Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 44, No. 1, pp. 165178 (2003) 

Smooth Lines on Projective Planes over TwoDimensional Algebras and Submanifolds with Degenerate Gauss MapsMaks A. Akivis and Vladislav V. GoldbergDepartment of Mathematics, Jerusalem College of Technology  Mahon Lev, Havaad Haleumi St., P. O. B. 16031, Jerusalem 91160, Israel; email: akivis@avoda.jct.ac.il; Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, N.J. 07102, U.S.A.; email: vlgold@m.njit.eduAbstract: The authors study smooth lines on projective planes over the algebra $\mathbb{C}$ of complex numbers, the algebra $\mathbb{C}^1$ of double numbers, and the algebra $\mathbb{C}^0$ of dual numbers. In the space $\mathbb{R} P^5$, to these smooth lines there correspond families of straight lines forming point threedimensional submanifolds $X^3$ with degenerate Gauss maps of rank $r \leq 2$. The authors study focal properties of these submanifolds and prove that they represent examples of different types of submanifolds $X^3$ with degenerate Gauss maps. Namely, the submanifold $X^3$, corresponding in $\mathbb{R} P^5$ to a smooth line $\gamma$ of the projective plane $\mathbb{C}P^2$, does not have real singular points, the submanifold $X^3$, corresponding in $\mathbb{R} P^5$ to a smooth line $\gamma$ of the projective plane $\mathbb{C}^1 P^2$, bears two plane singular lines, and finally the submanifold $X^3$, corresponding in $\mathbb{R} P^5$ to a smooth line $\gamma$ of the projective plane $\mathbb{C}^0 P^2$, bears one singular line. It is also proved that in all three cases, the rank of $X^3$ is equal to the rank of the curvature of the line $\gamma$. Keywords: smooth line, projective plane over an algebra, submanifold with degenerate Gauss map, hypersurface of Sacksteder, hypersurface of Bourgain Classification (MSC2000): 53A20; 14M99 Full text of the article:
Electronic version published on: 3 Apr 2003. This page was last modified: 4 May 2006.
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