Diffgeo 1 blatt 03 PDF

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Dr. Robert Helling Prof. Dr. Sebastian Hensel Dr. Jan Swoboda

Differentiable Manifolds Problem Set 3 1. (15 points) We consider the following subsets of R2 and R3 : (a) the coordinate cross X = {(x, y) ∈ R2 󳈌 xy = 0}; (b) the cone C = {(x, y, z) ∈ R3 󳈌 z ≥ 0 and x2 + y 2 = z}; (c) the double cone D = {(x, y, z) ∈ R3 󳈌 x2 + y 2 = z }. Each of these three sets X, C and D is given the relative topology. This is the topology in which a subset is open if and only if it arises as the intersection of an open subset of R2 , respectively R3 , with X, C or D. Decide for each of the topological spaces X, C and D whether or not it is a topological manifold. 2. (10 points) Let M be a smooth manifold with a group action G ↷ M. Suppose that the action satisfies the following: (i) It is free: If g ⋅ x = x for some x ∈ M and g ∈ G then g = e. (ii) It is properly discontinuous: Every x ∈ M has an open neighbourhood U such that for all g ≠ e we have that U ∩ gU = ∅. Let N = M󳆋G be the quotient space of M with respect to G, endowed with the quotient topology. Prove that N carries a unique differentiable structure such that the canonical projection π∶ M → N is a local diffeomorphism. By local diffeomorphism we mean that every p ∈ M has an open neighbourhood U such that the restricted map π󳈌U ∶ U → π(U ) is a diffeomorphism. Give an example which illustrates that π is in general not a global diffeomorphism. 3. (10 points) Let M m and N n be smooth manifolds. In problem set 2, you have shown that the product M × N (equipped with the product topology) is a smooth (m + n)-manifold. Prove that for every point (p, q) ∈ M × N the tangent space T(p,q) (M × N ) is canonically isomorphic (as a vector space) to the direct sum Tp M ⊕ Tq N . 4. (15 points) Let A be an R-algebra, i.e. A is a real vector space together with a bilinear map A × A → A. A derivation of the algebra A is an R-linear map D∶ A → A such that D(x ⋅ y) = D(x) ⋅ y + x ⋅ D(y) holds for all x, y ∈ A. Prove the following assertions: (a) The kernel of any derivation D is an R-algebra. (b) Every a ∈ A gives rise to a derivation Da defined by Da (b) = ab − ba for all b ∈ A. Such derivations are called inner derivations . (c) The set D of derivations is a Lie algebra with Lie bracket defined by [D1 , D2 ] = D1 ○ D2 − D2 ○ D1 for D1 , D2 ∈ D. I.e. show that the bracket is bilinear, antisymmetric and satisfies the Jacobi identity. Please hand in your solutions by Wednesday 2018/11/07, 12 a.m....