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)
Famous quotes containing the word definition:
“According to our social pyramid, all men who feel displaced racially, culturally, and/or because of economic hardships will turn on those whom they feel they can order and humiliate, usually women, children, and animalsjust as they have been ordered and humiliated by those privileged few who are in power. However, this definition does not explain why there are privileged men who behave this way toward women.”
—Ana Castillo (b. 1953)
“The very definition of the real becomes: that of which it is possible to give an equivalent reproduction.... The real is not only what can be reproduced, but that which is always already reproduced. The hyperreal.”
—Jean Baudrillard (b. 1929)
“It is very hard to give a just definition of love. The most we can say of it is this: that in the soul, it is a desire to rule; in the spirit, it is a sympathy; and in the body, it is but a hidden and subtle desire to possessafter many mysterieswhat one loves.”
—François, Duc De La Rochefoucauld (16131680)