Injectivity radius

Remarks

Example: on the unit sphere $S^n$, for any $p$, $\exp_p$ is a diffeomorphism from $B_\pi(0)$ onto $S^n$ minus the antipode $-p$. It maps the whole sphere $\set{\abs{v} = \pi}$ to $-p$, so no larger ball works, and $\operatorname{inj}(p) = \pi$.

Non-example: for the same $p$ on $S^n$, the radius $r = 2\pi$ does not belong to $A_p$, because $\exp_p$ is not injective on $B_{2\pi}(0)$.

For $\R^n$ with the Euclidean metric, $\operatorname{inj}(p) = \infty$ for every $p$. The injectivity radius is always positive at each point; if $M$ is compact, then $\operatorname{inj}(M) > 0$. If $M$ is complete, $\operatorname{inj}(p)$ is the distance from $p$ to its cut locus.