Riemannian Manifolds

The previous sections dealt with some basic notions for C∞ manifolds and thus belonged to differential topology. In this chapter another structure is added to manifolds, the Riemannian metric. This one structure will allow a remarkable number of geometric definitions: angles and lengths of vectors, a canonical volume form, lengths of curves on manifolds, the distance between two points, curvatures, and n-fold directional derivatives of functions and tensors.