Definition
Fix a point in a complete Riemannian manifold, and consider the tangent space . It is a standard result that for sufficiently small in, the curve defined by the Riemannian exponential map, for belonging to the interval is a minimizing geodesic, and is the unique minimizing geodesic connecting the two endpoints. Here denotes the exponential map from . The cut locus of in the tangent space is defined to be the set of all vectors in such that is a minimizing geodesic for but fails to be minimizing for for each . The cut locus of in is defined to be image of the cut locus of in the tangent space under the exponential map at . Thus, we may interpret the cut locus of in as the points in the manifold where the geodesics starting at stop being minimizing.
The least distance from p to the cut locus is the injectivity radius at p. On the open ball of this radius, the exponential map at p is a diffeomorphism from the tangent space to the manifold, and this is the largest such radius. The global injectivity radius is defined to be the infimum of the injectivity radius at p, over all points of the manifold.
Read more about this topic: Cut Locus (Riemannian Manifold)
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