Riemann curvature tensor

Summary

The Riemann curvature tensor of an affine connection, usually the Levi-Civita connection of a Riemannian metric, measures how far second covariant derivatives fail to commute. For the Levi-Civita connection, it vanishes exactly when the manifold is locally isometric to Euclidean space. Sectional, Ricci and scalar curvature are all obtained from it. In general relativity it describes the tidal effects of gravity.