pseudo-Riemannian Research Articles

Eta-Ricci solitons on Lorentzian para-Kenmotsu manifolds

This work introduces the investigation of ETA(η)-Ricci solitons on a Lorentzian para-Kenmotsu manifold. In this study, we investigate η-Ricci solitons on Lorentzian para-Kenmotsu manifolds satisfying the condition C.D=0. Additionally, we have constructed and thoroughly shown the findings about the harmonic and Weyl harmonic curvature tensor. Furthermore, our study focuses on the outcomes obtained by investigating Lorentzian para-kenmotsu manifolds that possess a η-Ricci soliton meeting the curvature requirement N•ψ=0. Finally, we provide an illustrative case of a manifold that demonstrates the presence of proper η-Ricci solitons.

Flexibility and Rigidity of Conformal Embeddings in Lorentzian Manifolds

Flexibility and Rigidity of Conformal Embeddings in Lorentzian Manifolds

The Master-Space Teleparallel Supergravity: Accelerated frames

Using Palatini’s formalism extended in a plausible fashion to the recent MS gp-Supergravity (TerKazarian, 2023c, 2024b), subject to certain rules, we reinterpret a flat MS gp-SG theory with Weitzenb¨ock torsion as the quantum field theory of Master-Space Teleparallel Supergravity (MS gp-TSG), having the gauge translation group in tangent bundle. Here the spin connection represents only inertial effects, but not gravitation at all. In order to recover the covariance, we introduce a 1-form of the Yang–Mills connection assuming values in the Lie algebra of the translation group. The Hilbert action vanishes and the gravitino action loses its spin connections, so that the accelerated reference frame has Weitzenb¨ock torsion induced by gravitinos. Due to the soldered character of the tangent bundle, torsion presents also the anholonomy of the translational covariant derivative. The gauge invariance of the tetrad provides torsion invariance under gauge transformations. The role of the Cartan-Killing metric usually comes, when it exists, from its being invariant under the group action. Here it does not exist, but we use the invariant Lorentz metric of Minkowski spacetime in its stead. The action of MS gp-TSG is invariant under local translations, under local super symmetry transformations and by construction is invariant under local Lorentz rotations and under diffeomorphisms. So that this action is invariant under the Poincar´e supergroup and under diffeomorphisms. We show the equivalence of the Teleparallel Gravity action with Hilbert action, which proves that the immediate cause of the fictitious Riemann curvature for the LeviCivita connection arises entirely due to the inertial properties of the Lorentz-rotated frame of interest. The curvature of Weitzenb¨ock connection vanishes identically, but for a tetrad involving a non-trivial translational gauge potential, the torsion is non-vanishing. We consider Weitzenb¨ock connection a kind of dual of the Levi-Civita connection, which is a connection with vanishing torsion, and non-vanishing fictitious curvature. The Weitzenb¨ock connection defines the acceleration through force equation, with torsion (or contortion) playing the role of force.

Correction: Alshehri, N.; Guediri, M. Projective Vector Fields on Semi-Riemannian Manifolds. Mathematics 2024, 12, 2914

In our recent paper [1], we stated in Theorem 10 that on an n-dimensional semi-Riemannian manifold (N,h) with n≥2, if P is a projective vector field that is also conformal, satisfying £Ph=2ψh, and the vector field ζ, dual to dψ, maintains a consistent causal character, then either P is homothetic or ζ is light-like [...]

ALMOST CONFORMAL RICCI SOLITONS ON LP-SASAKIAN MANIFOLDS

The object of the present paper is to classify almost conformal Ricci solitons on Lorentzian para-Sasakian manifolds. In this paper, we prove that such manifolds with infinitesimal contact vector field V is η-Einstein and the scalar curvature of the manifold is constant, where V is potential vector field. Moreover, we show that an almost conformal Ricci soliton on Lorentzian para-Sasakian manifold becomes a conformal Ricci soliton and it is shrinking, steady or expanding according as the dimension of the manifold is greater than 3 or equal to 3 or less than 3. Also we prove that V is strictly infinitesimal contact vector field.

The equivalence of smooth and synthetic notions of timelike sectional curvature bounds

Timelike sectional curvature bounds play an important role in spacetime geometry, both for the understanding of classical smooth spacetimes and for the study of Lorentzian (pre-)length spaces introduced by Kunzinger and Sämann [Ann. Global Anal. Geom. 54 (2018), pp. 399-447]. In the smooth setting, a bound on the sectional curvature of timelike planes can be formulated via the Riemann curvature tensor. In the synthetic setting, bounds are formulated by comparing various geometric configurations to the corresponding ones in constant curvature spaces. The first link between these notions in the Lorentzian context was established by Harris [Indiana Univ. Math. J. 31 (1982), pp. 289–308], which was instrumental in the proof of powerful results in spacetime geometry (see Beem et al. [Toponogov splitting theorem for Lorentzian manifolds, Springer, Berlin, 1985; J. Differential Geom. 22 (1985), pp. 29–42]; Galloway and Ling [Gen. Relativity Gravitation 50 (2018), p. 7]). For general semi-Riemannian manifolds, the equivalence between sectional curvature bounds and synthetic bounds was established by Alexander and Bishop [Comm. Anal. Geom. 16 (2008), pp. 251–282]; however in this approach the sectional curvatures of both timelike and spacelike planes have to be considered. In this article, we fill a gap in the literature by proving the full equivalence between sectional curvature bounds on timelike planes and synthetic timelike bounds on strongly causal spacetimes. As an essential tool, we establish Hessian comparison for the time separation and signed distance functions.

Silver Structures on the Riemann Extensions

In the present paper we deal with an $n-$dimensional differentiable manifold $M$ with a torsion-free linear connection $\nabla $. Here we study some properties of a silver structure on the cotangent bundle ${{T}^{*}}M$ equipped with the Riemannian extension ${{}^{R}}\nabla$ and obtain a necessary condition for which the silver semi-Riemannian manifold $\left( {{T}^{*}}M{{,}^{R}}\nabla ,S \right)$ to be a locally decomposable.

Isometric and anti-isometric classes of timelike minimal surfaces in Lorentz–Minkowski space

Isometric class of minimal surfaces in the Euclidean 3-space R3 has the rigidity: if two simply connected minimal surfaces are isometric, then one of them is congruent to a surface in the specific one-parameter family, called the associated family, of the other. On the other hand, the situation for surfaces with Lorentzian metrics is different. In this paper, we show that there exist two timelike minimal surfaces in the Lorentz-Minkowski 3-space R13 that are isometric each other but one of which does not belong to the congruent class of the associated family of the other. We also prove a rigidity theorem for isometric and anti-isometric classes of timelike minimal surfaces under the assumption that surfaces have no flat points.Moreover, we show how symmetries of such surfaces propagate for various deformations including isometric and anti-isometric deformations. In particular, some conservation laws of symmetry for Goursat transformations are discussed.

Characterizing Affine Vector Fields on Pseudo-Riemannian Manifolds

Characterizing Affine Vector Fields on Pseudo-Riemannian Manifolds

Geometric properties of almost pure metric plastic pseudo-Riemannian manifolds

This paper investigates the geometric and structural properties of almost plastic pseudo-Riemannian manifolds, with a specific focus on three-dimensional cases. We explore the interplay between an almost plastic structure and a pseudo-Riemannian metric, providing a comprehensive analysis of the conditions that define pure metric plastic P-Kählerian manifolds. In this context, the fundamental tensor field is symmetric and also represents another pure metric. A key finding is the necessary and sufficient condition for the integrability of the plastic structure by means of partial differential equations, which is characterized by a specific function depending solely on one variable. Furthermore, the necessary and sufficient condition for an almost pure metric plastic pseudo-Riemannian manifold to be pure metric plastic P-Kählerian is obtained. The study also reveals that the Riemannian curvature tensor vanishes under specific conditions, while the scalar curvature vanishes if a particular polynomial form is satisfied. The analysis extends to the properties of vector fields, identifying conditions under which they become Killing vector fields or form Ricci soliton structures. Additionally, we examine three-dimensional Walker manifolds, detailing the conditions for vanishing scalar curvature, Killing vector fields, and Ricci soliton structures. Our findings provide a detailed framework for understanding almost plastic manifolds, contributing to the broader field of differential geometry by clarifying the relationship between plastic structures and pseudo-Riemannian metrics.

Lie–Hamilton systems on Riemannian and Lorentzian spaces from conformal transformations and some of their applications

Abstract We propose a generalization of two classes of Lie–Hamilton systems on the Euclidean plane to two-dimensional curved spaces, leading to novel Lie–Hamilton systems on Riemannian spaces (flat 2-torus, product of hyperbolic lines, sphere and hyperbolic plane), pseudo-Riemannian spaces (anti-de Sitter, de Sitter, and Minkowski spacetimes), as well as to semi-Riemannian spaces (Newtonian or non-relativistic spacetimes). The vector fields, Hamiltonian functions, symplectic form and constants of the motion of the Euclidean classes are recovered by a contraction process. The construction is based on the structure of certain subalgebras of the so-called conformal algebras of the two-dimensional Cayley–Klein spaces. These curved Lie–Hamilton classes allow us to generalize naturally the Riccati, Kummer–Schwarz and Ermakov equations on the Euclidean plane to curved spaces, covering both the Riemannian and Lorentzian possibilities, and where the curvature can be considered as an integrable deformation parameter of the initial Euclidean system.

Wheeler-DeWitt canonical quantum gravity in hydrogenoid atoms according to de Broglie-Bohm and the geodesic hypothesis. Einstein’s Quantum Field Equation

We explore the geodesic hypothesis of orbital trajectories of the electrons in hydrogenoid atoms, in the frame of de Broglie-Bohm quantum theory. It is intended that the space-time can be curved, at very short distances, by the effect of the joint action of the energy content of the atomic system and the contribution of the electric and quantum potentials. The geodesic hypothesis would explain the non-lose of energy in the electron orbital trajectories. So we explore a conception where particles and waves interact in a closed system: the waves guide the particles and the particles generate the spacetime perturbation that acts as a wave, beyond the pilot-wave theory. We establish the equivalence, in a local neighborhood, between the electron trajectory of an hydrogenoid atom in the Minkowskian space where the de Broglie-Bohm can be cast with its movement in a Lorentzian manifold, according to the concept of metric tangent. Through the geodesic condition and the invariance of the elemental length, we establish a relationship between some components of the metrics. But as the particles in microphysics do not follow the Einstein’s field equation, we consider the 3+1 decomposition according to ADM and the quantization in the Wheeler De Witt theory and with a so-called quantum Einstein field equation, with a decomposition of spacetime into three-dimensional sheets of a spatial character. Then we derive from the moment-energy tensor a further equation between the components of the metrics. It opens the avenue to characterize the metric by an exact solution of the Einstein’s field equations.

Lorentzian G-manifolds of Constant Positive Curvature

We study the orbits arising from isometric actions of connected Lie groups on Lorentzian manifolds with constant positive curvature.

Compact locally homogeneous manifolds with parallel Weyl tensor

Abstract We construct new examples of compact ECS manifolds, that is, of pseudo-Riemannian manifolds with parallel Weyl tensor that are neither conformally flat nor locally symmetric. Every ECS manifold has rank 1 or 2, the rank being the dimension of a distinguished null parallel distribution discovered by Olszak. Previously known examples of compact ECS manifolds, in every dimension greater than 4, were all of rank 1, geodesically complete, and none of them was locally homogeneous. In contrast, our new examples — all of them geodesically incomplete — realize all odd dimensions starting from 5 and are of rank 2, as well as locally homogeneous.

The Interesting Characterizations of Some Solitons in Lorentzian Para-Kenmotsu Manifolds

The Interesting Characterizations of Some Solitons in Lorentzian Para-Kenmotsu Manifolds

Characterization of Bach and Cotton Tensors on a Class of Lorentzian Manifolds

In this work, we aim to investigate the characteristics of the Bach and Cotton tensors on Lorentzian manifolds, particularly those admitting a semi-symmetric metric ω-connection. First, we prove that a Lorentzian manifold admitting a semi-symmetric metric ω-connection with a parallel Cotton tensor is quasi-Einstein and Bach flat. Next, we show that any quasi-Einstein Lorentzian manifold admitting a semi-symmetric metric ω-connection is Bach flat.

Screen Cauchy–Riemann (SCR)-lightlike submanifolds of metallic semi-Riemannian manifolds

Screen Cauchy–Riemann (<i>SCR</i>)-lightlike submanifolds of metallic semi-Riemannian manifolds

Compact Locally Conformally Pseudo-Kähler Manifolds with Essential Conformal Transformations

A conformal transformation of a semi-Riemannian manifold is essential if there is no conformally equivalent metric for which it is an isometry. For Riemannian manifolds the existence of an essential conformal transformation forces the manifold to be conformally flat. This is false for pseudo-Riemannian manifolds, however compact examples of conformally curved manifolds with essential conformal transformation are scarce. Here we give examples of compact conformal manifolds in signature $(4n+2k,4n+2\ell)$ with essential conformal transformations that are locally conformally pseudo-Kähler and not conformally flat, where $n\ge 1$, $k, \ell \ge 0$. The corresponding local pseudo-Kähler metrics obtained by a local conformal rescaling are Ricci-flat.

Projective Vector Fields on Semi-Riemannian Manifolds

This paper explores the properties of projective vector fields on semi-Riemannian manifolds. The main result establishes that if a projective vector field P on such a manifold is also a conformal vector field with potential function ψ and the vector field ζ dual to dψ does not change its causal character, then P is homothetic, or ζ is a light-like vector field. Additionally, it is shown that a complete Riemannian manifold admits a projective vector field that is also conformal and non-Killing if and only if it is locally Euclidean. The paper also presents other results related to the characterization of Killing and parallel vector fields using the Ricci curvature and the Hessian of the function given by the inner product of the vector field.

On Spacelike Hypersurfaces in Generalized Robertson–Walker Spacetimes

This paper investigates generalized Robertson–Walker (GRW) spacetimes by analyzing Riemannian hypersurfaces within pseudo-Riemannian warped product manifolds of the form (M¯,g¯), where M¯=R×fM and g¯=ϵdt2+f2(t)gM. We focus on the scalar curvature of these hypersurfaces, establishing upper and lower bounds, particularly in the case where (M¯,g¯) is an Einstein manifold. These bounds facilitate the characterization of slices in GRW spacetimes. In addition, we use the vector field ∂t and the so-called support function θ to derive generalized Minkowski-type integral formulas for compact Riemannian and spacelike hypersurfaces. These formulas are applied to establish, under certain conditions, results concerning the existence or non-existence of such compact hypersurfaces with scalar curvature, either bounded from above or below.