Abstract
In [6] ROBERTSON introduced the notion of a transnormal manifold as a generalization of a compact connected closed hypersurface of constant width in Euclidean space. This paper includes a detailed proof of the fact that the projection of a transnormal manifold on its space of normal planes is a covering map. Furthermore we prove the following generalization of a property of closed convex hypersurfaces of constant width: If two points p, q of a transnormal manifold have the same normal plane, then (for a suitable choice) the sum of the corresponding radii of principal curvature in direction of the common normal line is equal to the distance of p and q. Finally there are given examples of transnormal manifolds, which do not possess minimal total absolute curvature.
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Die vorliegende Arbeit stellt im wesentlichen die ersten Kapitel der von der Fakultät für Allgemeine Ingenieurwissenschaften der TU Berlin genehmigten Dissertation [9] dar.
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Wegner, B. Krümmungseigenschaften transnormaler Mannigfaltigkeiten. Manuscripta Math 3, 375–390 (1970). https://doi.org/10.1007/BF01168293
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DOI: https://doi.org/10.1007/BF01168293