Exercise Sheet on Riemannian Geometry: Topics in Math 676, Assignments of Algebra

Download Exercise Sheet on Riemannian Geometry: Topics in Math 676 and more Assignments Algebra in PDF only on Docsity! MATH 676 Topics in Riemannian geometry : exercise sheet two 1. Let M ⊂ RN be a submanifold of Euclidean space. Then M inherits a Riemannian metric g and associated Levi-Civita connection ∇. Show that ∇ is the same as differentiating in RN and then projecting to TM . Specifically, let X and Y be vector fields on M which extend to vector fields X and Y on a neighbourhood of M in RN . Writing Y as (Y 1, . . . , Y N), the Levi-Civita connection on R N is given by ∇XY = (DXY 1, . . . , DXY N) ∈ TR N . Show that restricting to M and projecting from TRN |M to TM gives the Levi-Civita connection ∇XY of g on M . 2. Consider the upper half-plane R 2 + := {(x, y) ∈ R 2|y > 0} with the hyperbolic metric 1 y2 (dx2 + dy2). a) Calculate the Christoffel symbols of the hyperbolic metric. b) Let v0 be the vector (0, 1) ∈ T(0,1)R 2 + and let c : R → R 2 + be the curve t 7→ (t, 1). Find the parallel transport of v0 along c, i.e., find a formula for the vector v(t) ∈ T(t,1)R 2 + given by parallel transporting v0. 1