Riemannian Manifold - Riemannian Manifolds As Metric Spaces

Riemannian Manifolds As Metric Spaces

A connected Riemannian manifold carries the structure of a metric space whose distance function is the arclength of a minimizing geodesic.

Specifically, let (M,g) be a connected Riemannian manifold. Let c: → M be a parametrized curve in M, which is differentiable with velocity vector c′. The length of c is defined as

By change of variables, the arclength is independent of the chosen parametrization. In particular, a curve → M can be parametrized by its arc length. A curve is parametrized by arclength if and only if for all .

The distance function d : M×M → [0,∞) is defined by

where the infimum extends over all differentiable curves γ beginning at p∈M and ending at q∈M.

This function d satisfies the properties of a distance function for a metric space. The only property which is not completely straightforward is to show that d(p,q)=0 implies that p=q. For this property, one can use a normal coordinate system, which also allows one to show that the topology induced by d is the same as the original topology on M.