Injectivity radius of a manifold

The article defines a scalar value (that is, a single real number) associated with a differential manifold. The scalar value may be dependent on certain additional structure with which the differential manifold is equipped

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.