Local and global exposed points. (English) Zbl 0868.53022

Beltagy, M.

Commun. Fac. Sci. Univ. Ankara, Ser. A1 43, No. 1-2, 57-65 (1994).

In a complete simply connected \(C^\infty\)-smooth Riemannian manifold \(\widetilde{W}\) without focal (and therefore also without conjugate) points a point \(p\) of a subset \(B\subset \widetilde{W}\) is called (global) exposed point of \(B\) if for a suitable unit vector

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\(v\subset \widetilde{W}_p\) the horosphere \(H_v\) through \(p\), given by \(v\) and bounding the closed horodisc \(\overline{D}_v\), has the properties (i) \(B\subset \overline{D}_v\) and (ii) \(B\cap H_v= \{p\}\). If, however, there exists only a neighbourhood \(U\) of \(\widetilde{W}\) about \(p\) such that \(B\cap U\) has \(p\) as an exposed point then \(p\) is called a local exposed point of \(B\).
The main result of the paper is: \(B\) is a strictly (geodesically) convex body if either \(B\) is a connected open subset

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of \(\widetilde{W}\) with smooth boundary all the points of which are local exposed points of \(\overline{B}\) or if \(B\) is an open bounded subset \(\widetilde{W}\) with smooth boundary all the points of which are global exposed points of \(\overline{B}\).

Reviewer: K.Leichtweiß (Stuttgart)

MSC:

53C20	Global Riemannian geometry, including pinching
52A55	Spherical and hyperbolic convexity

Keywords:

global and local exposed points in a Riemannian manifold; horospheres and horodiscs; strict geodesic convexity