Given a Riemannian manifold \((M,g)\) (without boundary). For each point \(p\in M\) the injectivity radius of \(M\) at \(p\), denoted by \(\inj(p)\), is the supremum of all \(a>0\) such that \(\exp_p\) is a diffeomorphism from \(B_a(0)\subset T_pM\) onto its image.
The injectivity radius of \(M\), denoted by \(\inj(M)\), is the infimum over all injectivity radii at \(p\).
Remarks
- Every compact Riemannian manifold has positive injective radius [@lee2013smooth_manifolds, Lemma 6.16]