arXiv research

Study on 2-ruled hypersurfaces in a Walker 4-manifold.

New definition of skein lasagna module for specific 4-manifolds.

Extends 4D cornered skein theory to surfaces, proving gluing formulas.

New method uses Khovanov homology to distinguish exotic 4-manifolds.

Introduces Floer lasagna modules using link Floer homology.

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…

Link homology theories connect to 4-manifold invariants and TQFTs.

Formulae for 1-3 handle attachments in 4-manifolds.

New invariant for 4-manifolds with framed links, stronger than existing invariants.

The paper studies Einstein-like Walker metrics in Walker manifolds.

In this study, we defined Fermi-Walker derivative in Galilean space $\mathbb{G}^3$. Fermi-Walker transport and non-rotating frame by using Fermi- Walker derivative are given in $\mathbb{G}^3$. Being conditions of Fermi-Walker transport and non-rotating frame are investigated along any curve for Frenet frame and Darboux…

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…

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 of Ricci-Yamabe solitons on Walker 3-manifolds.

Research shows curves in Walker 3-manifolds can lie 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.

Study cosmetic surgeries on knots in homology spheres using Casson-Walker invariant.

The paper classifies and characterizes biconservative surfaces in Robertson-Walker spacetimes.

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 …

Formula for Casson-Walker invariant; applies to knot complements and cosmetic crossing conjecture.

Study maximal hypersurfaces in open spacetimes using a maximum principle.

Given an $n$-dimensional manifold $N$ with an affine connection $D$, we show that the associated Patterson-Walker metric $g$ on $T^*N$ admits a global and explicit Fefferman-Graham ambient metric. This provides a new and large class of conformal structures which are generically not conformally Einstein but for which th…

In this paper, we investigate geometric properties of some curvature tensors of a four-dimensional Walker manifold. Some characterization theorems are also obtained.

The study finds Ricci solitons on a specific type of 3D Lorentzian Walker manifold.

In this paper we study the invariant Walker structures over the conformally flat four-dimensional homogeneous manifolds according to the Seger types of the Ricci operator.

Study submanifolds in null hypersurfaces of generalized Robertson-Walker spacetimes.

New formula for 3-manifold invariants using combinatorial methods.

Study on Ricci-Yamabe solitons on Walker manifolds.

We use a locally constrained mean curvature flow to prove the isoperimetric inequality for spacelike domains in generalized Robertson-Walker spaces satisfying the null convergence condition.

We establish that, over certain ground fields, the set of osculating tangents of Cayley's ruled cubic surface gives rise to a (maximal partial) spread which is also a dual (maximal partial) spread. It is precisely the Betten-Walker spreads that allow for this construction. Every infinite Betten-Walker spread is not an …

Develops symmetric Cartan calculus linking to Patterson-Walker metric.

We give a local characterization of codimension two submanifolds which are marginally trapped in Robertson-Walker spaces, in terms of an algebraic equation to be satisfied by the height function. We prove the existence of a large number of local solutions. We refine the description in the case of curves with null accel…

Study analyzes derivative-free loss method for solving PDEs and fluid problems.

New rigidity results for specific hypersurfaces in spacetimes.

New rigidity results for submanifolds in GRW spacetimes.

Study Lie foliation of Walker manifolds in pseudo-Riemannian geometry.

Study translators in Generalised Robertson-Walker spacetimes, identifying warping functions and classifying examples.

Gauss diagram formulas are extensively used to study Vassiliev link invariants. Now we apply this approach to invariants of 3-manifolds, considering manifolds given by surgery on framed links in the 3-sphere. We study the lowest degree case - the celebrated Casson-Walker invariant of rational homology spheres. This pap…

We prove theorems about the Ricci and the Weyl tensors on generalized Robertson-Walker space-times of dimension $n\ge 3$. In particular, we show that the concircular vector introduced by Chen decomposes the Ricci tensor as a perfect fluid term plus a term linear in the contracted Weyl tensor. The Weyl tensor is harmoni…

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…

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 …

The study characterizes GRW spacetimes with gradient solitons and phantom era.

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.

In (Phys. Rev. D 62, 081501, 2000) we proposed a unified approach to description of continuous and discrete spacetime based on nonassociative geometry and described nonassociative smooth and discrete de Sitter models. In our paper we give the description of nonassociative Friedmann-Robertson-Walker spacetime.

Study finds conditions for stationary spacelike surfaces in a generalized Robertson-Walker spacetime.

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.

We give new necessary and sufficient conditions on the Weyl tensor for generalized Robertson-Walker (GRW) space-times to be perfect-fluid space-times. For GRW space-times, we determine the form of the Ricci tensor in all the O(n)-invariant subspaces provided by Gray's decomposition of the gradient of the Ricci tensor. …