scholarly journals Singularities for One-Parameter Null Hypersurfaces of Anti-de Sitter Spacelike Curves in Semi-Euclidean Space

2016 ◽  
Vol 2016 ◽  
pp. 1-8 ◽  
Author(s):  
Yongqiao Wang ◽  
Donghe Pei ◽  
Ruimei Gao
Keyword(s):  
Euclidean Space ◽  
Singularity Theory ◽  
De Sitter ◽  
Null Hypersurfaces ◽  
Index 2 ◽  
Generic Singularities

We consider one-parameter null hypersurfaces associated with spacelike curves. The spacelike curves are in anti-de Sitter 3-space while one-parameter null hypersurfaces lie in 4-dimensional semi-Euclidean space with index 2. We classify the generic singularities of the hypersurfaces, which are cuspidal edges and swallowtails. And we reveal the geometric meanings of the singularities of such hypersurfaces by the singularity theory.

scholarly journals Singularities of Indefinite Hypersurfaces and Indefinite Surfaces along Timelike Curves in Nullcone with Index Two

2018 ◽  
Vol 2018 ◽  
pp. 1-18
Author(s):  
Xue Song ◽  
Zhigang Wang
Keyword(s):  
Euclidean Space ◽  
Singularity Theory ◽  
Timelike Curve ◽  
Geometric Invariant ◽  
Index 2 ◽  
Definition Of

In this paper, a class of indefinite hypersurfaces and a class of indefinite surfaces generated by timelike curves located in nullcone in 4-dimensional semi-Euclidean space with index 2 are discussed. Using the unfolding theory in singularity theory, the singularities of the indefinite hypersurfaces and the indefinite surfaces are classified and the different kinds of singularities are estimated by means of a geometric invariant σ. Meanwhile, the definition of osculating nullcone is presented; the study shows that the differential geometric invariant σ of timelike curves measured also the order of the contact between a timelike curve and a osculating nullcone ONCv0⁎. Finally, some relevant counterexamples are indicated.

scholarly journals de Sitter symmetries and inflationary scalar-vector models

2016 ◽  
Vol 21 (3) ◽  
pp. 219 ◽  
Author(s):  
Cesar Alonso Valenzuela Toledo ◽  
Juan Beltrán Almeida ◽  
Josue Motoa-Manzano
Keyword(s):  
Field Theory ◽  
Euclidean Space ◽  
Correlation Functions ◽  
Vector Fields ◽  
Conformal Symmetry ◽  
De Sitter Space ◽  
Dual Theory ◽  
De Sitter ◽  
Curvature Perturbation ◽  
Scalar And Vector Fields

<div class="page" title="Page 1"><div class="section"><div class="layoutArea"><div class="column"><p><span>In this paper, we study the correspondence between a field theory in de Sitter space in D-dimensions and a dual conformal feld theory in a euclidean space in (D - 1)-dimensions. In particular, we investigate the form in which this correspondence is established for a system of interacting scalar and a vector fields propagating in de Sitter space. We analyze some necessary (but not sucient) conditions for which conformal symmetry is preserved in the dual theory in (D - 1)-dimensions, making possible the establishment of the correspondence. The discussion that we address in this paper is framed on the context of <em>inationary cosmology</em>. Thusly, the results obtained here pose some relevant possibilities of application to the calculation of the fields’s correlation functions and of the <em>primordial curvature perturbation</em> \zeta, in inationary models including coupled scalar and vector fields.</span></p></div></div></div></div>

scholarly journals Biharmonic Hypersurfaces in Pseudo-Riemannian Space Forms with at Most Two Distinct Principal Curvatures

2020 ◽  
Vol 2020 ◽  
pp. 1-9
Author(s):  
Chao Yang ◽  
Jiancheng Liu
Keyword(s):  
Euclidean Space ◽  
Mean Curvature ◽  
Riemannian Space ◽  
Constant Mean Curvature ◽  
Space Form ◽  
De Sitter Space ◽  
De Sitter ◽  
Principal Curvatures ◽  
Space Forms ◽  
Riemannian Space Forms

In this paper, we show that biharmonic hypersurfaces with at most two distinct principal curvatures in pseudo-Riemannian space form Nsn+1c with constant sectional curvature c and index s have constant mean curvature. Furthermore, we find that such biharmonic hypersurfaces Mr2k−1 in even-dimensional pseudo-Euclidean space Es2k, Ms−12k−1 in even-dimensional de Sitter space Ss2kcc>0, and Ms2k−1 in even-dimensional anti-de Sitter space ℍs2kcc<0 are minimal.

scholarly journals Null hypersurfaces in de Sitter and anti-de Sitter cosmologies

2018 ◽  
Vol 27 (04) ◽  
pp. 1850045
Author(s):  
P. A. Hogan
Keyword(s):  
Cosmological Constant ◽  
Gravitational Waves ◽  
Constant Curvature ◽  
Focus Attention ◽  
De Sitter ◽  
Einstein’S Field Equations ◽  
Field Equations ◽  
Null Hypersurfaces ◽  
Einstein's Field Equations ◽  
Line Elements

The study of gravitational waves in the presence of a cosmological constant has led to interesting forms of the de Sitter and anti-de Sitter line elements based on families of null hypersurfaces. The forms are interesting because they focus attention on the geometry of null hypersurfaces in spacetimes of constant curvature. Two examples are worked out in some detail. The first originated in the study of collisions of impulsive gravitational waves in which the post-collision spacetime is a solution of Einstein’s field equations with a cosmological constant, and the second originated in the generalization of plane fronted gravitational waves with parallel rays to include a cosmological constant.

scholarly journals Embedding with Vaidya geometry

2020 ◽  
Vol 80 (7) ◽  
Author(s):  
A. V. Nikolaev ◽  
S. D. Maharaj
Keyword(s):  
Equation Of State ◽  
Euclidean Space ◽  
Mass Function ◽  
Energy Conditions ◽  
Dimensional Euclidean Space ◽  
De Sitter ◽  
Radiating Star ◽  
Theory Of Gravity ◽  
Null String ◽  
Vaidya Metric

Abstract The Vaidya metric is important in describing the exterior spacetime of a radiating star and for describing astrophysical processes. In this paper we study embedding properties of the generalized Vaidya metric. We had obtained embedding conditions, for embedding into 5-dimensional Euclidean space, by two different methods and solved them in general. As a result we found the form of the mass function which generates a subclass of the generalized Vaidya metric. Our result is purely geometrical and may be applied to any theory of gravity. When we apply Einstein’s equations we find that the embedding generates an equation of state relating the null string density to the null string pressure. The energy conditions lead to particular metrics including the anti/de Sitter spacetimes.

Hyperbolic worldsheets and worldlines of null Cartan curves in de Sitter 3-space

2021 ◽  
pp. 2150026
Author(s):  
Qingxin Zhou ◽  
Jingbo Xu ◽  
Zhigang Wang
Keyword(s):  
Singularity Theory ◽  
Close Relation ◽  
Dual Relationships ◽  
Normal Curve ◽  
De Sitter ◽  
Geometric Invariant ◽  
Geometric Properties ◽  
Hyperbolic Quadric ◽  
Cuspidal Edge

The hyperbolic worldsheets and the hyperbolic worldline generated by null Cartan curves are defined and their geometric properties are investigated. As applications of singularity theory, the singularities of the hyperbolic worldsheets and the hyperbolic worldline are classified by using the approach of the unfolding theory in singularity theory. It is shown that under appropriate conditions, the hyperbolic worldsheet is diffeomorphic to cuspidal edge or swallowtail type of singularity and the hyperbolic worldline is diffeomorphic to cusp. An important geometric invariant which has a close relation with the singularities of the hyperbolic worldsheets and worldlines is found such that the singularities of the hyperbolic worldsheets and worldlines can be characterized by the invariant. Meanwhile, the contact of the spacelike normal curve of a null Cartan curve with hyperbolic quadric or world hypersheet is discussed in detail. In addition, the dual relationships between the spacelike normal curve of a null Cartan curve and the hyperbolic worldsheet are described. Moreover, it is demonstrated that the spacelike normal curve of a null Cartan curve and the hyperbolic worldsheet are [Formula: see text]-dual each other.

Wavefronts of traveling trajectories of geometric particles

2019 ◽  
Vol 16 (11) ◽  
pp. 1950175 ◽  
Author(s):  
Wanying Bi ◽  
Zhigang Wang
Keyword(s):  
Differential Geometry ◽  
Singularity Theory ◽  
Moving Frame ◽  
Two Dimensional ◽  
De Sitter ◽  
Duality Theorems ◽  
Space Forms ◽  
Lorentzian Space ◽  
Legendrian Singularity ◽  
Lorentzian Space Forms

Confining the traveling trajectory of a tachyon to the two-dimensional Lorentzian space forms, we describe the trajectory as a spacelike front in these Lorentzian space forms. Introducing the differential geometry of singular curves in Lorentzian space forms, that is, the hyperbolic space and de Sitter space, and applying the Legendrian duality theorems, we establish the moving frame along the front, whereby the definitions of the evolutes of spacelike fronts in Lorentzian space forms are presented and the geometric properties of these evolutes are investigated in detail. It is shown that these evolutes can be interpreted as wavefronts under the viewpoint of Legendrian singularity theory.

scholarly journals Singularities for One-Parameter Developable Surfaces of Curves

Symmetry ◽  
2019 ◽  
Vol 11 (1) ◽  
pp. 108 ◽  
Author(s):  
Qiming Zhao ◽  
Donghe Pei ◽  
Yongqiao Wang
Keyword(s):  
Singularity Theory ◽  
Geometric Properties ◽  
Developable Surfaces ◽  
Generic Singularities

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.

scholarly journals Singularities of Non-Developable Surfaces in Three-Dimensional Euclidean Space

Mathematics ◽  
2019 ◽  
Vol 7 (11) ◽  
pp. 1106
Author(s):  
Jie Huang ◽  
Donghe Pei
Keyword(s):  
Euclidean Space ◽  
Three Dimensional ◽  
Singularity Theory ◽  
Structure Functions ◽  
Ruled Surface ◽  
Singular Points ◽  
Dimensional Euclidean Space ◽  
Smooth Curves ◽  
Developable Surfaces

We study the singularity on principal normal and binormal surfaces generated by smooth curves with singular points in the Euclidean 3-space. We discover the existence of singular points on such binormal surfaces and study these singularities by the method of singularity theory. By using structure functions, we can characterize the ruled surface generated by special curves.

scholarly journals Osculating curves in 4-dimensional semi-Euclidean space with index 2

2017 ◽  
Vol 15 (1) ◽  
pp. 562-567
Author(s):  
Kazim İlarslan ◽  
Nihal Kiliç ◽  
Hatice Altin Erdem
Keyword(s):  
Euclidean Space ◽  
Sufficient Conditions ◽  
Necessary And Sufficient Conditions ◽  
Euclidian Space ◽  
Null Curves ◽  
Index 2 ◽  
Necessary And Sufficient

Abstract In this paper, we give the necessary and sufficient conditions for non-null curves with non-null normals in 4-dimensional Semi-Euclidian space with indeks 2 to be osculating curves. Also we give some examples of non-null osculating curves in $\mathbb{E}_{2}^{4}$ .

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