The Darboux mapping of canal hypersurfaces
Maks A. Akivis, Vladislav V. Goldberg
Abstract
The geometry of canal hypersurfaces of an n-dimensional conformal space Cn is studied. Such hypersurfaces are envelopes of r-parameter families of hyperspheres, 1 ≤ r ≤ n-2. In the present paper the conditions that characterize canal hypersurfaces, and which were known earlier, are made more precise. The main attention is given to the study of the Darboux maps of canal hypersurfaces in the de Sitter space M1n+1 and the projective space Pn+1. To canal hypersurfaces there correspond r-dimensional spacelike tangentially nondegenerate submanifolds in M1n+1 and tangentially degenerate hypersurfaces of rank r in Pn+1. In this connection the problem of existence of singular points on canal hypersurfaces is considered.
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