Injectivity radius

This article defines a scalar function on a manifold, viz a function from the manifold to real numbers. The scalar function may be intrinsic or defined in terms of some other structure/functions

The term injectivity radius is also used for injectivity radius of a manifold which is the infimum over the manifold of the injectivity radii at all points

Definition

The injectivity radius is a scalar function on a Riemannian manifold $M$ is defined as follows: the injectivity radius at $x\in M$ is the supremum of all values $r$ such that the exponential map from the unit ball $B_r(x)$ in $T_{x}M$, to the manifold $M$, is injective.

The fact that the injectivity radius at each point is strictly positive is one of the starting points of Riemannian geometry.