X-Nico

Flat manifold

Intuitively, a flat manifold is one that "locally looks like" Euclidean space in terms of distances and angles, e.g. the interior angles of a triangle add up to 180°.

•

A flat torus can be isometrically embedded in three-dimensional Euclidean space with a C¹ map (by the Nash embedding theorem) but not with a C² map, and the Clifford torus provides an isometric analytic embedding of a flat torus in four dimensions.

Ricci-flat manifold

Since Ricci curvature measures the amount by which the volume of a small geodesic ball deviates from the volume of a ball in Euclidean space, small geodesic balls will have no volume deviation, but their "shape" may vary from the shape of the standard ball in Euclidean space.

see also