Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 39, No. 2, pp. 395-411 (1998)

The Darboux Mapping of Canal Hypersurfaces

Maks A. Akivis, Vladislav V. Goldberg

Department of Mathematics and Computer Science, Ben-Gurion University of the Negev, P. O. Box 653, Beer-Sheva, 84105, Israel, e-mail:

Department of Mathematics, New Jersey Institute of Technology, Newark, NJ 07102, USA, e-mail:

Abstract: The geometry of canal hypersurfaces of an $n$-dimensional conformal space $C^n$ is studied. Such hypersurfaces are envelopes of $r$-parameter families of hyperspheres, $1\leq r\leq n-2$. 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 $M_1^{n+1}$ and the projective space $P^{n+1}$. To canal hypersurfaces there correspond $r$-dimensional spacelike tangentially nondegenerate submanifolds in $M_1^{n+1}$ and tangentially degenerate hypersurfaces of rank $r$ in $P^{n+1}$. In this connection the problem of existence of singular points on canal hypersurfaces is considered.

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