Riemannian Manifold as Metric Space

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

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Sources

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