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Riemannian metrics and connections
*October 27, 2009*

*Posted by Akhil Mathew in differential geometry.*

Tags: Christoffel symbols, connections, Riemannian metrics

8 comments

Tags: Christoffel symbols, connections, Riemannian metrics

8 comments

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 is defined as a covariant symmetric 2-tensor such that for all . For convenience, I will write for . In other words, a Riemannian metric is a collection of (positive) inner products on each of the tangent spaces such that if are (smooth) vector fields, the function defined by taking the inner product at each point, is smooth. There are several ways to get Riemannian metrics: (more…)