Equality of Radial Distance Function to Riemannian Distance

Theorem

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

Let $U_p = \map {\exp_p} {\map {B_\epsilon} 0 }$ be an open or closed geodesic ball around $p \in M$.

Let $r : U_p \to \R$ be the radial distance function.

Let $d_g$ be the Riemannian distance.

Then:

$\forall p \in M : \forall x \in U_p : \map r x = \map {d_g} {p, x}$

Proof

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Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 6$: Geodesics and Distance. Geodesics Are Locally Minimizing