Let \((M,g)\) be a Riemannian manifold. Then
\[ d(p,q) = \inf \{L(\gamma)\mid \text{\(\gamma\) is a curve segment from \(p\) to \(q\)}\} \]defines a metric on \(M\). The map \(d\) is sometimes called Riemannian distance.
Let \((M,g)\) be a Riemannian manifold. Then
\[ d(p,q) = \inf \{L(\gamma)\mid \text{\(\gamma\) is a curve segment from \(p\) to \(q\)}\} \]defines a metric on \(M\). The map \(d\) is sometimes called Riemannian distance.