Let $(M,g)$ and $(N,h)$ be Riemannian manifolds, and let $a,b\in\mathbb{R}$ with $a\le b$. For each $p\in M$, write $\|v\|_{g,p}:=\sqrt{g_p(v,v)}$ for $v\in T_pM$, and for each $q\in N$, write $\|u\|_{h,q}:=\sqrt{h_q(u,u)}$ for $u\in T_qN$. Let $\mathcal{L}^1$ denote one-dimensional [Lebesgue measure](/page/Lebesgue%20Measure), and let $L_g$ and $L_h$ denote the Riemannian length functionals obtained by integrating these speed norms over piecewise smooth subintervals with respect to $\mathcal{L}^1$, with length defined to be $0$ when the parameter interval is degenerate. Suppose $f:M\to N$ is a Riemannian isometry, meaning that $f$ is smooth and, for every $p\in M$ and every $v,w\in T_pM$,

\begin{align*} h_{f(p)}(df_p(v),df_p(w))=g_p(v,w). \end{align*}

If $\gamma:[a,b]\to M$ is a piecewise smooth curve, then $f\circ\gamma:[a,b]\to N$ is piecewise smooth and

\begin{align*} L_h(f\circ\gamma)=L_g(\gamma). \end{align*}