At a point p in a Riemannian manifold, the largest radius r>0 such that the exponential map exp_p restricts to a diffeomorphism from the open ball of radius r in T_pM onto its image in M; globally one may take the infimum over p for a manifold injectivity radius.

Principle

Measures the size of a normal coordinate neighborhood free of self-intersections, conjugate points and cut-locus obstructions; it quantifies how far geodesics from p remain minimizing and non-degenerate.

Demonstration

On the standard sphere of radius R, the injectivity radius at any point equals πR because geodesics minimize distance up to the antipodal point; on Euclidean space the injectivity radius is infinite.

Misapplication

Confusing injectivity radius with convexity radius or assuming injectivity guarantees all geodesics extend uniquely globally; injectivity is local to the exponential map and does not imply global metric simplicity beyond that radius.

Consequence

Within the injectivity radius one has unique geodesics from p to nearby points, smooth normal coordinates, and well-behaved Riemannian volumes expressed in those coordinates; many local estimates and comparison theorems use a positive injectivity radius.

Reversal

The cut locus of p is the complement of the maximal domain of injectivity of exp_p; reversing perspective, points at or beyond the injectivity radius encounter conjugate points or multiple minimizing geodesics.

Boundary

Depends on point p and requires a Riemannian metric; failure modes include conjugate points, self-intersections of the exponential map, and topological obstructions—injectivity radius is distinct from notions defined purely for metric spaces or embedded sets.

Semantic Tension

Often compared to convexity radius and reach; injectivity radius concerns the exponential map's diffeomorphism property, while convexity radius adds uniqueness of minimizers between any two points in the ball and reach is an extrinsic nearest-point property in Euclidean space.

Synthesis

The injectivity radius is the maximal radius around p for which the exponential map gives a single-sheeted smooth normal coordinate chart: it quantifies local geodesic uniqueness and the absence of conjugate or cut-locus singularities.