Mean Value function theorem on a Riemannian Manifold Suppose you have a Riemannian manifold $M$, a smooth function $\phi: M \longrightarrow \mathbb{R}$ and lastly, two points $p, q \in M$ so that $d(p, q) < \text{inj } M$ where $\text{inj } M$ is the injectivity radius of $M$. It seems that the following inequality is correct
$$
|\phi(p) - \phi(q)| \le d(p, q) \|\phi\|_{C^1}
$$
I had two questions.
How is

*

*$\|\phi\|_{C^1}$ defined?

*How is the inequality proved?

I think the paper by Hamilton, The inverse function theorem of Nash and Moser, might have my answer inside (Actually Theorem 5.1.3 seems close but yet different from what I want). Due to a limited time to read that paper (although I think there is something much more easier going on here), I would appreciate if you can help me understanding this. Mostly, what I need though is just a reference where this or something similar is discussed.
 A: The $\mathcal{C}^1$-norm on a Riemannian manifold $(M,g)$ is usually defined as $$\lvert\lvert f\rvert\rvert=\sup_{x\in M}\lvert f(x)\rvert+\sup_{x\in M}\lvert\lvert\nabla f(x)\rvert\rvert$$
where the second term is the norm of the gradient of $f$ with respect to $g$. See also this question for reference.
As for the second question, take any path $\gamma$ between $p$ and $q$. Then
\begin{equation}
\phi(p)-\phi(q)=\int_0^1\partial_t\left(\phi(\gamma(t)\right)\mathrm{d}t=\int_0^1g\big(\nabla\phi(\gamma(t)),\dot{\gamma}(t)\big)\mathrm{d}t.
\end{equation}
Taking absolute values and using the Cauchy-Schwarz inequality you find
\begin{equation}
\lvert\phi(p)-\phi(q)\rvert\leq\int_0^1\lvert\lvert\nabla\phi(\gamma(t))\rvert\rvert\,\lvert\lvert\dot{\gamma}(t)\rvert\rvert\,\mathrm{d}t\leq\mbox{length}(\gamma)\cdot\sup_M\lvert\lvert\nabla\phi\rvert\rvert.
\end{equation}
The hypothesis on the injectivity radius and the distance between $p$ and $q$ allows us to take in particular a path realizing the distance between the two points.
