Study examines null vector fields on Lorentzian manifolds.
problem Understanding the structure of null vector fields on Lorentzian manifolds.
method Investigates the bundle structure and ternary product of nowhere vanishing null vector fields.
result Null tangent bundle is a non-polynomial graded bundle with a para-associative ternary product.
Study null conformal Killing vector fields on complex surfaces.
problem Characterize pseudo-Hermitian surfaces with null vector fields.
method Analyze topological types and use vector fields to define para-hyperhermitian structures.
result Classify compact four-manifolds with orthogonal null Killing vector fields.
Study on null hypersurfaces with constant angle in Lorentzian manifolds.
problem Understanding constant angle null hypersurfaces in Lorentzian manifolds.
method Introduced constant angle null hypersurfaces, analyzed with respect to a given ambient vector field, and provided classification results.
result Null hypersurfaces have a canonical principal direction when the vector field is closed and conformal.
Symplectic forms derived from Lorentzian geometry null vector fields.
problem Constructing symplectic forms from Lorentzian geometry null vector fields.
method Using complete null vector fields with geodesic flow and Ricci curvature conditions.
result Symplectic forms can be constructed from Lorentzian geometry null vector fields, even if Ricci curvature conditions are not met.
The Einstein Equation on 4-dimensional Lorentzian manifolds admitting recurrent null vector fields is discussed. Several examples of a special form are constructed. The holonomy algebras, Petrov types and the Lie algebras of Killing vector fields of the obtained metrics are found.
Study slant null curves on specific 3-manifolds with parallel Reeb vector field.
problem Characterize slant null curves on 3-manifolds with parallel Reeb vector field.
method Analyze slant null curves using Frenet frames and Cartan Frenet frames.
result Existence and uniqueness of a distinguished Frenet frame for non-geodesic slant null curves.
We expound some results about the relationships between the Jacobi operators with respect to null vectors on a Lorentzian S \mathcal{S} S -manifold M M M and the Jacobi operators with respect to particular spacelike unit vectors on M M M . We study the number of the eigenvalues of such operators in a φ φ φ -null Osserman Lorentzi…
It is of interest to study supergravity solutions preserving a non-minimal fraction of supersymmetries. A necessary condition for supersymmetry to be preserved is that the spacetime admits a Killing spinor and hence a null or timelike Killing vector field. Any spacetime admitting a covariantly constant null vector fiel…
Given a null hypersurface L L L of a Lorentzian manifold, we construct a Riemannian metric g ~ \widetilde{g} g on it from a fixed transverse vector field ζ ζ ζ . We study the relationship between the ambient Lorentzian manifold, the Riemannian manifold ( L , g ~ ) (L,\widetilde{g}) ( L , g ) and the vector field ζ ζ ζ . As an application, we prove so…
Study principal configurations near special points on spacelike surfaces in null hypersurfaces.
problem Characterize principal configurations around η η η -umbilical points on spacelike surfaces in null hypersurfaces. method Analyzes principal configurations using a null vector field orthogonal to the surface.
result Recover local Darbouxian principal configurations for specific null rotation hypersurfaces.
On a Lorentzian manifold the existence of a parallel null vector field implies certain constraint conditions on the induced Riemannian geometry of a space-like hypersurface. We will derive these constraint conditions and, conversely, show that every real analytic Riemannian manifold satisfying the constraint conditions…
Study null hypersurfaces with privileged vector fields, extending surface gravity and defining new horizon types.
problem Characterize null hypersurfaces with privileged vector fields and extend surface gravity.
method Derive identities relating deformation tensor to intrinsic and extrinsic geometry, introduce generalized surface gravity, analyze Lie derivatives, and define new horizon types.
result Introduce three new horizon types that generalize existing concepts to arbitrary topologies and fixed points.
Study of Randers spacetimes and their Finsler gravity solutions.
problem Analyzing Finsler gravity field equations for Randers spacetimes.
method Examined Berwald-type Randers spacetimes and Finsler gravity field equations, showing equivalence to Einstein gravity.
result Found exact solutions for vacuum Finsler gravity are composed of pp-waves and 1-forms.
Local classification of 4D Ricci solitons with specific algebra properties.
problem Classifying 4D Ricci solitons with a 2D Abelian Killing algebra.
method Local classification under specific curvature and symmetry conditions.
result Classification of Ricci solitons with orthogonally intransitive 2D Abelian Killing algebra.
Researchers solved Einstein-Yang-Mills equations for arbitrary gauge groups.
problem Analyzing spacetime metrics and gauge fields with specific symmetries.
method Analytically integrated Einstein-Yang-Mills equations for a general gauge group.
result Solved Einstein-Yang-Mills equations for arbitrary gauge groups.
Perfect fluids in higher dimensions become generalized Robertson-Walker spaces under specific conditions.
problem Characterizing perfect-fluid space-times as generalized Robertson-Walker spaces.
method Analyzing conditions for a perfect-fluid space-time to be a generalized Robertson-Walker space-time.
result Conditions for a perfect-fluid space-time to be a generalized Robertson-Walker space-time are verified.
The study characterizes and analyzes spacelike surfaces with a canonical normal null direction in Minkowski 4-space.
problem Characterizing and analyzing spacelike surfaces with a specific null direction in Minkowski space.
method Using geometric properties, Gauss map, and a nonlinear partial differential equation, the study characterizes and analyzes these surfaces.
result Characterizations and properties of spacelike surfaces with a canonical normal null direction are obtained.
A function that optimally aligns a timelike vector field with its gradients
problem Finding a time function that aligns a timelike vector field with its gradients
method Introducing a functional that penalizes null gradients and minimizes misalignment
result Proving the existence of a unique alignment time function under suitable conditions
The paper describes timelike surfaces with a canonical null direction in Minkowski space.
problem Characterizing timelike surfaces with a canonical null direction.
method Analyzing ruled and non-ruled surfaces, using the Gauss map.
result Properties of timelike surfaces with a canonical null direction in different dimensions.
Study on special null submanifolds in indefinite Sasakian manifolds.
problem Characterizing null submanifolds in indefinite Sasakian manifolds.
method Proving properties of screen conformal null submanifolds and defining a new class.
result Existence of contact screen conformal r r r -null submanifolds in indefinite Sasakian space forms. The paper solves a Cauchy problem for Lorentzian manifolds and classifies manifolds with special holonomy.
problem Solving the Cauchy problem for parallel null vector fields on smooth Lorentzian manifolds.
method Deriving and analyzing hyperbolic evolution equations based on the Ricci tensor and geometric objects.
result Classification of Riemannian manifolds satisfying the Cauchy problem conditions and characterisation of holonomy reductions.
Study on geodesics in Kropina metrics with applications.
problem Existence of connecting and closed geodesics in Kropina metrics.
method Analytical proofs and applications to null geodesics and navigation problems.
result Proves existence of geodesics in Kropina metrics.
The study explores Lorentzian manifolds with specific null vector fields and their geometric properties.
problem Characterizing Lorentzian manifolds with shearfree null vector fields and their quotient structures.
method Analyzing quotient spaces and constructing metrics on total spaces of bundles.
result Existence of non-trivial generalized electromagnetic plane waves and Einstein metrics.
New types of null hypersurfaces found in Sasakian space-forms.
problem Identifying new structures in Sasakian space-forms.
method Defined and analyzed contact screen conformal and umbilic null hypersurfaces.
result Proved these hypersurfaces exist in Sasakian space forms with specific curvature.
New contact structures extend supergravity solutions.
problem Extend supergravity solutions using new contact structures.
method Introduce and investigate ε \varepsilon\, ε -contact metric structures, focusing on null contact structures. result Appropriate direct products of ε \varepsilon\, ε -Einstein structures produce solutions of six-dimensional minimal supergravity. We determine the geometry of supersymmetric heterotic string backgrounds for which all parallel spinors with respect to the connection ∇ ^ \hat\nabla ∇ ^ with torsion H H H , the NS ⊗ \otimes ⊗ NS three-form field strength, are Killing. We find that there are two classes of such backgrounds, the null and the timelike. The Killing s…
Proves symmetries of extremal horizons in spacetimes.
problem Proving symmetries of extremal horizons in arbitrary dimensions.
method Analyzes Killing vector fields and near-horizon geometry.
result Enhanced isometry groups and shifted Aretakis instability.
The paper studies canal hypersurfaces in Lorentz-Minkowski 4-space.
problem Characterizing canal hypersurfaces formed by pseudo null, partially null, and null curves.
method Obtained parametric expressions and geometric invariants of canal hypersurfaces.
result Characterizations of tubular hypersurfaces in E 1 4 E^4_1 E 1 4 . Proves non-existence of periodic vacuum spacetimes.
problem The existence of time-periodic vacuum spacetimes.
method Extending a candidate Killing vector field from null infinity using Carleman estimates.
result Smooth asymptotically flat solutions are stationary in a neighborhood of infinity.
New curvature obstruction for Killing vector fields on Lorentzian manifolds.
problem Existence of timelike or causal Killing vector fields on Lorentzian manifolds.
method New curvature obstruction in terms of timelike or null sectional curvature.
result Extension of Gauss-Bonnet-Chern obstruction to non-zero timelike sectional curvature.
The paper classifies 3D hypersurfaces with specific geometric properties.
problem Characterizing centro-affine hypersurfaces with J ~ \widetilde{J} J -tangent vector fields. method Local classification through detailed analysis of hypersurfaces' properties.
result Every nondegenerate hypersurface with specific null-directions is both an affine hypersphere and a hyperquadric.
Study of α \alpha α -associated metrics on null hypersurfaces.
problem Developing a method to construct α \alpha α -associated metrics on null hypersurfaces. method Introduce and study α \alpha α -associated metrics induced by a non-vanishing function α \alpha α on a rigging vector field. result Constructive method to find α \alpha α -associated metrics with Levi-Civita connections matching given null hypersurface. Study peels tensor equations on Schwarzschild spacetime.
problem Analyzing the asymptotic behavior of tensorial wave equations on Schwarzschild spacetime.
method Combining conformal compactification and vector field techniques to estimate tensorial field energies.
result Obtains optimal initial data for peeling at all orders.
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.
Extends Killing vector fields in electrovacuum spacetimes, proving non-extendibility.
problem Extension of Killing vector fields in electrovacuum spacetimes.
method Inspired by Ionescu-Klainerman's technique for Ricci flat manifolds, extends to strong null convex domains.
result Shows non-extendibility of Hawking vector field in Kerr-Newman solutions near horizons.
I present a construction of real or complex selfdual conformal 4-manifolds (of signature (2,2) in the real case) from a natural gauge field equation on a real or complex projective surface, the gauge group being the group of diffeomorphisms of a real or complex 2-manifold. The 4-manifolds obtained are characterized by …
In this paper we examine different aspects of the geometry of closed conformal vector fields on Riemannian manifolds. We begin by getting obstructions to the existence of closed conformal and nonparallel vector fields on complete manifolds with nonpositive Ricci curvature, thus generalizing a theorem of T. K. Pan. Then…
New approach to finite gauge transformations in doubled spacetime.
problem Complexity of finite gauge transformations in double field theory.
method Intuitive approach using untwisted vector fields and maximal null subspace.
result Finite transformation law automatically satisfies composition law and avoids the Papadopoulos problem.
Novel contact metric structures lead to supergravity solutions.
problem Developing new contact metric structures for supergravity.
method Introducing and studying ε η \varepsilonη\, ε η -Einstein structures. result Constructed families of six-dimensional supergravity solutions.
We study continuous groups of generalized Kerr-Schild transformations and the vector fields that generate them in any n-dimensional manifold with a Lorentzian metric. We prove that all these vector fields can be intrinsically characterized and that they constitute a Lie algebra if the null deformation direction is fixe…
We fully develop the concept of causal symmetry introduced in Class. Quant. Grav. 20 (2003) L139. A causal symmetry is a transformation of a Lorentzian manifold (V,g) which maps every future-directed vector onto a future-directed vector. We prove that the set of all causal symmetries is not a group under the usual comp…
We study the geometric nature of the Jacobi equation. In particular we prove that Jacobi vector fields (JVFs) along a solution of the Euler-Lagrange (EL) equations are themselves solutions of the EL equations but considered on a non-standard algebroid (different from the tangent bundle Lie algebroid). As a consequence …
Study transverse metric expansion on null hypersurfaces, proving uniqueness for Killing horizons.
problem Analyzing transverse expansion of metric on null hypersurfaces.
method Covariant approach, general geometric identities, generalized symmetry generators.
result Transverse expansion of spacetime metric uniquely determined at non-degenerate Killing horizons.
Unique global solutions found for specific initial data.
problem Einstein-scalar-field equations with specific initial conditions.
method Spherically symmetric analysis of small, slowly decaying data.
result Unique global solutions exist for the equations.
Unified description of string and brane worldvolumes using auto-parallel vector fields.
problem Describing worldvolumes of strings and branes in arbitrary backgrounds.
method Introducing auto-parallel generalised vector fields and their properties.
result Unified worldvolume equations for strings and branes.
Proves a rigidity result for Brinkmann spacetimes.
problem Geodesic completeness constraints in Brinkmann spacetimes.
method Analyzes Brinkmann spacetimes with null parallel vector fields.
result Proves a restricted rigidity result for Brinkmann spacetimes.
Theory for gravity coupled with fields on manifolds with null-boundary.
problem Formulating a theory for gravity coupled with scalar, SU(n), and spinor fields on manifolds with null-boundary.
method Symplectic reduction of boundary fields and constraints analysis.
result The set of constraints does not form a first class system for the three couplings.
This paper defines directional derivatives and solves Maxwell's equations in curved 3D space.
problem Analyzing electromagnetic fields in curved non-flat 3D space.
method Defined directional derivatives and used Frenet formulas to express Serret-Frenet relations. Solved Maxwell's equations for electric and magnetic fields.
result Solved Maxwell's equations for electromagnetic fields in curved 3D space.