Given a Riemannian manifold \((M,g)\). Then we can define a metric on \(M\) with the Riemannian distance function

\begin{equation*} d_g(p,q)=\inf_{\gamma \text{ curve connecting \(p\) and \(q\) }} L_g(\gamma), \end{equation*}

where \(L_g\) denotes the length of the curve segment \(\gamma\).

Remark
  • Since every Riemannian manifold may be endowed with the Riemannian distance function, it is also a metric space.
  • The topology induced \(d_g\) is identical to the given one of \(M\)