Injectivity radius

Summary

Let $p$ be a point of a Riemannian manifold. Its injectivity radius at $p$ is the largest $r$ such that the exponential map at $p$ is a diffeomorphism on the $r$-ball in $T_pM$. Within this radius, normal coordinates at $p$ exist and each point is joined to $p$ by a unique shortest geodesic. The injectivity radius of the whole manifold is the infimum over all points. Lower bounds for it are a key ingredient in compactness and finiteness theorems.