Beitr\ EMIS ELibM Electronic Journals Beiträge zur Algebra und Geometrie
Contributions to Algebra and Geometry
Vol. 50, No. 1, pp. 259-269 (2009)

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Cubic ruled surfaces with constant distribution parameter in $E_4$

Otto Röschel

Institute of Geometry, Graz University of Technology, Kopernikusgasse 24, A-8010 Graz, Austria, e-mail: roeschel@tugraz.a

Abstract: A first order invariant of ruled surfaces of $E_3$ is the so-called distribution parameter $d$ in a generator. It is defined as the limit of the quotient of the distance and the angle of the generator and its neighbour. Ruled surfaces with constant parameter of distribution are of special interest and have been studied by many authors. H. Brauner could prove that the only nontrivial cubic ruled surface with constant distribution parameter in $E_3$ is a special type of a Cayley surface. This paper is devoted to the investigation of these problems for higher dimensions. We will in fact determine all cubic ruled surfaces of $E_n$ with constant distribution parameter. Surprisingly, there is one class of such surfaces way beyond the $3$-dimensional Cayley surface case.

Keywords: ruled surfaces, constant distribution parameter, twisted cubic ruled surfaces in $E_4$, Cayley-surface

Classification (MSC2000): 53A25; 53A05

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Electronic version published on: 29 Dec 2008. This page was last modified: 28 Jan 2013.

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