Lorentzian Space Forms Research Articles

Conformally flat Willmore spacelike hypersurfaces inRn+1

In this paper, we give the equation satisfied by umbilics-free Willmore spacelike hypersurfaces using the conformal invariants in Lorentzian space forms. At the same time, we give the equation satisfied by hyperelastic spacelike curves in $2$-dimensional Lorentzian space forms and classify the closed hyperelastic spacelike curves. Finally conformally flat Willmore spacelike hypersurfaces are classified in terms of the hyperelastic spacelike curves in $2$-dimensional Lorentzian space forms.

Some Properties Curvture of Lorentzian Kenmotsu Manifolds

Abstract In this paper different curvature tensors on Lorentzian Kenmotsu manifod are studied. We investigate constant ϕ–holomorphic sectional curvature and ℒ-sectional curvature of Lorentzian Kenmotsu manifolds, obtaining conditions for them to be constant of Lorentzian Kenmotsu manifolds in such condition. We calculate the Ricci tensor and scalar curvature for all the cases. Moreover we investigate some properties of semi invariant submanifolds of a Lorentzian Kenmotsu space form. We show that if a semi-invariant submanifold of a Lorentzian Kenmotsu space form M is totally geodesic, then M is an η−Einstein manifold. We consider sectional curvature of semi invariant product of a Lorentzian Kenmotsu manifolds.

Two series of polyhedral fundamental domains for Lorentz bi-quotients

The main aim of this paper is to give two infinite series of examples of Lorentz space forms that can be obtained from Lorentz polyhedra by identification of faces. These Lorentz space forms are bi-quotients of G=SU(1,1)˜≅SL(2,R)˜ of the form Γ1\\G/Γ2, where Γ1 and Γ2 are discrete subgroups of G and the group Γ2 is cyclic. The simply connected Lie group G is equipped with the Lorentz metric given by the Killing form. A construction of polyhedral fundamental domains for the action of Γ1×Γ2 on G via (g,h)⋅x=gxh−1 was given in the earlier work of the second author. In this paper we give an explicit description of the fundamental domains obtained by this construction for two infinite series of groups. These results are connected to singularity theory as the bi-quotients Γ1\\G/Γ2 appear as links of certain quasi-homogeneous Q-Gorenstein surface singularities, i.e. the intersections of the singular variety with sufficiently small spheres around the isolated singular point.

Wavefronts of traveling trajectories of geometric particles

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.

Spacelike Möbius Hypersurfaces in Four Dimensional Lorentzian Space Form

In this paper, we first set up an alternative fundamental theory of Mobius geometry for any umbilic-free spacelike hypersurfaces in four dimensional Lorentzian space form, and prove the hypersurfaces can be determined completely by a system consisting of a function W and a tangent frame {Ei}. Then we give a complete classification for spacelike Mobius homogeneous hypersurfaces in four dimensional Lorentzian space form. They are either Mobius equivalent to spacelike Dupin hypersurfaces or to some cylinders constructed from logarithmic curves and hyperbolic logarithmic spirals. Some of them have parallel para-Blaschke tensors with non-vanishing Mobius form.

On the umbilicity of generalized linear Weingarten spacelike hypersurfaces in a Lorentzian space form

In this paper, we deal with complete generalized linear Weingarten spacelike hypersurfaces immersed in a Lorentzian space form, which means that there exists a linear relation involving some of the higher order mean curvatures. In this setting, we show that such a spacelike hypersurface must be totally umbilical, provided that its Gauss mapping has some appropriate behavior and that the tangential component of a fixed nonzero vector has integrable norm along it. Our approach is based on a suitable divergence formula jointly with an appropriate maximum principle at the infinity.

Null Screen Isoparametric Hypersurfaces in Lorentzian Space Forms

In this paper, we develop the notion of screen isoparametric hypersurface for null hypersurfaces of Robertson–Walker spacetimes. Using this formalism we derive Cartan identities for the screen principal curvatures of null screen isoparametric hypersurfaces in Lorentzian space forms and provide a local characterization of such hypersurfaces.

Classification of $$\eta $$ η -Biharmonic Surfaces in Non-flat Lorentz Space Forms

In this paper, we prove that $$\eta $$ -biharmonic surfaces in non-flat three-dimensional Lorentz space forms are isoparametric and give full classification results.

On the classification of contact metric (k,μ)-spaces via tangent hyperquadric bundles

We classify locally the contact metric -spaces whose Boeckx invariant is as tangent hyperquadric bundles of Lorentzian space forms.

Spacelike Dupin hypersurfaces in Lorentzian space forms

Similar to the definition in Riemannian space forms, we define the spacelike Dupin hypersurface in Lorentzian space forms. As conformal invariant objects, spacelike Dupin hypersurfaces are studied in this paper using the framework of the conformal geometry of spacelike hypersurfaces. Further we classify the spacelike Dupin hypersurfaces with constant Mobius curvatures, which are also called conformal isoparametric hypersurface.

Linear Weingarten spacelike hypersurfaces in Lorentz space forms with prescribed Gauss map

This paper address the geometry of complete linear Weingarten spacelike hypersurfaces in the Lorentz space forms. First, a divergence lemma concerning linear Weingarten spacelike hypersurfaces is obtained. Then, with the aid of this lemma, by supposing suitable restrictions on the Gauss map, we show that such hypersurfaces must be totally umbilical, which are some extension of the recent results of Aquino, Bezerra and Lima [7] and Aquino, Lima and Velasquez [11].

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

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.

On Non-degenerate Null Normal Sections of Codimension Two Spacelike Surfaces

In this paper, we develop a formula for spacelike surfaces in a four-dimensional Lorentzian space form which involves its mean curvature vector field, the Gauss curvature of the induced metric and the Gauss curvature of the second fundamental form associated to a non-degenerate null normal section. By means of this formula, we establish several sufficient conditions for a compact spacelike surface in a four-dimensional Lorentzian space form which has a null umbilical normal direction. As another application, we give a new proof of Liebmann rigidity theorems in Euclidean, hemispherical, hyperbolic spaces and in the De Sitter spacetime.

Characterizations of complete CMC spacelike hypersurfaces satisfying an Okumura type inequality

We deal with complete constant mean curvature spacelike hypersurfaces immersed in a Lorentzian space form and satisfying a suitable Okumura type inequality, which is a weaker hypothesis than to assume that the spacelike hypersurface has two distinct principal curvatures. In this setting, we obtain estimates for the norm of the traceless part of the second fundamental form and we show that these estimates are sharp in the sense that the hyperbolic cylinders realize them.

The DPW method for constant mean curvature surfaces in 3-dimensional Lorentzian spaceforms, with applications to Smyth type surfaces

We give criteria for singularities of spacelike constant mean curvature surfaces in 3-dimensional de Sitter and anti-de Sitter spaces constructed by the DPW method, which is a generalized Weierstrass representation. We also construct some examples of spacelike CMC surfaces, including analogs of Smyth surfaces with singularities, using appropriate models to visualize them.

Integrability aspects of the vortex filament equation for pseudo-null curves

An algebraic background in order to study the integrability properties of pseudo-null curve motions in a three-dimensional Lorentzian space form is developed. As an application, we delve into the relationship between the Burgers’ equation and the pseudo-null vortex filament equation. A recursion operator for the pseudo-null vortex filament equation is also provided.

Classification of product shaped hypersurfaces in Lorentz space forms

We define the product shaped hypersurfaces in Lorentz space forms by imposing the shape operator to be product type. Based on the classification of the isoparametric hypersurfaces, we obtain the whole families of the product shaped hypersurfaces in Minkowski, de Sitter and anti-de Sitter spaces.

Willmore spacelike submanifolds in an indefinite space form Nn+p q(c)

Let Nn+p q(c) be an (n+p)-dimensional connected indefinite space form of index q(1 ? q ? p) and of constant curvature c. Denote by ? : M ? Nn+p q (c) the n-dimensional spacelike submanifold in Nn+p q (c), ? : M ? Nn+p q(c) is called a Willmore spacelike submanifold in Nn+p q(c) if it is a critical submanifold to the Willmore functional W(?) = ?q M ?n dv =?M (S-nH2)n/2 dv, where S and H denote the norm square of the second fundamental form and the mean curvature of M and ?2 = S ? nH2. If q = p, in [14], we proved some integral inequalities of Simons? type and rigidity theorems for n-dimensional Willmore spacelike submanifolds in a Lorentzian space form Nn+p q(c). In this paper, we continue to study this topic and prove some integral inequalities of Simons? type and rigidity theorems for n-dimensional Willmore spacelike submanifolds in an indefinite space form Nn+p q(c) (1 ? q ? p).

On the equivalence of two curvature conditions for Lorentzian hypersurfaces

Let n≥3. We show that semi-symmetry and Ricci-semisymmetry conditions are equivalent for any n-dimensional Lorentzian hypersurface in a Lorentzian space form with nonzero curvature. We also show that these curvature conditions are equivalent for any n-dimensional Lorentzian isoparametric hypersurface in Minkowski space R1n+1, and we construct an example of a Ricci-semisymmetric 5-dimensional Lorentzian hypersurface in R16 which is not semi-symmetric.

On the geometry of trapped and marginally trapped submanifolds in Lorentzian space forms

In this paper, our aim is to study the geometry of [Formula: see text]-dimensional trapped and marginally trapped submanifolds immersed in a Lorentzian space form [Formula: see text] of constant sectional curvature [Formula: see text]. In this setting, we establish sufficient conditions to guarantee that a complete trapped submanifold with parallel mean curvature vector in [Formula: see text] must be pseudo-umbilical. Afterwards, we obtain a nonexistence result concerning complete trapped submanifolds in the Lorentz–Minkowski space. Furthermore, under suitable constraints on the Ricci curvature and the second fundamental form, we show that an [Formula: see text]-dimensional complete pseudo-umbilical marginally trapped submanifold of [Formula: see text] with parallel mean curvature vector is, in fact, totally umbilical.