NULL HELICES IN LORENTZIAN SPACE FORMS

2001 ◽  
Vol 16 (30) ◽  
pp. 4845-4863 ◽  
Author(s):  
ANGEL FERRÁNDEZ ◽  
ANGEL GIMÉNEZ ◽  
PASCUAL LUCAS
Keyword(s):  
Minkowski Space ◽  
Complete Classification ◽  
Space Forms ◽  
Lorentzian Space ◽  
Low Dimensions ◽  
Minkowski Space Time ◽  
Minimum Number ◽  
Null Curves ◽  
Lorentzian Space Forms

In this paper we introduce a reference along a null curve in an n-dimensional Lorentzian space with the minimum number of curvatures. That reference generalizes the reference of Bonnor for null curves in Minkowski space–time and it is called the Cartan frame of the curve. The associated curvature functions are called the Cartan curvatures of the curve. We characterize the null helices (that is, null curves with constant Cartan curvatures) in n-dimensional Lorentzian space forms and we obtain a complete classification of them in low dimensions.

scholarly journals Hypersurface Constrained Elasticae in Lorentzian Space Forms

2015 ◽  
Vol 2015 ◽  
pp. 1-13 ◽  
Author(s):  
Óscar J. Garay ◽  
Álvaro Pámpano ◽  
Changhwa Woo
Keyword(s):  
Space Form ◽  
Energy Functional ◽  
Bending Energy ◽  
Space Forms ◽  
Lorentzian Space ◽  
Totally Geodesic ◽  
Critical Curves ◽  
Different Types ◽  
Lorentzian Space Forms

We study geodesics in hypersurfaces of a Lorentzian space formM1n+1(c), which are critical curves of theM1n+1(c)-bending energy functional, for variations constrained to lie on the hypersurface. We characterize critical geodesics showing that they live fully immersed in a totally geodesicM13(c)and that they must be of three different types. Finally, we consider the classification of surfaces in the Minkowski 3-space foliated by critical geodesics.

Classification of the Blaschke Isoparametric Hypersurfaces in Lorentzian Space Forms

2015 ◽  
Vol 65 (3) ◽  
Author(s):  
Fengyun Zhang ◽  
Huafei Sun
Keyword(s):  
Conformal Transformation ◽  
Transformation Group ◽  
Second Fundamental Form ◽  
Conformal Metric ◽  
Space Forms ◽  
Isoparametric Hypersurfaces ◽  
Lorentzian Space ◽  
Blaschke Tensor ◽  
Lorentzian Space Forms

AbstractIn this paper, we study regular immersed hypersurfaces in Lorentzian space forms with a conformal metric, a conformal second fundamental form, the conformal Blaschke tensor and a conformal form, which are invariants under the conformal transformation group. We classify all the immersed hypersurfaces in Lorentzian space forms with two distinct constant Blaschke eigenvalues and vanishing conformal form.

The complete classification of a class of conformally flat Lorentzian hypersurfaces in ℝ14

2017 ◽  
Vol 28 (13) ◽  
pp. 1750092
Author(s):  
Zhenxiao Xie ◽  
Changping Wang ◽  
Xiaozhen Wang
Keyword(s):  
Three Dimensional ◽  
Complete Classification ◽  
Conformally Flat ◽  
Space Forms ◽  
Lorentzian Space ◽  
Lorentzian Metric ◽  
Cone Model ◽  
Shape Operators ◽  
Frenet Curves

A three-dimensional Lorentzian hypersurface [Formula: see text] is called conformally flat if its induced metric is conformal to the flat Lorentzian metric, this property is preserved under the conformal transformation of [Formula: see text]. In this paper, using the projective light-cone model, we give a complete classification of those ones whose shape operators have two distinct real eigenvalues and cannot be diagonalizable. These hypersurfaces are conformal equivalent to cones, cylinders, or rotational hypersurfaces generated by B-scrolls (over null Frenet curves) in three-dimensional Lorentzian space forms.

scholarly journals HYPERSURFACES IN NON-FLAT LORENTZIAN SPACE FORMS SATISFYING Lkψ = Aψ + b

2012 ◽  
Vol 16 (3) ◽  
pp. 1173-1203 ◽  
Author(s):  
Pascual Lucas ◽  
H. Fabian Ramirez-Ospina
Keyword(s):  
Space Forms ◽  
Lorentzian Space ◽  
Lorentzian Space Forms

On the quadric CMC spacelike hypersurfaces in Lorentzian space forms

2016 ◽  
pp. 1-10
Author(s):  
Cícero P. Aquino ◽  
Henrique F. de Lima ◽  
Fábio R. dos Santos
Keyword(s):  
Space Forms ◽  
Lorentzian Space ◽  
Spacelike Hypersurfaces ◽  
Lorentzian Space Forms

RELATIVISTIC PARTICLES ALONG NULL CURVES IN 3D LORENTZIAN SPACE FORMS

2010 ◽  
Vol 20 (09) ◽  
pp. 2851-2859 ◽  
Author(s):  
ÁNGEL GIMÉNEZ
Keyword(s):  
Vector Fields ◽  
Lagrange Equation ◽  
Killing Vector ◽  
Space Forms ◽  
Lorentzian Space ◽  
Motion Equations ◽  
Critical Curves ◽  
Null Curves ◽  
Cartan Curvature ◽  
Relativistic Particles

We study relativistic particles modeled by actions whose Lagrangians are arbitrary functions on the curvature of null paths in (2 + 1)-dimensions backgrounds with constant curvature. We obtain first integrals of the Euler–Lagrange equation by using geometrical methods involving the search for Killing vector fields along critical curves of the action. In the case in which Lagrangian density depends quadratically on Cartan curvature, it is shown that the mechanical system is governed by a stationary Korteweg–De Vries system. Motion equations are completely integrated by quadratures in terms of elliptic and hyperelliptic functions.

Complete classification of parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space

2010 ◽  
Vol 8 (4) ◽  
Author(s):  
Bang-Yen Chen
Keyword(s):  
Space Form ◽  
Complete Classification ◽  
Second Fundamental Form ◽  
Normal Space ◽  
Space Forms ◽  
Parallel Surface ◽  
Parallel Surfaces ◽  
Point To Point ◽  
Lorentz Surface

AbstractA Lorentz surface of an indefinite space form is called a parallel surface if its second fundamental form is parallel with respect to the Van der Waerden-Bortolotti connection. Such surfaces are locally invariant under the reflection with respect to the normal space at each point. Parallel surfaces are important in geometry as well as in general relativity since extrinsic invariants of such surfaces do not change from point to point. Recently, parallel Lorentz surfaces in 4D neutral pseudo Euclidean 4-space $$ \mathbb{E}_2^4 $$ and in neutral pseudo 4-sphere S 24 (1) were classified in [14] and in [10], respectively. In this paper, we completely classify parallel Lorentz surfaces in neutral pseudo hyperbolic 4-space H 24 (−1). Our main result states that there are 53 families of parallel Lorentz surfaces in H 24 (−1). Conversely, every parallel Lorentz surface in H 24 (−1) is obtained from the 53 families. As an immediate by-product, we achieve the complete classification of all parallel Lorentz surfaces in 4D neutral indefinite space forms.

Sign in / Sign up

Export Citation Format

Share Document