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.

Singularities of worldsheets in spherical space–times

2018 ◽  
Vol 33 (18n19) ◽  
pp. 1850114 ◽  
Author(s):  
Xuening Lian ◽  
Zhigang Wang ◽  
Huilai Li
Keyword(s):  
Three Dimensional ◽  
Singularity Theory ◽  
Space Time ◽  
De Sitter ◽  
Spherical Space ◽  
Geometric Invariant ◽  
Close Relationship ◽  
Theoretical Frame ◽  
Cuspidal Edge ◽  
Theoretical Results

In this paper, the singularities of the geometry for four classes of worldsheets, which are respectively, located in three-dimensional hyperbolic space and three-dimensional de Sitter space–time are considered. Under the theoretical frame of geometry of space–time and as applications of singularity theory, it is shown that these worldsheets have two classes of singularities, that is, in the local sense, these four classes of worldsheets are, respectively, diffeomorphic to the cuspidal edge and the swallowtail. The first hyperbolic worldsheet and the second hyperbolic worldsheet are [Formula: see text]-dual to the tangent curves of spacelike curves. Moreover, it is also revealed that there is a close relationship between the types of singularities of worldsheets and a geometric invariant [Formula: see text], depending on whether [Formula: see text] or [Formula: see text] and [Formula: see text], the singularities of these worldsheets can be characterized by the geometric invariant. We provide two explicit examples of worldsheets to illustrate the theoretical results.

Singularities of worldsheets associated with null Cartan curves in Lorentz–Minkowski space–time

2018 ◽  
Vol 33 (33) ◽  
pp. 1850192 ◽  
Author(s):  
Siyao Liu ◽  
Zhigang Wang
Keyword(s):  
Minkowski Space ◽  
Singularity Theory ◽  
Space Time ◽  
Dual Relationships ◽  
Edge Type ◽  
Minkowski Space Time ◽  
Model Surfaces ◽  
Cuspidal Edge ◽  
Theoretical Results

In this paper, as applications of singularity theory, we study the singularities of several worldsheets generated by null Cartan curves in Lorentz–Minkowski space–time. Using the approach of the unfolding theory in singularity theory, we establish the relationships between these worldsheets and invariants such that the cuspidal edge type of singularity and the swallowtail type of singularity can be characterized by these invariants, respectively. Meanwhile, the contact of the tangent curve of a null Cartan curve with some model surfaces are discussed in detail. In addition, we also describe the dual relationships between the tangent curve of a null Cartan curve and these worldsheets. Finally, some concrete examples are provided to explain our theoretical results.

scholarly journals Singularities of lightcone pedals of spacelike curves in Lorentz-Minkowski 3-space

2016 ◽  
Vol 14 (1) ◽  
pp. 889-896 ◽  
Author(s):  
Liang Chen
Keyword(s):  
Singularity Theory ◽  
Smooth Functions ◽  
Geometric Properties ◽  
Timelike Surface

AbstractIn this paper, geometric properties of spacelike curves on a timelike surface in Lorentz-Minkowski 3-space are investigated by applying the singularity theory of smooth functions from the contact viewpoint.

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 Geometric Invariants of Normal Curves under Conformal Transformation in $\mathbb{E}^3$

2021 ◽  
Vol 53 ◽  
Author(s):  
Mohamd Saleem Lone
Keyword(s):  
Conformal Transformation ◽  
Sufficient Condition ◽  
Normal Curve ◽  
Geometric Invariant ◽  
Geometric Invariants ◽  
Invariant Properties

In this paper, we investigate the geometric invariant properties of a normal curve on a smooth immersed surface under conformal transformation. We obtain an invariant-sufficient condition for the conformal image of a normal curve. We also find the deviations of normal and tangential components of the normal curve under the same motion.

Geometrical particles and dualities related to curves in null de Sitter 3-sphere

2021 ◽  
pp. 2150152
Author(s):  
Xue Song ◽  
Donghe Pei
Keyword(s):  
Duality Theory ◽  
Dimensional Space ◽  
Three Dimensional ◽  
Space Time ◽  
Dual Relationships ◽  
De Sitter ◽  
Contact Manifolds ◽  
Dimensional Space Time ◽  
Three Dimensional Space

In this paper, we confine the trajectory of geometrical particles to the de Sitter three-dimensional space–time, we model geometrical particles as spacelike tachyons. Using the Legendrian duality theory of pseudo-spheres and contact manifolds theory we establish the dual relationships between spacelike moving trajectory of geometrical particles and future nullcone hypersurfaces.

Lines, circles, focal and symmetry sets

1995 ◽  
Vol 118 (3) ◽  
pp. 411-436 ◽  
Author(s):  
J. W. Bruce
Keyword(s):  
Differential Geometry ◽  
Singularity Theory ◽  
Geometric Invariant ◽  
Circle Maps ◽  
Bifurcation Set ◽  
General Theme ◽  
Smooth Family ◽  
The Family ◽  
Parameter Values

Let X be a surface in Euclidean 3-space, hereafter denoted by ℝ3. In the paper [13] Montaldi considered the contact of the surface X with circles, and obtained some very attractive results. In this piece of work we want to address some more detailed questions concerning such contact. In keeping with a general theme within singularity theory we shall bundle the circles up into fibres of certain maps and consider the restriction of these mappings to our surface X. In other words we shall be interested in the simultaneous contact of the surface X with special families of circles. The particular families we shall consider are parameterized by the set K of all lines in ℝ3; associated to such a line we have the family of all circles lying in planes orthogonal to the line, and centred on the line. The line will be referred to as the axis of the circle. Suppose, for example, the line in question is given by x1 = x2 = 0. We can consider the map ℝ3 → ℝ2 given by . The fibres of this mapping are clearly the set of circles with the properties described above together, of course, with single points on the line itself. So the family of oriented lines parameterizes a family of mappings ℝ3 → ℝ2, and by restriction a family of mappings X → ℝ2. It is of interest to relate the singularities of this mapping to the differential geometry of X. The key geometric invariant of any smooth family is its bifurcation set, that is the set of parameter values for which the corresponding map fails to be stable. We shall see that for the family of circle maps the bifurcation set is of some interest.

Timelike sweeping surfaces and singularities

2020 ◽  
Vol 18 (01) ◽  
pp. 2150006
Author(s):  
Monia Fouad Naghi ◽  
Rashad A. Abdel-Baky
Keyword(s):  
Necessary Conditions ◽  
Ruled Surface ◽  
Ruled Surfaces ◽  
Geometric Invariant ◽  
Geometric Properties ◽  
Sufficient And Necessary Conditions

We consider a timelike sweeping surface with rotation minimizing frames in Minkowski 3-Space [Formula: see text]. We introduce a new geometric “invariant”, which demonstrates the geometric properties and local singularities of the surface. Subsequently, we give the sufficient and necessary conditions for this surface to be a developable ruled surface. Finally, the singularities of these ruled surfaces are investigated.

scholarly journals The positive energy theorem for asymptotically anti-de Sitter spacetimes

2015 ◽  
Vol 17 (04) ◽  
pp. 1550015 ◽  
Author(s):  
Yaohua Wang ◽  
Naqing Xie ◽  
Xiao Zhang
Keyword(s):  
Initial Data ◽  
Total Energy ◽  
Positive Energy ◽  
Data Sets ◽  
De Sitter Spacetime ◽  
De Sitter ◽  
Geometric Invariant ◽  
Sitter Spacetime

We establish the inequality for Henneaux–Teitelboim's total energy–momentum for asymptotically anti-de Sitter initial data sets which are asymptotic to arbitrary t-slice in anti-de Sitter spacetime. In particular, when t = 0, it generalizes Chruściel–Maerten–Tod's inequality in the center of AdS mass coordinates. We also show that the determinant of energy–momentum endomorphism Q is the geometric invariant of asymptotically anti-de Sitter spacetimes.

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.

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