Extended riemannian distance on a disconnected manifold

The extended Riemannian distance on arbitrary $M$ is the componentwise Riemannian distance when two points are in the same component, and $+\mathrm{\infty }$ otherwise.

Within each component use Riemannian distance is a metric. Components are open, since small coordinate balls are connected. A continuous curve cannot meet two components because its connected interval image is connected, so the cross-component curve family is empty, with $\inf \mathrm{\varnothing }=+\mathrm{\infty }$. This is an extended metric: if two endpoints are in different components, any third point is in a different component from at least one of them, so the triangle inequality has infinite right side. It is a finite metric precisely when there are no distinct components. Empty and singleton manifolds retain their unique distances.

Lee, Chapter 13, pp.337–340, Proposition 13.25, Lemma 13.28 and Theorem 13.29; finite piecewise $C^1$ refinements and pauses are treated explicitly here.