Geometry of Projective Planes over Two-Dimensional Algebras
Abstract
The authors study smooth lines on projective planes over the algebra C of complex numbers, the algebra C1 of double numbers, and the algebra C0 of dual numbers. In the space RP5, to these smooth lines there correspond families of straight lines describing point three-dimensional tangentially degenerate submanifolds X3 of rank 2. The authors study focal properties of these submanifolds and prove that they represent examples of different types of tangentially degenerate submanifolds. Namely, the submanifold X3, corresponding in RP5 to a smooth line γ of the projective plane C, does not have real singular points, the submanifold X3, corresponding in RP5 to a smooth line γ of the projective plane C1 P2, bears two plane singular lines, and finally the submanifold X3, corresponding in RP5 to a smooth line γ of the projective plane C0 P2, bears one singular line.
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