> [!definition] > > Let $(X, g)$ be a $C^p$-[[Riemannian Manifold|Riemannian manifold]] ($p \ge 1$), and $\gamma: [a, b] \to M$ be a piecewise $C^1$-[[Curve|curve segment]]. Define the **length** of $\gamma$ as > $ > L_g(\gamma) = \int_a^b \abs{\gamma'(t)}_gdt > $ > Let $\varphi: [c, d] \to [a, b]$ be a $C^1$-[[Diffeomorphism|diffeomorphism]], then $L_g(\gamma \circ \varphi) = L_g(\gamma)$. > > *Proof*. Over a change of variables, > $ > \begin{align*} > \int_{[a, b]} \abs{\gamma'(t)}_gdt &= \int_{[c, d]}\abs{\gamma' \circ \varphi(t)}_g \abs{\varphi'(t)}dt \\ > &= \int_{[c, d]}\abs{(\gamma \circ \varphi)'(t)}dt > \end{align*} > $ > [!definition] > > Let $(X, g)$ be a $C^p$-Riemannian manifold ($p \ge 1$). For any $p, q \in X$, define > $ > d_{g}(p, q) = \inf\bracs{L_g(\gamma): \gamma[p \to q] \text{ piecewise } C^p } > $ > where $\gamma[p \to q]$ is a piecewise $C^p$-curve segment from $p \to q$, then $d_g$ is a [[Metric Space|metric]], known as the **Riemannian distance function** on $X$. > > $d_g$ induces the same topology on $X$ as the final topology.