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On transnormal horizons of convex hypersurfaces. (English) Zbl 0643.53046

For a closed smooth hypersphere \(M\) of positive Gaussian curvature in Euclidean \(n\)-space \(E^n\) \((n\geq 3)\) and a given affine subspace \(e\) of \(E^n\) of dimension \(k\) \((1\leq k\leq n-2)\) the set of points in \(M\) where the tangent plane of \(M\) is parallel to \(e\) is a well-defined differentiable submanifold of dimension \(n-k-1\), called the \(e\)-horizon of \(M\). Here it is proved that round spheres are characterized by the property that all horizons of \(M\) of the same dimension are transnormal. This has been shown for the case \(k=1\) by F. J. Craveiro de Carvalho and S. A. Robertson in [Note Mat. 7, No. 2, 167–172 (1987; Zbl 0643.53046), see the preceding review], using different methods of proof. Additionally in the paper under review round spheres in 3-space are characterized by the existence of two transnormal horizons through each point of \(M\) if \(M\) is assumed to be of constant width. Generalizations to higher dimensions are obvious.

MSC:

53C42 Differential geometry of immersions (minimal, prescribed curvature, tight, etc.)
53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
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