG 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 informationVOLUME GROWTH AND HOLONOMY IN NONNEGATIVE CURVATURE
VOLUME GROWTH AND HOLONOMY IN NONNEGATIVE CURVATURE KRISTOPHER TAPP Abstract. The volume growth of an open manifold of nonnegative sectional curvature is proven to be bounded above by the difference between
More informationDifferential Geometry MTG 6257 Spring 2018 Problem Set 4 Duedate: Wednesday, 4/25/18
Differential Geometry MTG 6257 Spring 2018 Problem Set 4 Duedate: Wednesday, 4/25/18 Required problems (to be handed in): 2bc, 3, 5c, 5d(i). In doing any of these problems, you may assume the results
More informationAbstract. Jacobi curves are far going generalizations of the spaces of \Jacobi
Principal Invariants of Jacobi Curves Andrei Agrachev 1 and Igor Zelenko 2 1 S.I.S.S.A., Via Beirut 24, 34013 Trieste, Italy and Steklov Mathematical Institute, ul. Gubkina 8, 117966 Moscow, Russia; email:
More informationReduction of Homogeneous Riemannian structures
Geometric Structures in Mathematical Physics, 2011 Reduction of Homogeneous Riemannian structures M. Castrillón López 1 Ignacio Luján 2 1 ICMAT (CSICUAMUC3MUCM) Universidad Complutense de Madrid 2 Universidad
More informationTwisted Poisson manifolds and their almost symplectically complete isotropic realizations
Twisted Poisson manifolds and their almost symplectically complete isotropic realizations ChiKwong Fok National Center for Theoretical Sciences Math Division National Tsing Hua University (Joint work
More information1 Hermitian symmetric spaces: examples and basic properties
Contents 1 Hermitian symmetric spaces: examples and basic properties 1 1.1 Almost complex manifolds............................................ 1 1.2 Hermitian manifolds................................................
More informationLECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES
LECTURE 28: VECTOR BUNDLES AND FIBER BUNDLES 1. Vector Bundles In general, smooth manifolds are very nonlinear. However, there exist many smooth manifolds which admit very nice partial linear structures.
More informationM4P52 Manifolds, 2016 Problem Sheet 1
Problem Sheet. Let X and Y be ndimensional topological manifolds. Prove that the disjoint union X Y is an ndimensional topological manifold. Is S S 2 a topological manifold? 2. Recall that that the discrete
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 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 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 informationOBSTRUCTION TO POSITIVE CURVATURE ON HOMOGENEOUS BUNDLES
OBSTRUCTION TO POSITIVE CURVATURE ON HOMOGENEOUS BUNDLES KRISTOPHER TAPP Abstract. Examples of almostpositively and quasipositively curved spaces of the form M = H\((G, h) F ) were discovered recently
More informationDIFFERENTIAL GEOMETRY. LECTURE 1213,
DIFFERENTIAL GEOMETRY. LECTURE 1213, 3.07.08 5. Riemannian metrics. Examples. Connections 5.1. Length of a curve. Let γ : [a, b] R n be a parametried curve. Its length can be calculated as the limit of
More informationMATH 4030 Differential Geometry Lecture Notes Part 4 last revised on December 4, Elementary tensor calculus
MATH 4030 Differential Geometry Lecture Notes Part 4 last revised on December 4, 205 Elementary tensor calculus We will study in this section some basic multilinear algebra and operations on tensors. Let
More informationPatrick IglesiasZemmour
Mathematical Surveys and Monographs Volume 185 Diffeology Patrick IglesiasZemmour American Mathematical Society Contents Preface xvii Chapter 1. Diffeology and Diffeological Spaces 1 Linguistic Preliminaries
More informationclass # MATH 7711, AUTUMN 2017 MWF 3:00 p.m., BE 128 A DAYBYDAY LIST OF TOPICS
class # 34477 MATH 7711, AUTUMN 2017 MWF 3:00 p.m., BE 128 A DAYBYDAY LIST OF TOPICS [DG] stands for Differential Geometry at https://people.math.osu.edu/derdzinski.1/courses/851852notes.pdf [DFT]
More informationA CONSTRUCTION OF TRANSVERSE SUBMANIFOLDS
UNIVERSITATIS IAGELLONICAE ACTA MATHEMATICA, FASCICULUS XLI 2003 A CONSTRUCTION OF TRANSVERSE SUBMANIFOLDS by J. Szenthe Abstract. In case of Riemannian manifolds isometric actions admitting submanifolds
More informationSOME EXERCISES IN CHARACTERISTIC CLASSES
SOME EXERCISES IN CHARACTERISTIC CLASSES 1. GAUSSIAN CURVATURE AND GAUSSBONNET THEOREM Let S R 3 be a smooth surface with Riemannian metric g induced from R 3. Its LeviCivita connection can be defined
More informationON NATURALLY REDUCTIVE HOMOGENEOUS SPACES HARMONICALLY EMBEDDED INTO SPHERES
ON NATURALLY REDUCTIVE HOMOGENEOUS SPACES HARMONICALLY EMBEDDED INTO SPHERES GABOR TOTH 1. Introduction and preliminaries This note continues earlier studies [9, 1] concerning rigidity properties of harmonic
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 informationModern Geometric Structures and Fields
Modern Geometric Structures and Fields S. P. Novikov I.A.TaJmanov Translated by Dmitry Chibisov Graduate Studies in Mathematics Volume 71 American Mathematical Society Providence, Rhode Island Preface
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 informationComplex line bundles. Chapter Connections of line bundle. Consider a complex line bundle L M. For any integer k N, let
Chapter 1 Complex line bundles 1.1 Connections of line bundle Consider a complex line bundle L M. For any integer k N, let be the space of kforms with values in L. Ω k (M, L) = C (M, L k (T M)) Definition
More informationLifting Smooth Homotopies of Orbit Spaces of Proper Lie Group Actions
Journal of Lie Theory Volume 15 (2005) 447 456 c 2005 Heldermann Verlag Lifting Smooth Homotopies of Orbit Spaces of Proper Lie Group Actions Marja Kankaanrinta Communicated by J. D. Lawson Abstract. By
More informationDIFFERENTIAL GEOMETRY CLASS NOTES INSTRUCTOR: F. MARQUES. September 25, 2015
DIFFERENTIAL GEOMETRY CLASS NOTES INSTRUCTOR: F. MARQUES MAGGIE MILLER September 25, 2015 1. 09/16/2015 1.1. Textbooks. Textbooks relevant to this class are Riemannian Geometry by do Carmo Riemannian Geometry
More informationLECTURE 10: THE ATIYAHGUILLEMINSTERNBERG CONVEXITY THEOREM
LECTURE 10: THE ATIYAHGUILLEMINSTERNBERG CONVEXITY THEOREM Contents 1. The AtiyahGuilleminSternberg Convexity Theorem 1 2. Proof of the AtiyahGuilleminSternberg Convexity theorem 3 3. Morse theory
More informationTHE EULER CHARACTERISTIC OF A LIE GROUP
THE EULER CHARACTERISTIC OF A LIE GROUP JAY TAYLOR 1 Examples of Lie Groups The following is adapted from [2] We begin with the basic definition and some core examples Definition A Lie group is a smooth
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 informationComparison for infinitesimal automorphisms. of parabolic geometries
Comparison techniques for infinitesimal automorphisms of parabolic geometries University of Vienna Faculty of Mathematics Srni, January 2012 This talk reports on joint work in progress with Karin Melnick
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 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 informationLECTURE 11: SYMPLECTIC TORIC MANIFOLDS. Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8
LECTURE 11: SYMPLECTIC TORIC MANIFOLDS Contents 1. Symplectic toric manifolds 1 2. Delzant s theorem 4 3. Symplectic cut 8 1. Symplectic toric manifolds Orbit of torus actions. Recall that in lecture 9
More informationStable complex and Spin c structures
APPENDIX D Stable complex and Spin c structures In this book, Gmanifolds are often equipped with a stable complex structure or a Spin c structure. Specifically, we use these structures to define quantization.
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 informationBredon, Introduction to compact transformation groups, Academic Press
1 Introduction Outline Section 3: Topology of 2orbifolds: Compact group actions Compact group actions Orbit spaces. Tubes and slices. Pathlifting, covering homotopy Locally smooth actions Smooth actions
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 informationSymplectic Geometry versus Riemannian Geometry
Symplectic Geometry versus Riemannian Geometry Inês Cruz, CMUP 2010/10/20  seminar of PIUDM 1 necessarily very incomplete The aim of this talk is to give an overview of Symplectic Geometry (there is no
More informationLECTURE 1516: PROPER ACTIONS AND ORBIT SPACES
LECTURE 1516: PROPER ACTIONS AND ORBIT SPACES 1. Proper actions Suppose G acts on M smoothly, and m M. Then the orbit of G through m is G m = {g m g G}. If m, m lies in the same orbit, i.e. m = g m for
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 information4 Riemannian geometry
Classnotes for Introduction to Differential Geometry. Matthias Kawski. April 18, 2003 83 4 Riemannian geometry 4.1 Introduction A natural first step towards a general concept of curvature is to develop
More informationHadamard s Theorem. Rich Schwartz. September 10, The purpose of these notes is to prove the following theorem.
Hadamard s Theorem Rich Schwartz September 10, 013 1 The Result and Proof Outline The purpose of these notes is to prove the following theorem. Theorem 1.1 (Hadamard) Let M 1 and M be simply connected,
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 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 informationarxiv: v1 [math.sg] 6 Nov 2015
A CHIANGTYPE LAGRANGIAN IN CP ANA CANNAS DA SILVA Abstract. We analyse a simple Chiangtype lagrangian in CP which is topologically an RP but exhibits a distinguishing behaviour under reduction by one
More informationHYPERKÄHLER MANIFOLDS
HYPERKÄHLER MANIFOLDS PAVEL SAFRONOV, TALK AT 2011 TALBOT WORKSHOP 1.1. Basic definitions. 1. Hyperkähler manifolds Definition. A hyperkähler manifold is a C Riemannian manifold together with three covariantly
More informationConification of Kähler and hyperkähler manifolds and supergr
Conification of Kähler and hyperkähler manifolds and supergravity cmap Masaryk University, Brno, Czech Republic and Institute for Information Transmission Problems, Moscow, Russia Villasimius, September
More informationA Convexity Theorem For Isoparametric Submanifolds
A Convexity Theorem For Isoparametric Submanifolds Marco Zambon January 18, 2001 1 Introduction. The main objective of this paper is to discuss a convexity theorem for a certain class of Riemannian manifolds,
More informationStable bundles on CP 3 and special holonomies
Stable bundles on CP 3 and special holonomies Misha Verbitsky Géométrie des variétés complexes IV CIRM, Luminy, Oct 26, 2010 1 Hyperkähler manifolds DEFINITION: A hyperkähler structure on a manifold M
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 informationLecture XI: The nonkähler world
Lecture XI: The nonkähler world Jonathan Evans 2nd December 2010 Jonathan Evans () Lecture XI: The nonkähler world 2nd December 2010 1 / 21 We ve spent most of the course so far discussing examples of
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 blowingup operation amounts to replace a point in
More informationGeodesic Equivalence in subriemannian Geometry
03/27/14 Equivalence in subriemannian Geometry Supervisor: Dr.Igor Zelenko Texas A&M University, Mathematics Some Preliminaries: Riemannian Metrics Let M be a ndimensional surface in R N Some Preliminaries:
More informationk=0 /D : S + S /D = K 1 2 (3.5) consistently with the relation (1.75) and the RiemannRochHirzebruchAtiyahSinger index formula
20 VASILY PESTUN 3. Lecture: GrothendieckRiemannRochHirzebruchAtiyahSinger Index theorems 3.. Index for a holomorphic vector bundle. For a holomorphic vector bundle E over a complex variety of dim
More informationReduction of Symplectic Lie Algebroids by a Lie Subalgebroid and a Symmetry Lie Group
Symmetry, Integrability and Geometry: Methods and Applications SIGMA 3 (2007), 049, 28 pages Reduction of Symplectic Lie Algebroids by a Lie Subalgebroid and a Symmetry Lie Group David IGLESIAS, Juan Carlos
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 informationCohomology of the Mumford Quotient
Cohomology of the Mumford Quotient Maxim Braverman Abstract. Let X be a smooth projective variety acted on by a reductive group G. Let L be a positive Gequivariant line bundle over X. We use a Witten
More informationINSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD
INSTANTON MODULI AND COMPACTIFICATION MATTHEW MAHOWALD () Instanton (definition) (2) ADHM construction (3) Compactification. Instantons.. Notation. Throughout this talk, we will use the following notation:
More informationOrientation transport
Orientation transport Liviu I. Nicolaescu Dept. of Mathematics University of Notre Dame Notre Dame, IN 465564618 nicolaescu.1@nd.edu June 2004 1 S 1 bundles over 3manifolds: homological properties Let
More informationThe Atiyah bundle and connections on a principal bundle
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 120, No. 3, June 2010, pp. 299 316. Indian Academy of Sciences The Atiyah bundle and connections on a principal bundle INDRANIL BISWAS School of Mathematics, Tata
More informationQUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday September 21, 2004 (Day 1)
QUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday September 21, 2004 (Day 1) Each of the six questions is worth 10 points. 1) Let H be a (real or complex) Hilbert space. We say
More informationQUALIFYING EXAMINATION Harvard University Department of Mathematics Tuesday January 20, 2015 (Day 1)
Tuesday January 20, 2015 (Day 1) 1. (AG) Let C P 2 be a smooth plane curve of degree d. (a) Let K C be the canonical bundle of C. For what integer n is it the case that K C = OC (n)? (b) Prove that if
More informationMILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES
MILNOR SEMINAR: DIFFERENTIAL FORMS AND CHERN CLASSES NILAY KUMAR In these lectures I want to introduce the ChernWeil approach to characteristic classes on manifolds, and in particular, the Chern classes.
More informationDifferential Geometry Exercises
Differential Geometry Exercises Isaac Chavel Spring 2006 Jordan curve theorem We think of a regular C 2 simply closed path in the plane as a C 2 imbedding of the circle ω : S 1 R 2. Theorem. Given the
More informationEquivalence, Invariants, and Symmetry
Equivalence, Invariants, and Symmetry PETER J. OLVER University of Minnesota CAMBRIDGE UNIVERSITY PRESS Contents Preface xi Acknowledgments xv Introduction 1 1. Geometric Foundations 7 Manifolds 7 Functions
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 informationESSENTIAL KILLING FIELDS OF PARABOLIC GEOMETRIES: PROJECTIVE AND CONFORMAL STRUCTURES. 1. Introduction
ESSENTIAL KILLING FIELDS OF PARABOLIC GEOMETRIES: PROJECTIVE AND CONFORMAL STRUCTURES ANDREAS ČAP AND KARIN MELNICK Abstract. We use the general theory developed in our article [1] in the setting of parabolic
More informationMath 637 Topology Paulo LimaFilho. Problem List I. b. Show that a contractible space is path connected.
Problem List I Problem 1. A space X is said to be contractible if the identiy map i X : X X is nullhomotopic. a. Show that any convex subset of R n is contractible. b. Show that a contractible space 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 informationNotes on quotients and group actions
Notes on quotients and group actions Erik van den Ban Fall 2006 1 Quotients Let X be a topological space, and R an equivalence relation on X. The set of equivalence classes for this relation is denoted
More informationLecture Notes a posteriori for Math 201
Lecture Notes a posteriori for Math 201 Jeremy Kahn September 22, 2011 1 Tuesday, September 13 We defined the tangent space T p M of a manifold at a point p, and the tangent bundle T M. Zev Choroles gave
More informationSpin(10,1)metrics with a parallel null spinor and maximal holonomy
Spin(10,1)metrics with a parallel null spinor and maximal holonomy 0. Introduction. The purpose of this addendum to the earlier notes on spinors is to outline the construction of Lorentzian metrics in
More informationStratification of 3 3 Matrices
Stratification of 3 3 Matrices Meesue Yoo & Clay Shonkwiler March 2, 2006 1 Warmup with 2 2 Matrices { Best matrices of rank 2} = O(2) S 3 ( 2) { Best matrices of rank 1} S 3 (1) 1.1 Viewing O(2) S 3 (
More informationarxiv:math/ v1 [math.ag] 18 Oct 2003
Proc. Indian Acad. Sci. (Math. Sci.) Vol. 113, No. 2, May 2003, pp. 139 152. Printed in India The Jacobian of a nonorientable Klein surface arxiv:math/0310288v1 [math.ag] 18 Oct 2003 PABLO ARÉSGASTESI
More information1. Classifying Spaces. Classifying Spaces
Classifying Spaces 1. Classifying Spaces. To make our lives much easier, all topological spaces from now on will be homeomorphic to CW complexes. Fact: All smooth manifolds are homeomorphic to CW complexes.
More informationFundamentals of Differential Geometry
 Serge Lang Fundamentals of Differential Geometry With 22 luustrations Contents Foreword Acknowledgments v xi PARTI General Differential Theory 1 CHAPTERI Differential Calculus 3 1. Categories 4 2. Topological
More informationLECTURE 5: SURFACES IN PROJECTIVE SPACE. 1. Projective space
LECTURE 5: SURFACES IN PROJECTIVE SPACE. Projective space Definition: The ndimensional projective space P n is the set of lines through the origin in the vector space R n+. P n may be thought of as the
More informationHomework for Math , Spring 2012
Homework for Math 6170 1, Spring 2012 Andres Treibergs, Instructor April 24, 2012 Our main text this semester is Isaac Chavel, Riemannian Geometry: A Modern Introduction, 2nd. ed., Cambridge, 2006. Please
More information