Definition:Riemannian Distance

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

Let $p,q \in M$ be points.

Let $\gamma_{pq} : \closedint 0 1 \to M$ be an admissible curve such that $\map \gamma 0 = p$ and $\map \gamma 1 = q$.

Let $\map {L_g} {\gamma_{pq}}$ be the Riemannian length of $\gamma_{pq}$.

Then the Riemannian distance between $p$ and $q$, denoted by $\map {d_g} {p, q}$, is the infimum of all such curves $\gamma_{pq}$:

$\ds \map {d_g} {p, q} := \inf_{\gamma_{pq} \mathop \in M} \map {L_g} {\gamma_{pq}}$