Math 110b Spring 2004 Homework 1

A. Do Carmo, Chapter 1: 4,5,7.

B. Consider a metric g = dt2 + φ(t)22 on R2 -{0} in local coordinates (t,θ). What conditions on φ will ensure that g extends to a smooth metric on R2?

C. Show, as alleged in remark 2.3 (and generalized to the setting of connections on a vector bundle) that the notion of a connection is a local one. In other words, show that for a connection ∇ on a bundle E, given a vector field X and section σ of E, the value of ∇Xσ at a point p depends only on the section and the vector Xp. You may find it worthwhile to look at remark 5.7 of chapter 0. The fancy way to say this is that ∇, which is defined as a map C(TM) ⊗C(E) →C(E), is really a map C(TM ⊗E) →C(E). See if you can figure out the content of this apparently silly change of notation.