## Riemannian metrics and connectionsOctober 27, 2009

Posted by Akhil Mathew in differential geometry.
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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…)