# Injectivity radius

From Diffgeom

*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*Template:Radius notion

*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 is defined as follows: the injectivity radius at is the supremum of all values such that the exponential map from the unit ball in , to the manifold , is injective.

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