Bound on Riemannian Distance Inside Coordinate Neighborhood

Theorem

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

Let $d_g$ be the Riemannian distance.

Suppose $U \subseteq M$ is an open subset.

Let $p \in M$ be a point.

Then $p$ has a coordinate neighborhood $V \subseteq U$ such that:

$\forall q \in V : \exists C \in \R_{> 0} : \map {d_g} {p, q} \le C \map {d_{\bar g} } {p, q}$

where $\bar g$ is the Euclidean metric in the given coordinates on $V$.

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