Relating the norm of a diffeomorphism to the distance in a Riemannian manifold

Question

Suppose $M$ is a Riemannian manifold and $f: M \to M$ is a diffeo. I wonder whether or not there are ways to relate the operator norm of $df$ to the distances $d(x,y)$ and $d(f(x), f(y))$, like the mean value inequality does for applications $\mathbb{R}^m \to \mathbb{R}^n$, for instance.

I know, for example, that if $f:M \to \mathbb{R}$, then there is an analogue to the classical MVT that states that $$|f(x) - f(y)| \leq |df_p||x-y|,$$ where $p$ is a point in the geodesic joining $x$ to $y$ maximazing the norm of $df$. Is there anything similar in the case the range is not $\mathbb{R}$?

Answer 1

I will assume that $M$ is complete. Choose a geodesic $\gamma$ from $x$ to $y$ so that the length of $\gamma$ is $\ell(\gamma)=d(x,y)=L$. Then \begin{align*} d(f(x),f(y)) &\le \ell(f\circ\gamma) = \int_0^L \|(f\circ\gamma)'(t)\|dt = \int_0^L \|df_{\gamma(t)}(\gamma'(t))\|dt\\ &\le \max_{t\in [0,L]}\|df_{\gamma(t)}\|L =\max_{t\in [0,L]}\|df_{\gamma(t)}\|d(x,y). \end{align*}

I'll leave it to you to modify this slightly in the case the metric is not complete.

COMMENT: The hypothesis that $f$ is a diffeomorphism is only relevant if you want to get a double inequality.