arXiv research

In this paper, we introduce a new approach to non-lightlike curve pairs by using integral curves in Minkowski 3-space. We consider direction curve and donor curve to study non-lightlike curve couples such as involute-evolute curves, Mannheim partner curves and Bertrand partner curves. We obtain new methods to construct…

We study the geometry of curves in the Minkowski space and in the de Sitter space, specially at points where the tangent direction is lightlike (i.e. has length zero) called lightlike points of the curve. We define the focal sets of these curves and study the metric structure of them. At the lightlike points, the focal…

Study the geometry of lightlike loci on mixed type surfaces in Lorentz-Minkowski 3-space.

In this paper, we firstly introduce the group of similarity transformations in the Minkowski-3 space. We describe differential- geometric invariants of a non-lightlike curve according to the group of similarity transformations of the Minkowski 3-space. We show extension of fundamental theorem for non-lightlike curves u…

In this paper, the parallel transport frames over non-lightlike curves in Minkowski 3-space are introduced. Evolution equations of these frames with respect to arc length and time are calculated over the space of these curves. Then the equivalence of the non-linear Schrödinger equation and non-linear heat system to the…

In this work we provide necessary and sufficient conditions for the existence of a minimal timelike strip in Lorentz-Minkowski space $\mathbb{R}^4_1$ containing a given lightlike curve and prescribed normal bundle. We also discuss uniqueness of solutions.

Study pp-waves with lightlike parallel spinors in vacuum spacetimes.

Maximal causal curves for Lipschitz metrics are either lightlike or timelike.

Lightlike hypersurfaces in cone structures minimize time.

Survey of Fermat principle in general relativity and beyond.

Characterizes ruled translating solitons in Minkowski 3-space.

In this paper, we introduce the pseudo-torsion functions along spacelike curves whose curvature vector field has isolated lightlike points in Lorentz-Minkowski 3-space, and prove the fundamental theorem. Moreover, we analyze the behavior of the torsion function at such points. As a corollary, we obtain a necessary and …

Motivated by the classical Euler elastic curves, David A. Singer posed in 1999 the problem of determining a plane curve whose curvature is given in terms of its position. We propound the same question in Lorentz-Minkowski plane, focusing on spacelike and timelike curves. In this article, we study those curves in $\math…

Timelike Thomsen surfaces are timelike minimal surfaces that are also affine minimal. In this paper, we make use of both the Lorentz conformal coordinates and the null coordinates, and their respective representation theorems of timelike minimal surfaces, to obtain a complete global classification of these surfaces and…

We introduce a new class of lightlike submanifolds, namely, Screen Transversal Cauchy Riemann (STCR)-lightlike submanifolds, of indefinite Kaehler manifolds. We show that this new class is an umbrella of screen transversal lightlike, screen transversal totally real lightlike and CR-lightlike submanifolds. We give a few…

Lightlike manifolds studied via Cartan geometries in Lorentz-Minkowski spacetime.

Study of spacelike submanifolds with umbilical lightlike normals in Lorentzian spacetimes.

The paper extends curve deformation methods in Minkowski plane.

In our article, we introduce and study lightlike hypersurfaces of a metallic semi-Riemannian manifold. We examine some geometric properties of invariant lightlike hypersurfaces. We show that the induced structure on an invariant lightlike hypersurface is also metallic. We also define screen semi-invariant lightlike hyp…

We study lightlike submanifolds of indefinite statistical manifolds. Contrary to the classical theory of submanifolds of statistical manifolds, lightlike submanifolds of indefinite statistical manifolds need not to be statistical submanifold. Therefore we obtain some conditions for a lightlike submanifold of indefinite…

In this paper we study lightlike surfaces of Minkowski 3- space such that they have degenerate or non-degenerate planar normal sections. We first show that every lightlike surface of Minkowski $3-$ space has degenerate planar normal sections. Then we study lightlike surfaces with non-degenerate planar normal sections a…

In this paper, we give the Cartan's formula for half-lightlike submanifolds of Lorentzian manifolds and use it to show that a screen homothetic half-lightlike submanifolds of a Lorentzian space form, with a conformal co-screen distribution are locally a lightlike triple product manifolds. Then we give a classification …

Study on lightlike geometry in indefinite Sasakian statistical manifolds.

This paper studies lightlike Cartan geometries and their properties.

In this paper, we initiate the study of lightlike hypersurfaces of an $(ε)$-almost paracontact metric manifold which are tangent to the structure vector field. In particular, we give definitions of invariant lightlike hypersurfaces and screen semi-invariant lightlike hypersurfaces, and give some examples. Integrability…

Study on Dirac operators on lightlike hypersurfaces in 4D Lorentzian manifolds.

Study on lightlike submanifolds in statistical manifold geometry.

Lightlike hypersurfaces of a statistical manifold are studied. It is shown that a lightlike hypersurface of a statistical manifold is not a statistical manifold with respect to the induced connections, but the screen distribution has a canonical statistical structure. Some relations between induced geometric objects wi…

Extends Euler's problem to Lorentz-Minkowski plane.

Study on SASI-lightlike submanifolds in indefinite Kaehler manifolds.

The main purpose of the present paper is to study the geometry of screen transversal lightlike submanifolds and radical screen transversal lightlike submanifolds and screen transversal anti-invariant lightlike submanifolds of Golden Semi-Riemannian manifolds. We investigate the geometry of distributions and obtain nece…

The paper studies lightlike submanifolds in bronze semi-Riemannian manifolds with specific geometric properties.

In this work, we investigate the problem of finding surfaces in the Lorentz-Minkowski 3-space with prescribed skew ($S$) and mean ($H$) curvatures, which are defined through the discriminant of the characteristic polynomial of the shape operator and its trace, respectively. After showing that $H$ and $S$ can be interpr…

Study shows lightlike hypersurfaces in indefinite Sasakian manifolds are not symmetric.

In this paper, we introduce and study a new class of CR-lightlike submanifold of an indefinite nearly Sasakian manifold, called Quasi Generalized Cauchy-Riemann (QGCR) lightlike submanifold. We give some characterization theorems for the existence of QGCR-lightlike submanifolds and finally derive necessary and sufficie…

The authors study the geometry of lightlike hypersurfaces on manifolds $(M, c)$ endowed with a pseudoconformal structure $c = CO (n - 1, 1)$ of Lorentzian signature. Such hypersurfaces are of interest in general relativity since they can be models of different types of physical horizons. On a lightlike hypersurface, th…

Null geodesics in Kerr spacetimes cannot be closed or bounded.

Study on CR-lightlike submanifolds in golden semi-Riemannian manifolds.

The concept of quasi generalized CR-lightlike was first introduced by the authors in [18]. In this paper, we focus on ascreen and co-screen quasi generalized CR-lightlike submanifolds of indefinite nearly $μ$-Sasakian manifold. We prove an existence theorem for minimal ascreen quasi generalized CR-lightlike submanifold…

It is proved that the geometry of lightlike hypersurfaces of the de Sitter space S^{n+1}_1 is directly connected with the geometry of hypersurfaces of the conformal space C^n. This connection is applied for a construction of an invariant normalization and an invariant affine connection of lightlike hypersurfaces as wel…

The main purpose of this paper is to give fundamental properties of real lightlike hypersurfaces of paraquaternionic manifolds and to prove the non-existence of real lightlike hypersurfaces in paraquaternionic space forms under some conditions.

A connected regular surface in Lorentz-Minkowski 3-space is called a mixed type surface if the spacelike, timelike and lightlike point sets are all non-empty. Lightlike points on mixed type surfaces may be regarded as singular points of the induced metrics. In this paper, we introduce the L-Gauss map around non-degener…

A mixed type surface is a connected regular surface in a Lorentzian 3-manifold with non-empty spacelike and timelike point sets. The induced metric of a mixed type surface is a signature-changing metric, and their lightlike points may be regarded as singular points of such metrics. In this paper, we investigate the beh…

In this paper, we study Jacobi operators associated to algebraic curvature maps (tensors) on lightlike submanifolds M. We investigate conditions for an induced Rie- mann curvature tensor to be an algebraic curvature tensor on M. We introduce the notion of lightlike Osserman submanifolds and an example of 2-degenerate O…

In this study, we deal with the local structure of curves and surfaces immersed in a pseudo-isotropic space I_{p}^{3} that is a particular Cayley-Klein space. We provide the formulas of curvature, torsion and Frenet trihedron in order for spacelike and timelike curves. The causal character of all admissible surfaces in…

The authors study singular points of lightlike hypersurfaces of the de Sitter space S^{n+1}_1 and the geometry of hypersurfaces and use them for construction of an invariant normalization and an invariant affine connection of lightlike hypersurfaces.

In a metric $g.f.f$-manifold we study lightlike hypersurfaces $M$ tangent to the characteristic vector fields, and owing to the presence of the $f$-structure, we determine some decompositions of $TM$ and of a chosen screen distribution obtaining two distributions invariant with respect to the structure. We discuss the …

In the present paper, we introduce screen transversal lightlike submanifolds of metallic semi-Riemannian manifolds with its subclasses, namely screen transversal anti-invariant, radical screen transversal and isotropic screen transversal lightlike submanifolds, and give an example. We show that there do not exist co-is…