pseudo-Riemannian Research Articles

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.

Uniqueness and Non-Uniqueness Results for Spacetime Extensions

Abstract Given a function $f: A \to{\mathbb{R}}^{n}$ of a certain regularity defined on some open subset $A \subseteq{\mathbb{R}}^{m}$, it is a classical problem of analysis to investigate whether the function can be extended to all of ${\mathbb{R}}^{m}$ in a certain regularity class. If an extension exists and is continuous, then certainly it is uniquely determined on the closure of $A$. A similar problem arises in general relativity for Lorentzian manifolds instead of functions on ${\mathbb{R}}^{m}$. It is well-known, however, that even if the extension of a Lorentzian manifold $(M,g)$ is analytic, various choices are in general possible at the boundary. This paper establishes a uniqueness condition for extensions of globally hyperbolic Lorentzian manifolds $(M,g)$ with a focus on low regularities: any two extensions that are anchored by an inextendible causal curve $\gamma : [-1,0) \to M$ in the sense that $\gamma $ has limit points in both extensions must agree locally around those limit points on the boundary as long as the extensions are at least locally Lipschitz continuous. We also show that this is sharp: anchored extensions that are only Hölder continuous do in general not enjoy this local uniqueness result.

ON QUASI BI-SLANT RIEMANNIAN MAPS FROM LORENTZIAN PARA SASAKIAN MANIFOLDS

We first introduce quasi bi-slant Riemannian maps and study such Riemannian maps from Lorentzian para Sasakian manifolds into Riemannian manifolds. We give necessary and sufficient conditions for the integrability of the distributions which are involved in the definition of the quasi bi-slant Riemannian map and investigate their leaves. We also obtain totally geodesic conditions for such maps. Moreover, we provide some examples for this notion.

Magnetic trajectories in Walker 3-manifolds

A Walker manifold is a semi-Riemannian manifold admitting a parallel lightlike distribution. In this paper, geodesics, Killing magnetic trajectories and contact magnetic trajectories in Walker 3-manifolds are considered. Also, magnetic trajectories in foliated Walker 3-manifold are studied.

Degrees of Freedom in Modified Teleparallel Gravity

I discuss the issue of degrees of freedom in modified teleparallel gravity. These theories do have an extra structure on top of the usual (pseudo)Riemannian manifold, that of a flat parallel transport. This structure is absolutely abstract and unpredictable (pure gauge) in GRequivalent models, however, it becomes physical upon modifications. The problem is that, in the most popular models, this local symmetry is broken but not stably So, hence the infamous strong coupling issues. The Hamiltonian analyses become complicated and with contradictory results. A funny point is that what we see in available linear perturbation treatments of f (T) gravity is much closer to the analysis with less dynamical degrees of freedom which has got a well-known mistake in it, while the more accurate work predicts much more of dynamics than what has ever been seen till now. I discuss possible reasons behind this puzzle, and also argue in favor of studying the most general New GR models which are commonly ignored due to suspicion of ghosts.

Schrödinger evolution of a scalar field in Riemannian and pseudo-Riemannian expanding metrics

We study the quantum field theory (QFT) of a scalar field in the Schrödinger picture in the functional formulation. We derive a formula for the evolution kernel in a flat expanding metric. We discuss a transition between Riemannian and pseudo-Riemannian metrics (signature inversion). We express the real time Schrödinger evolution by the Brownian motion. We discuss the Feynman integral for a scalar field in a radiation background. We show that the unitary Schrödinger evolution for positive time can go over for negative time into a dissipative evolution as a consequence of the imaginary value of . The time evolution remains unitary if in the Hamiltonian is replaced by .

Control of waves on Lorentzian manifolds with curvature bounds

We prove boundary controllability results for wave equations (with lower-order terms) on Lorentzian manifolds with time-dependent geometry satisfying suitable curvature bounds. The main ingredient is a novel global Carleman estimate on Lorentzian manifolds that is supported in the exterior of a null (or characteristic) cone, which leads to both an observability inequality and bounds for the corresponding constant. The Carleman estimate also yields a unique continuation result on the null cone exterior, which has applications toward inverse problems for linear waves on Lorentzian backgrounds.

Generalized Bach solitons on (κ,μ)′-almost-Kenmotsu manifolds and its applications in perfect fluid spacetimes

The principal aim of this paper is to find the expression of *-Bach tensors on [Formula: see text]-almost-Kenmotsu manifolds. It is shown that if the metric of a [Formula: see text]-almost-Kenmotsu manifold admits a generalized closed *-Bach soliton, then the metric is *-Bach flat under certain restrictions. We also prove that the metric of a generalized gradient *-Bach soliton on a [Formula: see text]-almost-Kenmotsu manifold is *-Bach flat, for a soliton function which is invariant under the Reeb vector field. Further, we characterize generalized *-Bach solitons with trace-free *-Bach tensors on [Formula: see text]-almost-Kenmotsu manifolds and results are verified by an example. As an application of these solitons, we study generalized almost-Bach solitons whose metrics are associated with the Lorentzian metric of the perfect fluid spacetimes.