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The fundamental theorem of Riemannian geometry and the Levi-Civita connection
*November 10, 2009*

*Posted by Akhil Mathew in differential geometry, MaBloWriMo.*

Tags: connections, Levi-Civita connection, Riemannian metrics

8 comments

Tags: connections, Levi-Civita connection, Riemannian metrics

8 comments

Ok, now onto the Levi-Civita connection. Fix a manifold with the pseudo-metric . This means essentially a metric, except that as a bilinear form on the tangent spaces is still symmetric and nondegenerate but not necessarily positive definite. It is still possible to say that a pseudo-metric is compatible with a given connection.

This is the fundamental theorem of Riemannian geometry:

**Theorem 1** *There is a unique symmetric connection on compatible with . (more…)*