Beiträge zur Algebra und Geometrie Contributions to Algebra and Geometry Vol. 44, No. 1, pp. 165-178 (2003) |
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Smooth Lines on Projective Planes over Two-Dimensional 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; e-mail: akivis@avoda.jct.ac.il; Department of Mathematical Sciences, New Jersey Institute of Technology, University Heights, Newark, N.J. 07102, U.S.A.; e-mail: 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 three-dimensional 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.
© 2003 Heldermann Verlag
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