Definition:Injectivity Radius of Riemannian Manifold

Definition

Let $\struct {M, g}$ be a Riemannian manifold without boundary.

Let $\map {\operatorname {inj} } p$ be the injectivity radius at $p \in M$.

Suppose $\map {\operatorname {inj} } p = \infty$.

Then the infimum of $\map {\operatorname {inj} } p$ over $p \in M$ is called the injectivity radius of $M$ and is denoted by $\map {\operatorname{inj} } M$:

$\ds \map {\operatorname{inj} } M = \inf_{p \mathop \in M} {\map {\operatorname {inj} } p}$

Sources

• 2018: John M. Lee: Introduction to Riemannian Manifolds (2nd ed.) ... (previous) ... (next): $\S 6$: Geodesics and Distance. Uniformly Normal Neighborhoods