arXiv research

Similar to the definition of Dupin hypersurface 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 conformal geometry. Further we classify the spacelike Dupin hype…

Study proves spacelike self-shrinkers are hyperplanes under certain conditions.

In the present note, first we derive an intrinsic inequality for Pseudo-umbilical spacelike submanifold in an indefinite space form. We use this inequality to show that such submanifold is totally geodesic. In the rest part of this paper, using a result of Aiyama [1], we prove that Pseudo-umbilical spacelike subamnifol…

Spacelike Willmore surfaces in 4-dimensional Lorentzian space forms, a topic in Lorentzian conformal geometry which parallels the theory of Willmore surfaces in $S^4$, are studied in this paper. We define two kinds of transforms for such a surface, which produce the so-called left/right polar surfaces and the adjoint s…

Let $M^n$ be an $n$-dimensional umbilic-free hypersurface in the $(n+1)$-dimensional Lorentzian space form $M^{n+1}_1(c)$. Three basic invariants of $M^n$ under the conformal transformation group of $M^{n+1}_1(c)$ are a $1$-form $C$, called conformal $1$-form, a symmetric $(0,2)$ tensor $B$, called conformal second fun…

Formula for spacelike submanifolds in warped products.

The conformal geometry of spacelike surfaces in 4-dimensional Lorentzian space forms has been studied by the authors in a previous paper, where the so-called polar transform was introduced. Here it is shown that this transform preserves spacelike conformal isothermic surfaces. We relate this new transform with the know…

We first describe the numerical invariants attached to the second fundamental form of a spacelike surface in four-dimensional Minkowski space. We then study the configuration of the nu-principal curvature lines on a spacelike surface, when the normal field nu is lightlike (the lightcone configuration). Some observation…

Paper proves rigidity for certain spacelike hypersurfaces in de Sitter space.

We present some examples of curvature homogeneous pseudo-Riemannian manifolds which are k-spacelike Jordan Stanilov; their higher order curvature operator has constant Jordan normal form on the Grassmannian of unoriented k-dimensional spacelike subspaces of the tangent plane.

A spacelike surface in four-dimensional Lorentz-Minkowski spacetime through the lightcone has a meaningful lightlike normal vector field $η$. Several sufficient assumptions on such a surface with non-degenerate $η$-second fundamental form are established to prove that it must be a totally umbilical round sphere. With t…

The paper extends Bernstein Theorem for minimal spacelike surfaces in 4D Minkowski space.

Anti-de Sitter space is the Lorentzian space form with negative curvature. In this paper we consider lightlike hypersurfaces along spacelike submanifolds in anti-de Sitter space with general codimension. In particular, we investigate the singularities of lightlike hypersurfaces as an application of the theory of Legend…

In this paper, we get the time evolution equations of the curvature and torsion of the evolving spacelike curves in the Minkowski space. Also, we give inextensible evolutions of timelike ruled surfaces that are produced by the timelike normal and spacelike binormal vector fields of spacelike curve and derive the necess…

A pseudo-Riemannian manifold is said to be spacelike Jordan IP if the Jordan normal form of the skew-symmetric curvature operator depends upon the point of the manifold, but not upon the particular spacelike 2-plane in the tangent bundle at that point. We use methods of algebraic topology to classify connected spacelik…

Study null surfaces of pseudo-spherical curves in anti-de Sitter space.

The paper decomposes spacelike hypersurface properties for general relativistic vacuum equations.

A world sheet in anti-de Sitter space is a timelike submanifold consisting of a one-parameter family of spacelike submanifolds. We consider the family of lightlike hypersurfaces along spacelike submanifolds in the world sheet. The locus of the singularities of lightlike hypersurfaces along spacelike submanifolds forms …

In this paper we study weakly irreducible holonomy representations of the normal connection of a spacelike submanifold in a pseudo-Riemannian space from. We associate screen representations to weakly irreducible normal holonomy groups and classify the screen representations having the Borel-Lichnérowicz property. In pa…

Study proves behaviors of CMC surfaces near future null-infinity in Schwarzschild spacetime.

Study Lorentz harmonic maps and spacelike surfaces in anti-de Sitter space.

The paper calculates curvatures in a specific type of warped product space.

Study rigidity of spacelike LW-submanifolds in locally symmetric semi-Riemannian spaces.

In this paper, we generalize the polar transforms of spacelike isothermic surfaces in $Q^4_1$ to n-dimensional pseudo-Riemannian space forms $Q^n_r$. We show that there exist $c-$polar spacelike isothermic surfaces derived from a spacelike isothermic surface in $Q^n_r$, which are into $S^{n+1}_r(c)$, $H^{n+1}_{r-1}(c)$…

Study of spacelike hypersurfaces in twisted product spacetimes with specific conditions.

Study of spacelike singularities in spherical spacetimes with scalar matter.

The classical Minkowski formula is extended to spacelike codimension-two submanifolds in spacetimes which admit "hidden symmetry" from conformal Killing-Yano two-forms. As an application, we obtain an Alexandrov type theorem for spacelike codimension-two submanifolds in a static spherically symmetric spacetime: a codim…

The Ahlfors Laplacian is applied to solve geometric and relativistic problems.

We study the existence of a non-spacelike isometry, ζ, in higher dimensional Kundt spacetimes with constant scalar curvature invariants (CSI). We present the particular forms for the null or timelike Killing vectors and a set of constraints for the metric functions in each case. Within the class of N dimensional CSI Ku…

We investigate the twistor space and the Grassmannian fibre bundle of a Lorentzian 4-space with natural almost optical structures and its induced CR-structures. The twistor spaces of the Lorentzian space forms $\R^4_1, \Di{S}^4_1$ and $\Di{H}^4_1$ are explicitly discussed. The given twistor construction is applied to s…

Develops weak formulation for spacelike flows in pseudo-Euclidean space.

In the Friedmann Model of the universe, cosmologists assume that spacelike slices of the universe are Riemannian manifolds of constant sectional curvature. This assumption is justified via Schur's Theorem by stating that the spacelike universe is locally isotropic. Here we define a Riemannian manifold as almost locally…

A normal field on a spacelike surface in $R^4_1$ is called bi-normal if $K^ν$, the determinant of Weingarten map associated with $ν$, is zero. In this paper we give a relationship between the spacelike pseudo-planar surfaces and spacelike pseudo-umbilical surfaces, then study the bi-normal fields on spacelike ruled sur…

We give local, explicit representation formulas for n-dimensional spacelike submanifolds which are marginally trapped in the Minkowski space, the de Sitter and anti de Sitter spaces and the Lorentzian products of the sphere and the hyperbolic space by the real line.

We show that isothermic surfaces and S-Willmore surfaces are also the solutions to the corresponding Blaschke's problem for both spacelike and timelike surfaces in pseudo-Riemannian space forms. For timelike surfaces both Willmore and isothermic, we obtain a description by minimal surfaces similar to the classical resu…

In this paper, we define dual geodesic trihedron(dual Darboux frame) of a spacelike ruled surface. Then, we study Mannheim offsets of spacelike ruled surfaces in dual Lorentzian space by considering the E. Study Mapping. We represent spacelike ruled surfaces by dual Lorentzian unit spherical curves and define Mannheim …

Study spacelike loxodromes on helicoidal surfaces in Lorentzian n-space.

The paper studies mean curvature flow of spacelike-convex submanifolds in pseudo-Euclidean space.

In this paper, we give the characterizations of spacelike B2-slant helix by means of curvatures of the spacelike curve in Minkowski 4-space. Furthermore, we give the integral characterization of the spacelike B2-slant helix.

Three explicit families of spacelike Zoll surface admitting a Killing field are provided. It allows to prove the existence of spacelike Zoll surface not smoothly conformal to a cover of de Sitter space as well as the existence of Lorentzian Möbius strips of non constant curvature all of whose spacelike geodesics are cl…

In this paper, we investigate the parametric version and non-parametric version of rigidity theorem of spacelike translating solitons in pseudo-Euclidean space $\mathbb{R}^{m+n}_{n}$. Firstly, we classify $m$-dimensional complete spacelike translating solitons in $\mathbb{R}^{m+n}_{n}$ by affine technique and classical…

In this paper, we study entire spacelike translating solitons in Minkowski space. By constructing convex spacelike solutions to (1.3) in bounded convex domains, we obtain many entire smooth convex strictly spacelike translating solitons by prescribing boundary data at infinity.

In this paper, using the classifications of timelike and spacelike ruled surfaces, we define and study the Mannheim offsets of spacelike ruled surfaces in Minkowski 3-space. We give the conditions for spacelike offset surfaces to be developable.

We first define a complex angle between two oriented spacelike planes in 4-dimensional Minkowski space, and then study the constant angle surfaces in that space, i.e. the oriented spacelike surfaces whose tangent planes form a constant complex angle with respect to a fixed spacelike plane. This notion is the natural Lo…

Estimates for spacelike hypersurfaces in de Sitter space.

In this paper, we investigate the relations between the pitch, the angle of pitch and drall of parallel ruled surface of a closed spacelike curve with a spacelike binormal in dual Lorentzian space.

Proves inequality for maximal spacelike submanifolds in Minkowski space.

We consider spacelike graphs $Γ_f$ of simple products $(M\times N, g\times -h)$ where $(M,g)$ and $(N,h)$ are Riemannian manifolds and $f:M\to N$ is a smooth map. Under the condition of the Cheeger constant of $M$ to be zero and some condition on the second fundamental form at infinity, we conclude that if $Γ_f \subset…