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The test case: flat Riemannian manifolds November 12, 2009

Posted by Akhil Mathew in differential geometry, MaBloWriMo.
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Recall that two Riemannian manifolds {M,N} are isometric if there exists a diffeomorphism {f: M \rightarrow N} that preserves the metric on the tangent spaces. The curvature tensor  (associated to the Levi-Civita connection) measures the deviation from flatness, where a manifold is flat if it is locally isometric to a neighborhood of {\mathbb{R}^n}.

Theorem 1 (The Test Case) The Riemannian manifold {M} is locally isometric to {\mathbb{R}^n} if and only if the curvature tensor vanishes. (more…)