Definition:Metrically Complete Connected Riemannian Manifold

Definition

Let $\struct {M, g}$ be a connected Riemannian manifold.

Let $d_g$ be the Riemannian distance.

Then $M$ is called the (metrically) complete manifold if every Cauchy sequence in $M$ converges with respect to the distance function $d_g$.

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 2$: Riemannian Metrics. Lengths and Distances