# Injectivity radius of a manifold

From Diffgeom

*This article defines a scalar value (viz, a real number) associated with a Riemannian manifold. This real number depends both on the underlying differential manifold and the Riemannian metric*

## Definition

The **injectivity radius** of a Riemannian manifold is defined as the infimum over all points, of the injectivity radius at that point.

## Facts

### Klingenberg's theorem

`Further information: Klingenberg's theorem`

A useful way of bounding the injectivity radius of a manifold is the so-called Klingenberg's theorem which states that the injectivity radius of the manifold is bounded from below by the minimum of two values, one being inversely related to any upper bound on sectional curvature, and the other being related to the minimum length of a smooth closed geodesic.