Let $M$ be a closed compact Riemannian manifold.

The exponential map $\mathrm{exp}:TM\to M\times M$ takes $(p,v)$ to $(p,\gamma_v(1))$, where $\gamma_v$ is the geodesic flow at $p$ in the direction of $v$. The exponential map $\mathrm{exp}_p:T_pM\to M$ is the projection to the second coordinate of the restriction of $\mathrm{exp}$.

By the inverse function theorem, $\mathrm{exp}$ is a diffeomorphism on a neighborhood of the zero section in $TM$. So in particular there is a value $\epsilon_{\mathrm{Diff}}$ which is the supremum of values so that $\mathrm{exp}$ restricted to disk bundles in $TM$ of radius less than $\epsilon_{\mathrm{Diff}}$ is a diffeomorphism.

On the other hand, the injectivity radius at $p$, $\epsilon_p$, is the supremum of $\epsilon$ such that $\mathrm{exp}_p$ is a diffeomorphism from the radius $\epsilon$ ball around $0$ in $T_pM$ onto its image. The injectivity radius of $M$ is $\epsilon_{\mathrm{inj}}=\min_p \epsilon_p$.

Clearly $\epsilon_{\mathrm{Diff}}\le \epsilon_{\mathrm{inj}}$. Is the converse true? Is that obvious? If not, what can go wrong? Is there a reference? I glanced at several books and couldn't find an explicit answer anywhere. In Berger's "A Panoramic View of Riemannian Geometry" there was something somewhat parenthetical along these lines but with no argument or reference.

globalstatement followed from some version of yourlocalstatement. I'm satisfied with Theo's answer. $\endgroup$ – Gabriel C. Drummond-Cole May 23 '15 at 2:00