# Riemannian Manifold as Metric Space

Jump to navigation
Jump to search

## Theorem

Let $\struct {M, g}$ be a connected Riemannian manifold with or without boundary.

Let $d_g$ be the Riemannian distance.

Then $\struct {M, d_g}$ is a metric space whose metric topology is the same as the given manifold topology.

## Proof

This theorem requires a proof.You can help $\mathsf{Pr} \infty \mathsf{fWiki}$ by crafting such a proof.To discuss this page in more detail, feel free to use the talk page.When this work has been completed, you may remove this instance of `{{ProofWanted}}` from the code.If you would welcome a second opinion as to whether your work is correct, add a call to `{{Proofread}}` the page. |

## Sources

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