Mean value theorem on Riemannian manifold?

Question

Is there some generalisation of the classical mean value theorem for real-valued functions on an interval $$|f(x)-f(y)| \leq |\nabla f(c)||x-y|$$ for some $c$ between $(x,y)$ to the case where $f:M \to \mathbb{R}$ where $M$ is a compact Riemannian manifold? This MVT would involve the Riemannian gradient $\nabla_g$.

Answer 1

If "between" means "$c$ lies on a minimizing geodesic from $x$ to $y$", the answer is yes.

Let $\gamma:[0, \ell] \to M$ be a unit-speed geodesic with $\gamma(0) = x$ and $\gamma(\ell) = y$, and let $t_{0}$ be a point of $[0, \ell]$ at which the real-valued function $\left|g\left(\nabla f\bigl(\gamma(t)\bigr), \gamma'(t)\right)\right|$ achieves a maximum. Putting $c = \gamma(t_{0})$, we have $$ \left|g\left(\nabla f\bigl(\gamma(t)\bigr), \gamma'(t)\right)\right| \leq \left|g\left(\nabla f\bigl(\gamma(t_{0})\bigr), \gamma'(t_{0})\right)\right| \leq \left|\nabla f(c)\right|,\quad\text{for all $t$ in $[0, \ell]$,} $$ and elementary calculus gives \begin{align*} \left|f(y) - f(x)\right| &= \left|\int_{0}^{\ell} g\left(\nabla f\bigl(\gamma(t)\bigr), \gamma'(t)\right)\, dt\right| \\ &\leq \int_{0}^{\ell} \left|g\left(\nabla f\bigl(\gamma(t)\bigr), \gamma'(t)\right)\right|\, dt \\ &\leq \int_{0}^{\ell} \left|g\left(\nabla f\bigl(\gamma(t_{0})\bigr), \gamma'(t_{0})\right)\right|\, dt \\ &\leq \left|\nabla f(c)\right| \int_{0}^{\ell} dt \\ &= \left|\nabla f(c)\right|\, \left|x - y\right|. \end{align*} (The final equality uses the fact that $\gamma$ is minimizing. Clearly this hypothesis is crucial.)