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Zbl 0874.53032
Shiohama, Katsuhiro; Tanaka, Minoru
Cut loci and distance spheres on Alexandrov surfaces. (English)
[A] Besse, Arthur L. (ed.), Actes de la table ronde de géométrie différentielle en l'honneur de Marcel Berger, Luminy, France, 12--18 juillet, 1992. Paris: Société Mathématique de France. Sémin. Congr. 1, 531-559 (1996). ISBN 2-85629-047-7/pbk

Let $X$ be a complete Alexandrov surface with curvature bounded below, and let $K$ be a compact subset of $X$. The authors investigate the structure of the distance spheres (i.e., sets at a constant distance from $K$) and of the cut locus of $K$. An important aspect of this paper is that it shows that many properties of the distance spheres and cut loci, familiar in two--dimensional Riemannian manifolds, hold with hardly any differentiability assumptions at all. Their methods also apply to the cut locus of a point at infinity of a noncompact $X$ and to the Busemann functions on it. In the course of the paper, they also obtain a generalized Sard's theorem for a continuous real valued function of one variable.
[J.Hebda (St.Louis)]

MSC 2000:: *53C20 Riemannian manifolds (global)
53C45 Global surface theory (a la A.D. Aleksandrov)

Keywords: cut loci; distance sphere; Alexandrov surface

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