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…)