ABSTRACT
Loosely speaking, curvature measures the extent to which a manifold differs in the way it bends from flat space.
There are two kinds of curvature: extrinsic and intrinsic. Extrinsic curvature describes the manifold from the point of view of an external observer, who compares the bending of curves that lie on the manifold with the straight lines that go off it. Intrinsic curvature describes the manifold from the point of view of an observer confined to the manifold, who performs measurements only along paths that lie on the manifold.
