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Riemannian metrics and connections October 27, 2009

Posted by Akhil Mathew in differential geometry.
Tags: , ,

Wow.  Blogging is definitely way harder during the academic year. 

Ok, so I’m aiming to change things around a bit here and take a break from algebraic number theory to do some differential geometry. I’ll assume basic familiarity with what manifolds are, the tangent bundle and its variants, but generally no more. I eventually want to get to some real theorems, but this post will focus primarily on definitions.

Riemannian Metrics

A Riemannian metric on a smooth manifold {M} is defined as a covariant symmetric 2-tensor {(\cdot, \cdot)_p, p \in M} such that {(v,v)_p > 0} for all {v \in T_p(M)}. For convenience, I will write {(v,w)} for {(v,w)_p}. In other words, a Riemannian metric is a collection of (positive) inner products on each of the tangent spaces {T_p(M)} such that if {X,Y} are (smooth) vector fields, the function {(X,Y): M \rightarrow \mathbb{R}} defined by taking the inner product at each point, is smooth. There are several ways to get Riemannian metrics: (more…)