Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 50, No. 1, pp. 259269 (2009) 

Cubic ruled surfaces with constant distribution parameter in $E_4$Otto RöschelInstitute of Geometry, Graz University of Technology, Kopernikusgasse 24, A8010 Graz, Austria, email: roeschel@tugraz.aAbstract: A first order invariant of ruled surfaces of $E_3$ is the socalled 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 $E_4$, Cayleysurface Classification (MSC2000): 53A25; 53A05 Full text of the article:
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