Cut locus

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 [2]); if $W$ is analytic of arbitrary dimension, then it is a polyhedron of analytic submanifolds (see [3]). 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 [4]) and in two-dimensional manifolds of bounded curvature.

References

[1]	D. Gromoll, W. Klingenberg, W. Meyer, "Riemannsche Geometrie im Grossen" , Springer (1968)
[2]	S.B. Myers, "Connections between differential geometry and topology. I Simply connected surfaces" Duke Math. J. , 1 (1935) pp. 376–391
[3]	M.A. Buchner, "Simplicial structure of the real analytic cut locus" Proc. Amer. Math. Soc. , 64 : 1 (1977) pp. 118–121
[4]	J. Kunze, "Der Schnittort auf konvexen Verheftungsflächen" , Springer (1969)

Comments

"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 $Ox\subset Ox'$.