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## The fundamental theorem of Riemannian geometry and the Levi-Civita connectionNovember 10, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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8 comments

Ok, now onto the Levi-Civita connection. Fix a manifold ${M}$ with the pseudo-metric ${g}$. This means essentially a metric, except that ${g}$ 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 ${\nabla}$ on ${M}$ compatible with ${g}$. (more…)