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The test case: flat Riemannian manifolds
*November 12, 2009*

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

Tags: curvature tensor, flat manifolds, Riemannian metrics, test case

1 comment so far

Tags: curvature tensor, flat manifolds, Riemannian metrics, test case

1 comment so far

Recall that two Riemannian manifolds are isometric if there exists a diffeomorphism 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 .

Theorem 1 (The Test Case)The Riemannian manifold is locally isometric to if and only if the curvature tensor vanishes. (more…)