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

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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 one-form 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 non-empty 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 n-dimensional 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 2-torus 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 Levi-Civita 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 1-forms 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 2-forms 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. Gauß-Bonnet 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 Gauß-Bonnet 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 2-torus T 2 [3 points] a) Show that the standard 2-torus 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 non-positive sectional curvature [3 points] Let (M, g) be a Riemannian manifold with non-positive 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. Fermi-Walker 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 Fermi-Walker 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 Fermi-Walker field V along γ whose value at t 0 is the given value V (t 0 ). b) Show that T is a Fermi-Walker field along γ. c) Show that g(v, W ) is constant along γ if V and W are Fermi-Walker fields along γ. d) Show that Fermi-Walker 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 t-derivatives of J(t) 2 at t = 0 in two different ways using the Jacobi-equation.

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 semi-continuous.

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 rank-1 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 G-principal 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 H-principal bundle and H G a Lie subgroup. Show that the associated bundle P H G M is naturally a G-principal 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 G-principal 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 G-space. 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 G-invariant 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 G-principal 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 G-action. 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 non-vanishing 1-form α on a 2-dimensional 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 2-dimensional manifold can be written locally as ω = dq dp. 2. Symplectomorphisms of the cotangent bundle [4 points] Show that the group of fiber-preserving 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 1-forms on Q, where D acts on Z by pull-back. 3. Hamiltonian functions on a presymplectic manifold [4 points] Let ω Ω 2 (M) be a closed 2-form (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 well-defined 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 1-form 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.