Download Geometry Homework Assignments for MATH 6490-351: Topics in Geometry - Prof. Weiping Li and more Assignments Geometry in PDF only on Docsity! MATH 6490-351 TOPICS IN GEOMETRY HOMEWORK ASSIGNMENTS 1. Homework 1 (1) Let M be the image of the curve c : R→ R4 defined by c(t) = (cos t, sin t, cos(at), sin(at)). (a) Prove that M is a submanifold of R4 if and only if a is a rational number. (b) Prove that M is not a manifold as induced topology from R4 if a is irrational; M can be a 1-dimensional manifold from the abstract definition. (2) Define the Mobius band MB as the topological quotient of [0, 1] × R by the equivalence relation which identifies the pairs (0, t) and (1,−t). (a) Prove that MB can be equipped with a structure of a smooth manifold consistent with its topology. (b) Prove that MB as a total space of a vector bundle over a circle is non-trivial. (3) Prove that the tangent bundle TSO(3) is trivial. (4) Define the bracket [X,Y ] of two vector fields X and Y is the vector field corresponding to the derivation LXLY − LY LX , where LXf(m) = ∑n i=1X i(m)( ∂f∂xi ) for X = ∑n i=1X i · ∂∂xi . (a) If f ∈ C2, show that [X,Y ]i = n∑ j=1 (Xj ∂Y i ∂xj − Y j ∂X i ∂xj ), for [X,Y ] = ∑n i=1[X,Y ] i · ∂∂xi . (b) Using (a), prove the Jacobi identity, [X, [Y, Z]] + [Y, [Z,X]] + [Z, [X,Y ]] = 0. 1 2 MATH 6490-351 TOPICS IN GEOMETRY 2. Homework 2 (1) Let H3 = {x ∈ R4 : 〈x, x〉 = −x20 + x21 + x22 + x23 = −1, x0 > 0} be the submanifold of R4. Prove that the quadratic form −dx20 + dx21 + dx22 + dx23 induces on H3 a positive definite symmetric bilinear form g. (2) Let f(x) = s − 2(x−s)〈x−s,x−s〉 be the pseudo-inversio with pole s = (−1, 0, 0, 0). Show that f is a diffeomorphism from H3 onto the unit disk D3 = {x ∈ R3 : ‖x‖ < 1}, and the pull-back metric (f−1)∗g = 4 ∑3 i=1 dx2i (1−‖x‖2)2 . (This is the Poincaré disk model of hyperbolic space) (3) Let T 2 be embedded in R3 as image of R2 by the map Φ(θ, φ) = ((2 + cos θ) cosφ, (2 + cos θ) sinφ, sin θ). Let g be the metric induced on T 2 by the Euclidean metric of R3. Let g0 be the product metric on (T 2, g0) = (S1, can)× (S1, can). Express both g and g0 in terms of θ, φ, and prove that g is not the product metric. (4) Let (e1, e2) be a basis of (R2, can), and let G be the group of diffeomorphisms generated by r1(x, y) = (x+ 1,−y), r2(x, y) = (x, y + 1). Show that R2 is diffeomorphic to the Klein bottle K (the flat Klein bottle), but the Euclidean metric can only go to the quotient in case of the basis (e1, e2) is orthogonal. Show that a flat Klein bottle has a 2-folded Riemannian covering, which is a rectangle flat torus. 3. Homework 3 (1) Let ∇ be the Levi-Civita connection of the induced metric of S2 from R3. For the spherical coordinate (θ, φ), Compute ∇ ∂ ∂θ ∂ ∂θ , ∇ ∂ ∂φ ∂ ∂φ , ∇ ∂ ∂θ ∂ ∂φ , (2) Let f : (M, g)→ R be a function of a Riemannian manifold with its first covariant derivative df . Define the Hessian Hessf as the second covariant derivative of f . So (Hessf)((ξ, η) = ξ(df(Y ))− (df)(∇ξY ), where Y is any extension of η. (a) Prove that Hessf is symmetric in ξ, η; (b) a function with positive Hessian has no local maxima.
