Lorentzian Hypersurfaces Research Articles

A new approach to the study of spacelike submanifolds in a spherical Friedmann–Lemaître–Robertson–Walker spacetime: characterization of the stationary spacelike submanifolds as an application

A natural codimension one isometric embedding of each -dimensional spherical Friedmann–Lemaître–Robertson–Walker (FLRW) spacetime in the -dimensional Lorentz–Minkowski spacetime permits to contemplate as a rotation Lorentzian hypersurface in . After a detailed study of such Lorentzian hypersurfaces, any k-dimensional spacelike submanifold of such an FLRW spacetime can be contemplated as a spacelike submanifold of . Then, we use that situation to study k-dimensional stationary (i.e. of zero mean curvature vector field) spacelike submanifolds of the FLRW spacetime. In particular, we prove a wide extension of the Lorentzian version of the classical Takahashi theorem, giving a characterization of stationary spacelike submanifolds of when contemplating them as spacelike submanifolds of .

An extension of biconservative timelike hypersurfaces in Einstein space

A well-known conjecture of Bang-Yen Chen says that the only biharmonic Euclidean submanifolds are minimal ones, which affirmed by himself for surfaces in 3-dimensional Euclidean space, E³. We consider an extended version of Chen conjecture (namely, Lk-conjecture) on Lorentzian hypersurfaces of the pseudo-Euclidean space E⁴₁ (i.e. the Einstein space). The biconservative submanifolds in the Euclidean spaces are submanifolds with conservative stress-energy with respect to the bienergy functional. In this paper, we consider an extended condition (namely, Lk-biconservativity) on non-degenerate timelike hypersurfaces of the Einstein space E⁴₁ . A Lorentzian hypersurface x : M³₁ → E⁴₁ is called Lk-biconservative if the tangent part of L²k x vanishes identically. We show that Lk-biconservativity of a timelike hypersurface of E⁴₁ (with constant kth mean curvature and some additional conditions) implies that its (k + 1) th mean curvature is constant.

Ricci solitons on pseudo–Riemannian hypersurfaces of 4–dimensional Minkowski space

In this paper, we get classification theorems for Ricci solitons on pseudo–Riemannian hypersurfaces of a Minkowski space E14 by taking the potential vector field as the tangent component of the position vector of the pseudo–Riemannian hypersurfaces. First, we obtain the necessary and sufficient condition for a pseudo–Riemannian hypersurface in E14 which admits such a Ricci soliton. According to the forms of the shape operator of a pseudo–Riemannian hypersurface, we obtain characterization results about Ricci solitons on the pseudo–Riemannian hypersurfaces in E14. More precisely, we show that totally umbilical isoparametric hypersurfaces, hyperbolic and a pseudo–spherical cylinder in E14 admit shrinking Ricci solitons whose potential vector fields are the tangent part of the position vectors in E14. Furthermore, we also get that a generalized umbilical Lorentzian hypersurface in E14 admits a shrinking Ricci soliton. Finally, we obtain that there do not exist Ricci solitons on the Lorentzian hypersurfaces in E14 whose minimal polynomials have complex roots or real roots with multiplicity three.

Conformally flat Lorentzian hypersurfaces in Lorentzian 4-space with special shape operator

In the conformal (Möbius) geometry of submanifolds, using algebraic invariants of the shape operator to construct conformal invariants is a frequently used method. But it does not apply to [Formula: see text]-dim Lorentzian hypersurfaces of the last type, the minimal polynomial of whose shape operator has a triple root. In this paper, using the obstruction of some distribution to be integrable, a new method to construct conformal invariants is introduced. Using this method, a complete conformal invariant system is constructed for generic conformally flat Lorentzian hypersurfaces of the last type. We find that such kind of hypersurfaces allows an infinite parameter family of non-equivalent deformations, which implies they are more abundant than the other three types. For non-generic conformally flat Lorentzian hypersurfaces of the last type, the isometric geometry is studied and a fundamental theorem is obtained in this paper.

On $L_1$-biharmonic timelike hypersurfaces in pseudo-Euclidean space $E_1^4$

A well-known conjecture of Bang Yen-Chen says that the only biharmonic Euclidean submanifolds are minimal ones. In this paper, we consider an extended condition (namely, $L_1$-biharmonicity) on non-degenerate timelike hypersurfaces of the pseudo-Euclidean space $E_1^4$. A Lorentzian hypersurface $x: M_1^3\rightarrow\E_1^4$ is called $L_1$-biharmonic if it satisfies the condition $L_1^2x=0$, where $L_1$ is the linearized operator associated to the first variation of 2-th mean curvature vector field on $M_1^3$. According to the multiplicities of principal curvatures, the $L_1$-extension of Chen's conjecture is affirmed for Lorentzian hypersurfaces with constant ordinary mean curvature in pseudo-Euclidean space $E_1^4$. Additionally, we show that there is no proper $L_1$-biharmonic $L_1$-finite type connected orientable Lorentzian hypersurface in $E_1^4$.

Pseudo‐hyperbolic Gauss maps of Lorentzian surfaces in anti‐de Sitter space

AbstractIn this paper, we determine the type numbers of the pseudo‐hyperbolic Gauss maps of all oriented Lorentzian surfaces of constant mean and Gaussian curvatures and non‐diagonalizable shape operator in the 3‐dimensional anti‐de Sitter space. Also, we investigate the behavior of type numbers of the pseudo‐hyperbolic Gauss map along the parallel family of such oriented Lorentzian surfaces in the 3‐dimensional anti‐de Sitter space. Furthermore, we investigate the type number of the pseudo‐hyperbolic Gauss map of one of Lorentzian hypersurfaces of B‐scroll type in a general dimensional anti‐de Sitter space.

Conformally flat Lorentzian hypersurfaces inR14with a pair of complex conjugate principal curvatures

Abstract A three dimensional Lorentzian hypersurface x : M 1 3 → R 1 4 is called conformally flat if its induced metric is conformal to the flat Lorentzian metric, this property is preserved under the conformal transformation of R 1 4 . Using the projective light-cone model, for those ones whose shape operators have a pair of complex conjugate eigenvalues, we study the integrability condition by constructing a scalar conformal invariant and a canonical moving frame in this paper. It follows that these hypersurfaces can be determined by the solutions to a system of three-order partial differential equations.

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.

Conformally flat Lorentzian hypersurfaces in ℝ14 with three distinct principal curvatures

A three dimensional Lorentzian hypersurface x: M 1 3 → ℝ 1 4 is called conformally flat if its induced metric is conformal to the flat Lorentzian metric, and this property is preserved under the conformal transformation of ℝ 1 4 . Using the projective light-cone model, for those whose shape operators have three distinct real eigenvalues, we calculate the integrability conditions by constructing a scalar conformal invariant and a canonical moving frame in this paper. Similar to the Riemannian case, these hypersurfaces can be determined by the solutions to some system of partial differential equations.

Lorentzian CR structures and nonembeddability

In this paper we construct examples of CR deformations of Lorentzian hypersurfaces which are CR embeddable at all points outside an arbitrarily small compact set whose interior contains a point where CR embeddablity is not possible.

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.

Some Classifications of Biharmonic Lorentzian Hypersurfaces in Minkowski 5-Space

In this paper, we study Lorentzian hypersurfaces in Minkowski 5-space with non-diagonalizable shape operator whose characteristic polynomial is (t − k 1)2(t − k 3)(t − k 4) or (t − k 1)3(t − k 4). We prove that in these cases, a hypersurface is biharmonic if and only if it is minimal.

Singularity analysis of Lorentzian hypersurfaces on pseudo n‐spheres

This paper considers the one parameter families of extrinsic differential geometries of Lorentzian hypersurfaces on pseudo n‐spheres. We give a continuous relationship among the three types Gauss indicatrices by one parameter map. Meanwhile, we also give the singularity analysis of the one parameter Gauss indicatrices of Lorentzian hypersurfaces on pseudo n‐spheres by the Legendrian singularity theory. Copyright © 2014 John Wiley & Sons, Ltd.

Semi-symmetric Lorentzian hypersurfaces in Lorentzian space forms

In the present paper, we shall investigate and classify Lorentzian hypersurfaces in Lorentzian space forms satisfying some curvature conditions. We shall focus especially on semi-symmetric Lorentzian hypersurfaces. Those of constant curvature and those so-called “good” are explicitly described and classified. We shall also classify Einstein–Lorentzian hypersurfaces.

Families of Gauss indicatrices on Lorentzian hypersurfaces in pseudo-spheres in semi-Euclidean 4-space

We consider the one-parameter families of Gauss indicatrices on Lorentzian hypersurfaces in pseudo-spheres in semi-Euclidean 4-space with index 2 and give the types of singularities of the Lorentzian hypersurfaces by the contact theory.

ON GEODESIC COMPLETENESS OF NONDEGENERATE SUBMANIFOLDS IN SEMI-EUCLIDEAN SPACES

In this paper, we study the geodesic completeness of nondegenerate submanifolds in semi-Euclidean spaces by extending the study of Beem and Ehrlich [1] to semi-Euclidean spaces. From the physical point of view, this extend may have a significance that a semi-Euclidean space contains more variety of Lorentzian submanifolds rather than those of Lorentzian hypersurfaces in a Minkowski space as in [1]. From mathematical point of view, since there is no distinction in the analysis of geodesic completeness of Lorentzian submanifolds and nondegenerate submanifolds in a semi-Euclidean space, we treat the mathematically more general case of nondegenerate submanifolds in a semi-Euclidean space. The new ideas leading to this generalization are the sufficient conditions for algorithms in the proofs of the results in [1]. Indeed these sufficient conditions for the algorithms also work well for the nondegenerate submanifolds in a semi-Euclidean space.

Conformally flat Lorentzian hypersurfaces in the conformal compactification of Lorentz space

We study conformally flat Lorentzian hypersurfaces in the conformal compactification of Lorentz space R 1 n + 1 , which is the projectivized light cone R ̂ 1 n + 1 ⊂ R P n + 2 induced from R 2 n + 3 . We establish a Lorentzian version of the local classification theorem of Cartan, in terms of branched channel hypersurfaces for n ≥ 4 , and for n = 3 , in terms of the conformal fundamental forms. For hypersurfaces whose shape operator has complex eigenvalues, we give a necessary condition for being conformally flat in terms of local integrability of distributions.

Conformally flat Lorentzian hypersurfaces and curved flats

Conformally flat Lorentzian hypersurfaces and curved flats

Conformally flat Lorentzian hypersurfaces and curved flats

Conformally flat Lorentzian hypersurfaces and curved flats

Pseudo-isotropic Lorentzian hypersurfaces in Minkowski space

We show that a Lorentzian hypersurface M n of the Minkowski space R n+1 1 is pseudoisotropic if and only if M n is flat and minimal. Next we obtain a classification of all pseudo-isotropic Lorentzian hypersurfaces in R 3 1 and R 4 1.