from a point $O$
The set of points $x$ of a Riemannian manifold $W$ on the geodesic rays emanating from $O$ for which the ray $Ox$ is not extendable as a geodesic beyond the point $x$. In the two-dimensional case the cut locus is a one-dimensional graph with no cycles (see ); if $W$ is analytic of arbitrary dimension, then it is a polyhedron of analytic submanifolds (see ). The cut locus depends continuously on $O$. The cut locus is defined not only with respect to a point but also with respect to other subsets, for example, the boundary $\partial W$, and also in spaces other than Riemannian manifolds, for example, on convex surfaces (see ) and in two-dimensional manifolds of bounded curvature.
|||D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)|
|||S.B. Myers, "Connections between differential geometry and topology. I Simply connected surfaces" Duke Math. J. , 1 (1935) pp. 376–391|
|||M.A. Buchner, "Simplicial structure of the real analytic cut locus" Proc. Amer. Math. Soc. , 64 : 1 (1977) pp. 118–121|
|||J. Kunze, "Der Schnittort auf konvexen Verheftungsflächen" , Springer (1969)|
"Non-extendable as a geodesic" means that $Ox$ looses the property of minimality after the point $x$, i.e. $Ox'$ is no longer the minimal path from $O$ to $x'$ if $Ox\subset Ox'$.
Cut locus. Encyclopedia of Mathematics. URL: http://encyclopediaofmath.org/index.php?title=Cut_locus&oldid=31715