A manifold with a Riemannian metric that has zero curvature is a flat manifold. The basic example is Euclidean space with the usual metric d s^2 = sum_i d x_i^2. In fact, any point on a flat manifold has a neighborhood isometric to a neighborhood in Euclidean space. A flat manifold is locally Euclidean in terms of distances and angles, as well as merely topologically locally Euclidean, as all manifolds are.

curvature | exponential map | flat | isometry | torus | universal cover