Study the geometry of lightlike loci on mixed type surfaces in Lorentz-Minkowski 3-space.
problem Characterize the differential geometric properties of lightlike loci on mixed type surfaces.
method Define a frame field and lightlike ruled surfaces along the lightlike locus, analyze their singularities and intersections.
result Establish a relationship between the singularities of lightlike ruled surfaces and the differential geometric properties of the lightlike locus.
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…
The paper explores reflection principles for lightlike line segments on maximal surfaces.
problem Reflection property does not hold for lightlike line segments on maximal surfaces.
method Analyzes reflection properties for lightlike line segments connecting shrinking singularities.
result Shows a kind of reflection principle for lightlike line segments on maximal surfaces.
The paper studies deformations of mixed type surfaces in Lorentz-Minkowski space.
problem Deformations of mixed type surfaces with singular points.
method Introduced L-Gauss map around non-degenerate lightlike points to prove fundamental theorem of surface theory.
result Real analytic mixed type surfaces admit non-trivial isometric deformations around generic lightlike points.
Study on bounded Gaussian curvature in mixed type surfaces of Lorentzian manifolds.
problem Characterizing the behavior of Gaussian curvature at non-degenerate lightlike points of mixed type surfaces.
method Introducing invariants and using Gauss-Bonnet type formula results.
result Gauss-Bonnet type formula for mixed type surfaces with bounded Gaussian curvature.
We investigate the geometric properties of lightlike surfaces in the Minkowski space R2,1, using Cartan's method of moving frames to compute a complete set of local invariants for such surfaces. Using these invariants, we give a complete local classification of lightlike surfaces of constant type in R2,1 an…
The paper studies helicoidal surfaces of non-lightlike frontals in Lorentz-Minkowski 3-space.
problem Investigating the properties and singularities of helicoidal surfaces in Lorentz-Minkowski space.
method Defining and analyzing two types of helicoidal surfaces, using diffeomorphic transformations and criteria for cusps and cuspidal edges.
result Identification theorems for the singular types of both 1-type and 2-type helicoidal surfaces.
The paper classifies pseudo-isotropic and lightlike pseudo-isotropic Lagrangian surfaces.
problem Characterizing Lagrangian surfaces in complex pseudo spaces.
method Analyzing properties of Lagrangian surfaces in complex pseudo spaces.
result Classification of Lagrangian surfaces based on pseudo-isotropy.
Paper classifies ruled surfaces in Lorentz-Minkowski space for a specific flow.
problem Classifying ruled surfaces in Lorentz-Minkowski space.
method Examining homothetic self-similar solutions of the inverse mean curvature flow.
result Existence of two classes of non-cylindrical homothetic solitons.
Study magnetic geodesics on surfaces avoiding lightlike points.
problem Understanding geodesics on surfaces with singularities.
method Analytical proof and numerical solutions.
result Magnetic geodesics avoid points with lightlike tangent planes.
The lightlike geometry of codimension two spacelike submanifolds in Lorentz-Minkowski space has been developed in [Izumiya, S. and Romero Fuster, M. C. Selecta Mathematica (NS), 13 23--55 (2007)] which is a natural Lorentzian analogue of the classical Euclidean differential geometry of hypersurfaces. In this paper we i…
Paper classifies timelike Thomsen surfaces and reveals their geometric properties.
problem Classifying timelike Thomsen surfaces and understanding their geometric properties.
method Use of Lorentz conformal and null coordinates, representation theorems, and geometric invariants.
result Revealed the relationship between timelike Thomsen surfaces and timelike minimal surfaces with planar curvature lines.
Paper connects minimal and maximal surfaces' boundary value problems.
problem Existence and uniqueness of minimal surfaces' boundary value problems.
method One-to-one correspondence between minimal and maximal surfaces' solutions.
result One-to-one correspondence between minimal and maximal surfaces' boundary value problems.
A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. We introduce meridian surfaces of parabolic type as one-parameter systems of meridians of a rotational hypersurface with lightlike axis in Minkowski 4-space and find their …
Characterizes ruled translating solitons in Minkowski 3-space.
problem Classifying translating solitons in Minkowski 3-space.
method Analyzing vector fields and curves in Minkowski space.
result Found non-cylindrical ruled translating solitons.
The study classifies special Lorentz surfaces in a 4-space with neutral metric.
problem Classifying meridian surfaces in a pseudo-Euclidean 4-space.
method Constructing and classifying meridian surfaces with specific properties.
result There exist meridian surfaces with parallel normalized mean curvature vector field but not parallel mean curvature vector.
The family of Willmore immersions from a Riemann surface into Sn+2 can be divided naturally into the subfamily of Willmore surfaces conformally equivalent to a minimal surface in Rn+2 and those which are not conformally equivalent to a minimal surface in Rn+2. On the level of their conformal Gauss maps…
Paper proves rigidity of de-Sitter tori with conical singularities.
problem Global rigidity of de-Sitter tori with singularities.
method Introduced constant curvature Lorentzian surfaces with conical singularities and proved rigidity via topological dynamics.
result De-Sitter tori with a single singularity are determined by their lightlike bi-foliation.
In this paper we deal with the uniqueness of the Lorentzian helicoid and Enneper's surface among properly embedded maximal surfaces with lightlike boundary of mirror symmetry in the Lorentz-Minkowski space L3.
Introduces new lightlike submanifolds in indefinite Kaehler manifolds.
problem Characterizing and understanding new types of lightlike submanifolds.
method Definition and analysis of Screen Transversal Cauchy Riemann (STCR) lightlike submanifolds.
result Characterizes STCR lightlike submanifolds as umbrella of other types.
Lightlike manifolds studied via Cartan geometries in Lorentz-Minkowski spacetime.
problem Characterizing and constructing lightlike metrics and hypersurfaces.
method Introducing lightlike Cartan geometries and constructing ambient Lorentzian manifolds.
result Every lightlike Cartan geometry on a manifold provides a lightlike metric with a globally spanned radical distribution.
In this paper we construct an example of a weakly complete maximal surface in the Lorentz-Minkowski space L^3, which is bounded by a hyperboloid. Moreover, all the singularities of our example are of lightlike type.
Proves compact Cauchy horizons have constant surface gravity under null energy condition.
problem Proving compact Cauchy horizons have constant surface gravity.
method Combines ergodic theory, Hodge theory, and Riemannian flow theory.
result Compact Cauchy horizons admit a smooth lightlike tangent vector field of constant surface gravity.
The paper studies half-lightlike submanifolds in Lorentzian manifolds with specific distributions.
problem Characterizing half-lightlike submanifolds in Lorentzian manifolds with conformal co-screen distributions.
method Using Cartan's formula, the authors classify half-lightlike submanifolds of Lorentzian space forms with constant screen principal curvatures.
result Screen homothetic half-lightlike submanifolds of a Lorentzian space form with a conformal co-screen distribution are locally lightlike triple product manifolds.
Study on lightlike hypersurfaces in metallic semi-Riemannian manifolds.
problem Exploring geometric properties of lightlike hypersurfaces in metallic semi-Riemannian manifolds.
method Investigation of invariant and screen semi-invariant lightlike hypersurfaces, examination of integrability conditions.
result Induced structure on invariant lightlike hypersurfaces is metallic.
The study proves that certain spacelike surfaces in 4D Lorentz-Minkowski space are round spheres.
problem Characterizing compact spacelike surfaces with a non-degenerate lightlike normal direction.
method Developed a new formula relating Gauss curvatures and characterized totally umbilical round spheres.
result Totally umbilical round spheres are the only compact spacelike surfaces with non-degenerate η-second fundamental form and constant Gauss curvature two. The geometry and topology of complete nonorientable maximal surfaces with lightlike singularities in the Lorentz-Minkowski 3-space are studied. Some topological congruence formulae for surfaces of this kind are obtained. As a consequence, some existence and uniqueness results for maximal Moebius strips and maximal Klei…
In the four-dimensional pseudo-Euclidean space with neutral metric there are three types of rotational surfaces with two-dimensional axis - rotational surfaces of elliptic, hyperbolic or parabolic type. A surface whose mean curvature vector field is lightlike is said to be quasi-minimal. In this paper we classify all q…
Study lightlike submanifolds in indefinite statistical manifolds, finding conditions and curvature expressions.
problem Characterize lightlike submanifolds in indefinite statistical manifolds.
method Analyze conditions for lightlike submanifolds to be lightlike statistical submanifolds, derive statistical sectional curvature, and investigate induced statistical Ricci tensor symmetry.
result Conditions for lightlike submanifolds to be lightlike statistical submanifolds and expressions for statistical sectional curvature and induced Ricci tensor symmetry.
A Lorentz surface in the four-dimensional pseudo-Euclidean space with neutral metric is called quasi-minimal if its mean curvature vector is lightlike at each point. In the present paper we obtain the complete classification of quasi-minimal Lorentz surfaces with pointwise 1-type Gauss map.
Study of spacelike submanifolds with umbilical lightlike normals in Lorentzian spacetimes.
problem Geometric and topological constraints on codimension-two spacelike submanifolds.
method Analysis of submanifolds with umbilical lightlike normal directions, using geometric and topological constraints.
result Any such submanifold is contained in a lightlike hypersurface, which is totally umbilical if the lightlike normal direction is umbilical.
Lightlike hypersurfaces in statistical manifolds have unique geometric properties.
problem Characterizing lightlike hypersurfaces in statistical manifolds.
method Analyzing geometric properties and induced structures of lightlike hypersurfaces.
result Lightlike hypersurfaces are not statistical manifolds but have a canonical screen distribution.
Study new CR-lightlike submanifolds in indefinite nearly Sasakian manifolds.
problem Characterizing new CR-lightlike submanifolds in indefinite nearly Sasakian manifolds.
method Introduce and study Quasi Generalized Cauchy-Riemann (QGCR) lightlike submanifolds.
result Derive necessary and sufficient conditions for integrability of distributions.
A marginally trapped surface in the four-dimensional Minkowski space is a spacelike surface whose mean curvature vector is lightlike at each point. In the present paper we find all marginally trapped surfaces with pointwise 1-type Gauss map. We prove that a marginally trapped surface is of pointwise 1-type Gauss map if…
Study on lightlike geometry in indefinite Sasakian statistical manifolds.
problem Exploring lightlike hypersurfaces and their properties in indefinite Sasakian statistical manifolds.
method Introducing indefinite Sasakian statistical manifolds and analyzing lightlike hypersurfaces with respect to dual connections.
result An invariant lightlike submanifold of an indefinite Sasakian statistical manifold is itself an indefinite Sasakian statistical manifold.
This paper studies lightlike Cartan geometries and their properties.
problem Understanding geometric structures on lightlike cones in spacetime.
method Develops Cartan geometries on the future lightlike cone of Lorentz-Minkowski spacetime.
result Lightlike Cartan geometries induce a lightlike metric and compatible structures.
New method to construct partner curves of non-lightlike curves.
problem Constructing partner curves for non-lightlike curves.
method Using integral curves in Minkowski 3-space, direction curve, and donor curve.
result New methods to construct partner curves of a unit speed non-lightlike curve.
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.
problem Investigating Dirac operators on hypersurfaces with degenerate metrics.
method Spinorial Gauss formula, investigation of Dirac operator, relation with Riemannian curvatures.
result Established relation between Dirac operators and curvatures of the manifold and hypersurface.
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…
Study on lightlike submanifolds in statistical manifold geometry.
problem Characterizing contact CR and SCR-lightlike submanifolds.
method Developed characterization theorems on integrability and geodesicity.
result Obtained results on geometry of contact CR and SCR-lightlike submanifolds.
Study on lightlike submanifolds in metallic semi-Riemannian manifolds.
problem Characterizing and investigating properties of lightlike submanifolds in metallic semi-Riemannian manifolds.
method Introduced and analyzed subclasses of screen transversal lightlike submanifolds and investigated their geometric properties.
result Necessary and sufficient condition for an isotropic screen transversal lightlike submanifold to be totally geodesic.
Study the geometry of specific submanifolds in Golden Semi-Riemannian manifolds.
problem Investigate the geometry of specific submanifolds in Golden Semi-Riemannian manifolds.
method Investigate the geometry of distributions and induced connections, provide necessary and sufficient conditions for metric connections, and characterize submanifolds.
result Characterization of screen transversal anti-invariant lightlike submanifolds of Golden Semi-Riemannian manifolds.
Study on SASI-lightlike submanifolds in indefinite Kaehler manifolds.
problem Characterizing and understanding SASI-lightlike submanifolds in indefinite Kaehler manifolds.
method Introduced and analyzed SASI-lightlike submanifolds, derived conditions for metric connection, and investigated integrability of distributions.
result Found necessary and sufficient conditions for the induced connection to be a metric connection on SASI-lightlike submanifolds.
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…
The paper studies lightlike submanifolds in bronze semi-Riemannian manifolds with specific geometric properties.
problem Characterizing and understanding lightlike submanifolds in bronze semi-Riemannian manifolds.
method Characterization theorems on geodesicity, integrability, and parallelism of distributions.
result No coisotropic, isotropic, or totally proper screen generic lightlike submanifolds exist.
Study shows lightlike hypersurfaces in indefinite Sasakian manifolds are not symmetric.
problem Characterizing curvature symmetries in lightlike hypersurfaces of indefinite Sasakian manifolds.
method Analyzing properties of lightlike hypersurfaces in indefinite Sasakian manifolds.
result Lightlike hypersurfaces are not locally symmetric, semi-symmetric, or semi-parallel.
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…