Definition:Injectivity Radius at Point of Riemannian Manifold

Definition

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

Let $T_ p M$ be the tangent space of $M$ at $p \in M$.

Let $\exp_p$ be the restricted exponential map at $p \in M$.

Let $\map {B_a} 0 \in T_p M$ be an open ball.

Suppose $A_p$ is the set of all $a \in \R_{> 0}$ for which $\exp_p$ is a diffeomorphism from $\map {B_a} 0 \subseteq T_p M$ onto its image.

Then the supremum of $A_p$ is called the injectivity radius of $M$ at $p$ and is denoted by $\map {\operatorname{inj} } p$:

$\map {\operatorname{inj} } p = \sup {A_p}$

Sources

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