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 one 1. A proper affine function is a map f : R → R given by f(t) = bt+a for a ∈ R and b ∈ R>0. The set of proper affine functions forms a Lie group: as a smooth manifold it is just the upper half plane {(a, b) ∈ R2|b > 0}, and the group operation is given by composition. a) Find a formula for the product (a, b)(c, d). b) The identity element is e = (0, 1). Find a formula for the left-invariant metric which takes the value g = ( 1 0 0 1 ) at e. 1