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Singularities of non-degenerate n-ruled (n + 1)-manifolds in Euclidean space

Kentaro Saji — 2004

Banach Center Publications

The objective of this paper is to study singularities of n-ruled (n + 1)-manifolds in Euclidean space. They are one-parameter families of n-dimensional affine subspaces in Euclidean space. After defining a non-degenerate n-ruled (n + 1)-manifold we will give a necessary and sufficient condition for such a map germ to be right-left equivalent to the cross cap × interval. The behavior of a generic n-ruled (n + 1)-manifold is also discussed.

Legendrian dual surfaces in hyperbolic 3-space

Kentaro Saji;  Handan Yıldırım — 2015

Annales Polonici Mathematici

We consider surfaces in hyperbolic 3-space and their duals. We study flat dual surfaces in hyperbolic 3-space by using extended Legendrian dualities between pseudo-hyperspheres in Lorentz-Minkowski 4-space. We define the flatness of a surface in hyperbolic 3-space by the degeneracy of its dual, which is similar to the case of the Gauss map of a surface in Euclidean 3-space. Such surfaces are a kind of ruled surfaces. Moreover, we investigate the singularities of these surfaces and the dualities...

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