Singularity properties of one parameter lightlike hypersurfaces in Minkowski 4-space

Sun, Jianguo; Pei, Donghe

doi:10.22436/jnsa.008.05.02

Cited by 3 publications

(1 citation statement)

References 17 publications

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“…In the present paper, we investigate one-parameter developable surfaces, which are related to the space curves, as a fundamental case for the research of the highest dimensional manifolds in Euclidean 3-space. We investigated the singularities of hypersurfaces in semi-Euclidean space [7][8][9][10]. However, at least to the best of our knowledge, there exists little literature concerning the singularities of one-parameter developable surfaces related to regular space curves in Euclidean space.…”

Section: Introductionmentioning

confidence: 99%

Singularities for One-Parameter Developable Surfaces of Curves

2019

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Developable surfaces, which are important objects of study, have attracted a lot of attention from many mathematicians. In this paper, we study the geometric properties of one-parameter developable surfaces associated with regular curves. According to singularity theory, the generic singularities of these developable surfaces are classified-they are swallowtails and cuspidal edges. In addition, we give some examples of developable surfaces which have symmetric singularity models.Symmetry 2019, 11, 108 2 of 15 surfaces are sections of one-parameter developable surfaces. We also define the one-parameter support functions on regular space curves, which can be used to study the geometric properties of one-parameter developable surfaces. In fact, one-parameter developable surfaces are the discriminant sets of these functions. The main result, Theorem 2, shows that the singularities of developable surfaces are A k -singularities (k = 2, 3) of these functions.The organization of this paper is as follows: We review the concepts of ruled surfaces in Euclidean space in Section 2. In Section 3, the one-parameter developable surfaces of a space curve are defined, and we obtain two geometric invariants of the curve. We also get singularities of one-parameter developable surfaces (Theorem 1), and Theorem 2 gives the classification of these singularities in this section. The preparations for the proof of Theorem 2 are in Sections 4 and 5. In the last section, we give some examples to illustrate the main results in this paper. Basic NotationLet R 3 be 3-dimensional Euclidean space and x = (x 1 , x 2 , x 3 ), y = (y 1 , y 2 , y 3 ) ∈ R 3 . We denote their standard inner product by x · y, and the norm of x is denoted by x . Let γ : I → R 3 be a curve and the tangent vector respect to t isγ(t) = dγ/dt(t). The arc-length is s(t) = t t 0

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Section: Introductionmentioning

confidence: 99%

Singularities for One-Parameter Developable Surfaces of Curves

2019

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Finite Determinacy of High Codimension Smooth Function Germs

Gan

Pei

et al. 2019

Appl. Math. J. Chin. Univ.

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Geometry of Developable Surfaces of Frenet Type Framed Base Curves from the Singularity Theory Viewpoint

2022

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In this paper, we consider Frenet type framed base curves that may have singular points and define one-parameter developable surfaces associated with such curves. By using the singularity theory, we classify the generic singularities of the developable surfaces, which are cuspidal edges and swallowtails. In order to characterize these singularities, two geometric invariants are discovered. At last, an example is given to demonstrate the main results.

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Singularity properties of one parameter lightlike hypersurfaces in Minkowski 4-space

Cited by 3 publications

References 17 publications

Singularities for One-Parameter Developable Surfaces of Curves

Singularities for One-Parameter Developable Surfaces of Curves

Finite Determinacy of High Codimension Smooth Function Germs

Geometry of Developable Surfaces of Frenet Type Framed Base Curves from the Singularity Theory Viewpoint

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