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.
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
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.
New rigidity results for specific hypersurfaces in spacetimes.
problem Characterizing maximal hypersurfaces in Generalized Robertson-Walker spacetimes.
method Applying rigidity results under geometric assumptions and the Null Energy Condition.
result New parametric uniqueness and nonexistence results for maximal hypersurfaces.
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.
A perfect-fluid space-time of dimension n>3 with 1) irrotational velocity vector field, 2) null divergence of the Weyl tensor, is a generalised Robertson-Walker space-time with Einstein fiber. Condition 1) is verified whenever pressure and energy density are related by an equation of state. The contraction of the Weyl …
New conditions for GRW space-times to be perfect-fluid space-times.
problem Conditions for GRW space-times to be perfect-fluid.
method Gray's decomposition of the gradient of the Ricci tensor, determining Ricci tensor forms in invariant subspaces.
result For most GRW space-times, the Ricci tensor is Einstein or perfect fluid.
The paper finds conditions for pseudosymmetric spacetimes to be perfect fluids.
problem Characterizing pseudosymmetric spacetimes as perfect fluids.
method Analyzes generalized Robertson-Walker spacetimes, conformally flat spacetimes, and dust fluids.
result Conditions for pseudosymmetric spacetimes to be perfect fluids are established.
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 finds conditions for certain warped product manifolds to be quasi-Einstein.
problem Conditions for quasi-Einstein sequential warped product manifolds.
method Investigated necessary and sufficient conditions for specific types of manifolds.
result Identified conditions for sequential warped product manifolds to be quasi-Einstein.
The paper examines gradient ρ-Einstein solitons on specific manifolds and spacetimes.
problem Characterizing gradient ρ-Einstein solitons on doubly warped product manifolds.
method Analyzing necessary and sufficient conditions for doubly warped product manifolds to be gradient ρ-Einstein solitons, applying results to specific spacetime models.
result No 3-dimensional essentially conformally symmetric gradient ρ-Einstein soliton exists.
Walker manifolds of signature (2,2) have been used to provide examples of Osserman and of conformal Osserman manifolds of signature (2,2). We study questions of geodesic completeness and Ricci blowup in this context.
Study submanifolds in null hypersurfaces of generalized Robertson-Walker spacetimes.
problem Characterize submanifolds in null hypersurfaces of generalized Robertson-Walker spacetimes.
method Analyze light cones, lightlike cylinders, and null cones in specific spacetimes; provide conditions for conformal diffeomorphism.
result Conditions guaranteeing conformal diffeomorphism to hyperbolic space, round cylinder, and sphere.
Let p be an odd prime and G the finite cyclic group of order p. We use the Casson-Walker-Lescop invariant to find a necessary condition for a three-manifold to have an action of G with a circle as the set of fixed points.
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.
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.
The study characterizes GRW spacetimes with gradient solitons and phantom era.
problem Characterizing generalized Robertson-Walker spacetimes with gradient solitons.
method Examined gradient type Ricci solitons and (m,τ)-quasi Einstein solitons in GRW spacetimes. result Demonstrated that GRW spacetimes can be Robertson-Walker or phantom era spacetimes under certain conditions.
Study on gradient solitons on specific manifolds.
problem Existence and properties of gradient solitons on warped product manifolds.
method Analyzing necessary and sufficient conditions for the existence of generalized quasi Yamabe gradient solitons.
result Existence of non-trivial gradient Yamabe solitons on specific spacetimes.
Study space-like surfaces in Robertson-Walker spacetimes with specific geometric conditions.
problem Characterize space-like surfaces in Robertson-Walker spacetimes with given geometric conditions.
method Investigate surfaces satisfying specific conditions on tangential and normal parts of the unit vector field, using shape operators and minimal surfaces.
result Classification theorem and parametrizations of space-like class A surfaces in L14(f,0). We study constant mean curvature spacelike hypersurfaces in generalized Robertson-Walker spacetimes which are spatially parabolic covered (i.e. its fiber F is a (non- compact) complete Riemannian manifold whose universal covering is parabolic) and satisfy the null convergence condition. In particular, we provide severa…
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.
Study finds conditions for stationary spacelike surfaces in a generalized Robertson-Walker spacetime.
problem Characterize complete stationary spacelike surfaces in a generalized Robertson-Walker spacetime.
method Assume a natural inequality involving curvature and warping function, use parabolicity and superharmonic function properties.
result Non-compact complete stationary spacelike surfaces are totally geodesic under given conditions.
We consider the inverse mean curvature flow in Robertson-Walker spacetimes that satisfy the Einstein equations and have a big crunch singularity and prove that under natural conditions the rescaled inverse mean curvature flow provides a smooth transition from big crunch to big bang. We also construct an example showing…
Study proves unique foliation of spacetimes by constant mean curvature surfaces.
problem Proving unique foliation of open spacetimes by constant mean curvature surfaces.
method Spatially asymptotic to Robertson-Walker spacetime, proving existence and smoothness of mean curvature function.
result Existence and smoothness of unique foliation by constant mean curvature surfaces.
Our purpose in this paper is to apply some maximum principles in order to study the rigidity of complete spacelike hypersurfaces immersed in a spatially weighted generalized Robertson-Walker (GRW) spacetime, which is supposed to obey the so called strong null convergence condition. Under natural constraints on the weig…
In this paper we study the problem of uniqueness for spacelike hypersurfaces with constant higher order mean curvature in generalized Robertson-Walker (GRW) spacetimes. In particular, we consider the following question: Under what conditions must a compact spacelike hypersurface with constant higher order mean curvatur…
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…
In this paper, we extend a technique due to Romero, Rubio and Salamanca establishing sufficient conditions to guarantee the parabolicity of complete spacelike hypersurfaces immersed in a weighted generalized Robertson-Walker spacetime whose fiber has phi-parabolic universal Riemannian covering. As some applications of …
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.
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 …
Study space-like and time-like surfaces in Robertson-Walker space-times with positive nullity.
problem Characterize space-like and time-like surfaces in Robertson-Walker space-times with positive relative nullity.
method Provide necessary and sufficient conditions, local classification theorems, and analyze special spaces.
result Local classification theorems for space-like and time-like surfaces in L14(f,0) with positive relative nullity. 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.
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.
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.
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.
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.
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.
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.
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 …
Proves Witten-Reshetikhin-Turaev 3-TQFT as a boundary condition of Crane-Yetter 4-TQFT.
problem Proving a boundary condition for Crane-Yetter 4-TQFT.
method Extending ideas of Crane-Yetter and Jordan, proving Crane-Yetter 4-TQFT and its non-semisimple version are once-extended TQFTs, defining a boundary condition.
result Reconstructs Witten-Reshetikhin-Turaev 3-TQFT and its non-semisimple versions using Crane-Yetter 4-TQFT.
The paper studies how certain surfaces evolve in space-time.
problem Preserving the space-like condition of non-compact hypersurfaces.
method Prescribed mean curvature flow in generalized Robertson-Walker spaces.
result The flow preserves space-like condition and exists for infinite time.
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.
The paper studies the past inextendibility of FLRW spacetimes using the VDR asymptote.
problem Investigating the past inextendibility of FLRW spacetimes.
method Using the volume-distance-ratio (VDR) asymptote to assess spacetime inextendibility criteria.
result Conditions for past inextendibility of FLRW spacetimes are identified.
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.