arXiv research

The paper classifies Weyl tensors in Riemannian 4-manifolds via Lorentzian deformation.

Conditions for the existence of Kähler-Einstein metrics and central Kähler metrics [MS] along with examples, both old and new, are given on classes of Lorentzian $4$-manifolds with two distinguished vector fields. The results utilize the general construction [AM] of Kähler metrics on such manifolds. The examples includ…

Study torsion parallel spinors on Lorentzian 4-manifolds and their evolution flows.

We investigate the twistor space and the Grassmannian fibre bundle of a Lorentzian 4-space with natural almost optical structures and its induced CR-structures. The twistor spaces of the Lorentzian space forms $\R^4_1, \Di{S}^4_1$ and $\Di{H}^4_1$ are explicitly discussed. The given twistor construction is applied to s…

We observe that, in dimension four, symplectic forms may be obtained via Lorentzian geometry; in particular, null vector fields can give rise to exact symplectic forms. That a null vector field is nowhere vanishing yet orthogonal to itself is essential to this construction. Specifically, we show that on a Lorentzian 4-…

On an oriented 4-manifold, we examine the geometry that arises when the curvature operator of a Riemannian or Lorentzian metric $g$ commutes, not with its own Hodge star operator, but rather with that of another semi-Riemannian metric $h$ that is a suitable deformation of $g$. We classify the case when one of these met…

Modeling wormhole creation without singularities in relativity.

The fundamental equations of Gauss, Codazzi and Ricci provide the conditions for local isometric embeddability. In general, the three fundamental equations are independent for surfaces in Riemannian 4-manifolds. In contrast, we prove in this article that for arbitrary Lorentz surfaces in Lorentzian Kaehler surfaces the…

Chernov-Nemirovski observed that the existence of a globally hyperbolic Lorentzian metric on a (3 + 1)-spacetime pins down a smooth structure on the underlying 4-manifold. In this paper, we point out that the diffeomorphism type of a globally hyperbolic (n + 1)-spacetime is determined by the h-cobordism class of its Ca…

We work on a parallelizable time-orientable Lorentzian 4-manifold and prove that in this case the notion of spin structure can be equivalently defined in a purely analytic fashion. Our analytic definition relies on the use of the concept of a non-degenerate two-by-two formally self-adjoint first order linear differenti…

Charge measurements for instantons and gravitational perturbations.

An immense class of physical counterexamples to the four dimensional strong cosmic censor conjecture---in its usual broad formulation---is exhibited. More precisely, out of any closed and simply connected 4-manifold an open Ricci-flat Lorentzian 4-manifold is constructed which is not globally hyperbolic and no perturba…

Geometric approach to Dirac operator evolution on spacetimes.

We work on a 4-manifold equipped with Lorentzian metric $g$ and consider a volume-preserving diffeomorphism which is the unknown quantity of our mathematical model. The diffeomorphism defines a second Lorentzian metric $h$, the pullback of $g$. Motivated by elasticity theory, we introduce a Lagrangian expressed algebra…

In this article we study minimal flat Lorentzian surfaces in Lorentzian complex space forms. First we prove that, for minimal flat Lorentzian surfaces in a Lorentzian complex form, the equation of Ricci is a consequence of the equations of Gauss and Codazzi. Then we classify minimal flat Lorentzian surfaces in the Lore…

Defines metrics for Lorentzian spaces and explores maximal developments.

Study of a series of Lorentzian structures on SL(2,R) with SO(1,1) symmetry.

Study left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.

It is shown how one can apply the classification of the holonomy algebras of Lorentzian manifolds to solve some problems. In particular, a new proof to the classification of Lorentzian manifolds with recurrent curvature tensor is given; the classification of two-symmetric Lorentzian manifolds is explained; conformally …

Study gluing of Lorentzian length spaces and their causal ladder properties.

Defines doubling conditions for Lorentzian spaces linked to curvature bounds.

Research explores Lorentzian distances on a specific geometric plane.

Finds a limit on Lorentzian manifolds.

The study proves a transverse diameter theorem for Lorentzian foliations.

The paper classifies 3D Lorentzian $Ein(2)$ Lie groups.

Timelike minimal surfaces in Lorentzian Heisenberg group have singular points.

Study of 2D Lorentzian anti-de Sitter plane using geometric control theory.

Defines new metrics for Lorentzian spaces and their convergence.

Compact group found for special Lorentzian manifolds.

Study on Lorentzian spaces with curvature bounds, proving comparison theorems.

Characterizes Lorentzian manifolds embeddable in Minkowski spacetime.

Diagonalizes metrics of 3D Lorentzian manifolds.

We study necessary and sufficient conditions for the existence of Lorentzian and weak Lorentzian cobordisms between closed smooth manifolds of arbitrary dimension such that the structure group of the frame bundle of the cobordism is $\Spin(1, n)_0$. This extends a result of Gibbons-Hawking on $\Sl(2, \C)$-Lorentzian co…

Proves a synthetic Lorentzian Cartan-Hadamard theorem.

In this paper, we study the invariant and noninvariant hypersurfaces of (1,1,1) almost contact manifolds, Lorentzian almost paracontact manifolds and Lorentzian para-Sasakian manifolds, respectively. We show that a noninvariant hypersurface of an (1,1,1) almost contact manifold admits an almost product structure. We in…

Study classifies Riemann solitons on specific 3D Lorentzian groups.

If an $m+2$-manifold $M$ is locally modeled on $\RR^{m+2}$ with coordinate changes lying in the subgroup $G=\RR^{m+2}\rtimes ({\rO}(m+1,1)\times \RR^+)$ of the affine group ${\rA}(m+2)$, then $M$ is said to be a \emph{Lorentzian similarity manifold}. A Lorentzian similarity manifold is also a conformally flat Lorentzia…

Proves existence of longest paths in sub-Lorentzian problems.

Three functors link Lorentzian geometry concepts.

The Lorentzian length, which is one of the most significant functions in Lorentzian geometry, is a complex-valued function. Its square gives a real-valued non-degenerate quadratic function. In this paper, we define naturally extended mappings of Lorentzian distance-squared functions, wherein each component is a Lorentz…

Holonomy of Weyl connections in Lorentzian space classified.

Explains non-lorentzian theories and their dynamics.

The paper reconstructs Lorentzian spacetimes from causal sets.

Two flat sub-Lorentzian problems on Martinet distribution differ in attainable set intersections.

In this paper, a survey of the recent results about the classification of the connected holonomy groups of the Lorentzian manifolds is given. A simplification of the construction of the Lorentzian metrics with all possible connected holonomy groups is obtained. As the applications, the Einstein equation, Lorentzian man…

Third-order symmetric Lorentzian manifolds, i.e. Lorentzian manifold with zero third derivative of the curvature tensor, are classified. These manifolds are exhausted by a special type of pp-waves, they generalize Cahen-Wallach spaces and second-order symmetric Lorentzian spaces.

In this paper we define and study the slant and semi-slant submanifolds of a Lorentzian almost paracontact manifold. Some necessary and sufficient conditions for a submanifold of a Lorentzian almost paracontact manifold to be slant are obtained. We give some examples for the slant and semi-slant submanifolds of a Loren…

The paper classifies Ricci solitons on specific Lorentzian Lie groups.