The paper studies Einstein-like Walker metrics in Walker manifolds.
problem Characterizing Walker metrics with specific curvature properties.
method Analyzing four-dimensional Walker manifolds with a parallel degenerate plane field.
result Characterization of Walker metrics that are Einstein-like.
New ambient metrics reveal properties of Walker metrics.
problem Characterizing Walker metrics using ambient metrics.
method Developed Fefferman-Graham ambient metrics for Walker metrics.
result Walker metrics have vanishing Q-curvature.
The classical Patterson-Walker construction of a split-signature (pseudo-)Riemannian structure from a given torsion-free affine connection is generalized to a construction of a split-signature conformal structure from a given projective class of connections. A characterization of the induced structures is obtained. We …
A generalized Robertson-Walker spacetime is the warped product with base an open interval of the real line endowed with the opposite of its metric and base any Riemannian manifold. The family of generalized Robertson-Walker spacetimes widely extends the one of classical Robertson-Walker spacetimes. In this article we p…
Develops symmetric Cartan calculus linking to Patterson-Walker metric.
problem No specific problem stated; focuses on developing a new calculus.
method Symmetric Cartan calculus, using torsion-free affine connections.
result Symmetric Cartan calculus is a complete analogue of classical Cartan calculus.
A four-dimensional Walker geometry is a four-dimensional manifold M with a neutral metric g and a parallel distribution of totally null two-planes. This distribution has a natural characterization as a projective spinor field subject to a certain constraint. Spinors therefore provide a natural tool for studying Walker …
Developed method to find explicit conformal metrics with Ricci-flat ambient metrics.
problem Finding explicit conformal metrics with Ricci-flat ambient metrics.
method Method to find ambient metrics for conformal classes of metrics with two-step nilpotent Schouten tensor.
result Obtained explicit ambient metrics for certain types of Walker metrics.
The paper constructs and classifies 3D Walker manifolds with specific structures.
problem Classifying 3D Walker manifolds with specific paracontact structures.
method Constructing structures using a unit space-like vector field and a function, characterizing the Lorentzian metric.
result Necessary and sufficient conditions for the manifold to belong to specific classes of almost paracontact metric manifolds.
Study of Ricci-Yamabe solitons on Walker 3-manifolds.
problem Characterizing Ricci-Yamabe solitons on Walker 3-manifolds.
method Using Hodge decomposition of De-Rham, the soliton field is found from the potential function.
result Classification of all Ricci-Yamabe and gradient Ricci-Yamabe solitons in a Walker 3-manifold.
In this paper, we prove a Kastler-Kalau-Walze type theorem for 4-dimensional and 6-dimensional spin manifolds with boundary associated with the conformal Robertson-Walker metric. And we give two kinds of operator theoretic explanations of the gravitational action for boundary in the case of 4-dimensional manifolds with…
Study on Ricci-Yamabe solitons on Walker manifolds.
problem Characterizing Walker manifolds for Ricci-Yamabe solitons.
method Explicit calculation of Ricci tensor, scalar curvature, and Hessian Perelman potential; solving partial differential equations.
result Identifying constraints on functions and vector field for soliton existence.
Study proves mean curvature flow in GRW spacetimes with perpendicular boundary condition.
problem Longtime existence of mean curvature flow in GRW spacetimes.
method Proved longtime existence using perpendicular Neumann boundary condition and null convergence condition.
result Metric of solution is conformal to GRW leaf's metric in asymptotic time.
The study finds Ricci solitons on a specific type of 3D Lorentzian Walker manifold.
problem Existence of non-trivial Ricci solitons on specific manifolds.
method Defined Ricci solitons on pseudo-Riemannian manifolds and applied to a family of 3D Lorentzian Walker manifolds.
result Existence of non-trivial Ricci solitons on a family of 3D Lorentzian Walker manifolds.
We give necessary and sufficient conditions for warped product manifolds with 1-dimensional base, and in particular, for generalized Robertson-Walker spacetimes, to satisfy some generalized Einstein metric condition. We also construct suitable examples of such manifolds. They are quasi-Einstein or not.
Notation for spin coefficients for metrics of neutral signature in four dimensions is introduced. The utility and interpretation of spin coefficients is explored through themes in null geometry familiar from (complex) general relativity. Four-dimensional Walker geometry is exploited to provide examples and the generali…
We provide a coordinate-free version of the local classification, due to A. G. Walker [Quart. J. Math. Oxford (2) 1, 69 (1950)], of null parallel distributions on pseudo-Riemannian manifolds. The underlying manifold is realized, locally, as the total space of a fibre bundle, each fibre of which is an affine principal b…
k-Curvature homogeneous three-dimensional Walker metrics are described for k=0,1,2. This allows a complete description of locally homogeneous three-dimensional Walker metrics, showing that there exist exactly three isometry classes of such manifolds. As an application one obtains a complete description of all locally h…
We exhibit Walker manifolds of signature (2,2) with various commutativity properties for the Ricci operator, the skew-symmetric curvature operator, and the Jacobi operator. If the Walker metric is a Riemannian extension of an underlying affine structure A, these properties are related to the Ricci tensor of A.
Survey of connected holonomy groups in Lorentzian manifolds.
problem Classifying connected holonomy groups of Lorentzian manifolds.
method Simplified construction of Lorentzian metrics and applications to specific manifolds.
result Obtained a simplification in constructing Lorentzian metrics with all possible connected holonomy groups.
The paper constructs almost para-Kähler-Einstein metrics on cotangent bundles.
problem Developing metrics on cotangent bundles associated with geometric structures.
method Using a construction involving geometric structures on a manifold M to associate an almost para-Kähler-Einstein metric on T∗M. result Explicit formulae for these metrics are derived in specific geometric cases.
Classifies cosmological Finsler spacetimes, finding viable non-stationary models.
problem Locating viable non-stationary Finsler spacetimes in cosmology.
method Locally classified all possible cosmological homogeneous and isotropic Landsberg-type Finsler structures in 4-dimensions.
result Identified unique Finsler, non-Berwaldian Landsberg generalization of Friedmann-Lemaitre-Robertson-Walker geometry.
The local classification of conformally flat Lorentzian manifolds with special holonomy groups is obtained. The corresponding local metrics are certain extensions of Riemannian spaces of constant sectional curvature to Walker metrics.
The paper classifies critical closed RW spacetimes using elliptic functions.
problem Classifying critical closed Robertson-Walker spacetimes.
method Study of critical RW spacetimes via volume-preserving variations and action functionals.
result Complete classification of critical RW spacetimes with explicit solutions.
Defined Fermi-Walker derivative in Galilean space and its applications.
problem Defining and applying Fermi-Walker derivative in Galilean space.
method Defined Fermi-Walker derivative in Galilean space G3, and investigated conditions for Fermi-Walker transport and non-rotating frame along curves. result Conditions for Fermi-Walker transport and non-rotating frame were investigated in Galilean space.
By a real alphabeta-geometry we mean a four-dimensional manifold M equipped with a neutral metric h such that (M,h) admits both an integrable distribution of alpha-planes and an integrable distribution of beta-planes. We obtain a local characterization of the metric when at least one of the distributions is parallel (i…
The paper studies geometric structures in perfect fluid spacetimes with specific metrics.
problem Analyzing the geometric properties of perfect fluid spacetimes with specific metrics.
method Investigates conditions for conformal Ricci-Yamabe soliton and derives Laplace equations.
result Conditions for expanding, steady, or shrinking conformal Ricci-Yamabe solitons are identified.
The purpose of this article is to review some recent results on the geometry of neutral signature metrics in dimension four and their twistor spaces. The following topics are considered: Neutral Kähler and hyperkähler surfaces, Walker metrics, Neutral anti-self-dual 4-manifolds and projective structures, Twistor spaces…
Novel estimation methods improve MAR model accuracy for high-dimensional time series.
problem Limited estimation techniques for Matrix Autoregressive (MAR) models.
method Adapted Yule-Walker equations and Burg's method.
result Proposed methods achieve comparable model fit to VAR models.
Einstein's equation, in its standard form, breaks down at the Big Bang singularity. A new version, equivalent to Einstein's whenever the latter is defined, but applicable in wider situations, is proposed. The new equation remains smooth at the Big Bang singularity of the Friedmann-Lemaitre-Robertson-Walker model. It is…
While the Lorenzian and Riemanian metrics for which all polynomial scalar curvature invariants vanish (the VSI property) are well-studied, less is known about the four-dimensional neutral signature metrics with the VSI property. Recently it was shown that the neutral signature metrics belong to two distinct subclasses:…
Embeds Lorentzian manifolds in \(\mathbb{R}^{n+2}\) with SO(2,n) compatibility.
problem Embedding Lorentzian manifolds in \(\mathbb{R}^{n+2}\) with specific metric properties.
method Embedding using SO(2,n) compatible metrics.
result Conformal transformations on submanifolds inherited from ambient space.
Proves isoperimetric inequality for spacelike domains in generalized Robertson-Walker spaces.
problem Proving the isoperimetric inequality for spacelike domains in generalized Robertson-Walker spaces.
method Locally constrained mean curvature flow
result Proves the isoperimetric inequality
Research shows curves in Walker 3-manifolds can lie in flat cylinders.
problem Understanding curves in Walker 3-manifolds.
method Showed curves lie in flat cylinders, constructed an example.
result Curves in Walker 3-manifolds can be contained in flat cylinders.
Symmetric Poisson structures linked to geodesic foliations and Jordan algebras.
problem Understanding geometric structures related to geodesic foliations and dynamics.
method Introducing symmetric Poisson structures, proving correspondences with geodesic foliations and Jordan algebras.
result Symmetric Poisson structures correspond to totally geodesic foliations and Jacobi-Jordan algebras.
The purpose of the present article is to study and characterize sev- eral types of symmetries of generalized Robertson-Walker space-times. Con- formal vector fields, curvature and Ricci collineations are studied. Many im- plications for existence of these symmetries on generalied Robertson-Walker spacetimes are obtaine…
Study curves of constant breadth in a specific 3D manifold.
problem Differential geometry of curves in Walker 3-manifolds.
method Investigate curves of constant breadth using Darboux frame.
result Properties of curves of constant breadth in Walker 3-manifolds.
Derives curvature conditions for spatial isotropy without field equations.
problem Conditions for spatial isotropy in cosmological models.
method Geometric derivation of curvature conditions independent of field equations.
result Local isometry between space and Robertson-Walker space-time.
Lorentzian LCS spaces are equivalent to Generalized Robertson-Walker spaces.
problem Identifying equivalent space-time structures.
method Direct demonstration of equivalence between LCS and GRW spaces.
result Lorentzian LCS spaces are equivalent to Generalized Robertson-Walker spaces.
Exploring distance functions on spacetime models.
problem No canonical distance function exists for Lorentzian manifolds.
method Comparing Riemannianization and null distance function approaches.
result Concrete comparison of distance functions in GRW setting.
We study a Fefferman-type construction based on the inclusion of Lie groups SL(n+1) into Spin(n+1,n+1). The construction associates a split-signature (n,n)-conformal spin structure to a projective structure of dimension n. We prove the existence of a canonical pure twistor spinor and a light-like co…
Study cosmetic surgeries on knots in homology spheres using Casson-Walker invariant.
problem Cosmetic surgeries on knots in homology spheres and their constraints.
method Rational surgery formula of the Casson-Walker invariant for 2-component links.
result Constraints on knots and surgery slopes for cosmetic surgeries.
The paper classifies and characterizes biconservative surfaces in Robertson-Walker spacetimes.
problem Characterizing biconservative surfaces in Robertson-Walker spacetimes.
method Deriving geometrical properties and classifying surfaces in specific spacetimes.
result Complete local classifications of biconservative surfaces in L14(f,0), L15(f,0) and L15(1,±1). Study on 2-ruled hypersurfaces in a Walker 4-manifold.
problem Characterize and analyze 2-ruled hypersurfaces in a Walker 4-manifold.
method Define and analyze three types of 2-ruled hypersurfaces, compute Gaussian and mean curvatures, and study Laplace-Beltrami operators.
result Characterizations and properties of 2-ruled hypersurfaces in a Walker 4-manifold.
The paper analyzes tensors in generalized Robertson-Walker space-times.
problem Analyzing tensors in generalized Robertson-Walker space-times.
method Proving theorems about Ricci and Weyl tensors, decomposing Ricci tensor, showing conditions for harmonic Weyl tensor, and generalizing Riemann tensor structure.
result Conditions for a GRW space-time to be a quasi-Einstein manifold and the structure of Riemann tensor.
Study on quasi-statistical structures on manifolds and their properties.
problem Characterizing quasi-statistical structures and their integrability.
method Introducing generalized quasi-statistical structures and proving their integrability conditions.
result Any quasi-statistical structure on a manifold induces generalized quasi-statistical structures on its tangent and cotangent bundles.
Formula for Casson-Walker invariant; applies to knot complements and cosmetic crossing conjecture.
problem Determining knots in L-space complements and verifying the cosmetic crossing conjecture.
method Rational surgery formula for Casson-Walker invariant of 2-component links.
result Examples of non-hyperbolic L-space complements where knots are determined by their complements.
Study maximal hypersurfaces in open spacetimes using a maximum principle.
problem Characterize maximal hypersurfaces in open spacetimes.
method Use a generalized maximum principle to analyze hypersurfaces in spatially open Generalized Robertson-Walker spacetimes.
result Provide new uniqueness and non-existence results for complete maximal hypersurfaces in open Robertson-Walker spacetimes.
We show that there are isometrically nonequivalent Robertson-Walker metrics which have the same set of geodesics. While one of these metrics satisfies the Einstein equations of pure dust without a cosmological constant, all the other describe pure dust with additional energy momentum tensor of cosmological constant typ…