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Geometry of pseudo-non-degenerate two-ruled hypersurfaces

Junzhen Li, Kentaro Saji

Abstract

We investigate the singularities of two-ruled hypersurfaces in the Euclidean four-space. By considering the points that minimize the distance between adjacent rulings, we obtain a characterization the striction curve. We introduce the notion of pseudo-non-degenerate two-ruled hypersurfaces and examine their fundamental properties. We show that two-ruled hypersurfaces constructed from a curve equipped with a Frenet-type frame, via height functions, are pseudo-non-degenerate. Furthermore, we study properties of the original curve through the striction curves and the singularities of pseudo-non-degenerate two-ruled hypersurfaces constructed in this manner.

Geometry of pseudo-non-degenerate two-ruled hypersurfaces

Abstract

We investigate the singularities of two-ruled hypersurfaces in the Euclidean four-space. By considering the points that minimize the distance between adjacent rulings, we obtain a characterization the striction curve. We introduce the notion of pseudo-non-degenerate two-ruled hypersurfaces and examine their fundamental properties. We show that two-ruled hypersurfaces constructed from a curve equipped with a Frenet-type frame, via height functions, are pseudo-non-degenerate. Furthermore, we study properties of the original curve through the striction curves and the singularities of pseudo-non-degenerate two-ruled hypersurfaces constructed in this manner.
Paper Structure (15 sections, 17 theorems, 85 equations)

This paper contains 15 sections, 17 theorems, 85 equations.

Table of Contents

  1. Introduction
  2. Two-ruled hypersurfaces
  3. Basic notions and definitions
  4. Striction surfaces and striction curves
  5. Singularities and their criteria
  6. Singularities of pseudo-non-degenerate two-ruled hypersurface
  7. Generic singularities of pseudo-non-degenerate two-ruled frontals
  8. Two-ruled hypersurfaces constructed from height functions
  9. A Frenet-type frame along a curve
  10. Developable hypersurfaces along a curve
  11. Singularities of the two-ruled frontals $S_i$ ($i=1,2,3,4$)
  12. Properties of the frontal $S_1$
  13. Properties of the frontal $S_2$
  14. Properties of the frontal $S_3$
  15. Properties of the frontal $S_4$

Key Result

Lemma 2.1

(sing2ruledhypersurface) For any germs $X,Y:(\boldsymbol{R},0)\to(\boldsymbol{R}^4,0)$ that are linearly independent for all $t\in(\boldsymbol{R},0)$, there exists $\tilde{X},\tilde{Y}:(\boldsymbol{R},0)\to(\boldsymbol{R}^4,0)$ satisfying $\langle X,Y\rangle_{\boldsymbol{R}}=\langle \tilde{X},\tilde

Theorems & Definitions (34)

  • Lemma 2.1
  • Definition 2.2
  • Proposition 2.3
  • proof
  • Proposition 2.4
  • proof
  • Theorem 2.5
  • proof
  • Definition 2.6
  • Lemma 2.7
  • ...and 24 more