pseudo-Riemannian Research Articles

Generalized Ricci Solitons on Three-Dimensional Lorentzian Walker Manifolds

In this paper, we characterize the generalized Ricci soliton equation on the three-dimensional Lorentzian Walker manifolds. We prove that every generalized Ricci soliton with C, beta ,mu ne 0 on a three-dimensional Lorentzian Walker manifold is steady. Moreover, non-trivial solutions for strictly Lorentzian Walker manifolds are derived. Finally, we give some conditions on the defining function f under which a generalized Ricci soliton on a three-dimensional Lorentzian Walker manifold to be gradient.

Certain Results on the Lifts from an LP-Sasakian Manifold to Its Tangent Bundle Associated with a Quarter-Symmetric Metric Connection

The purpose of this study is to examine the complete lifts from the symmetric and concircular symmetric n-dimensional Lorentzian para-Sasakian manifolds (briefly, (LPS)n) to its tangent bundle TM associated with a Riemannian connection DC and a quarter-symmetric metric connection (QSMC) D¯C.

Properties of Anti-Invariant Submersions and Some Applications to Number Theory

In this article, we investigate anti-invariant Riemannian and Lagrangian submersions onto Riemannian manifolds from the Lorentzian para-Sasakian manifold. We demonstrate that, for these submersions, horizontal distributions are not integrable and their fibers are not totally geodesic. As a result, they are not totally geodesic maps. The harmonicity of such submersions is also examined. We specifically prove that they are not harmonic when the Reeb vector field is horizontal. Finally, we provide an illustration of our findings and mention some number-theoretic applications for the same submersions.

Quasi hemi-slant pseudo-Riemannian submersions in para-complex geometry

We introduce a new class of pseudo-Riemannian submersions which are called quasi hemi-slant pseudo-Riemannian submersions from para-Kaehler manifolds to pseudo-Riemannian manifolds as a natural generalization of slant submersions, semi-invariant submersions, semi-slant submersions and hemislant Riemannian submersions in our study. Also, we give non-trivial examples of such submersions. Further, some geometric properties with two types of quasi hemi-slant pseudo-Riemannian submersions are investigated

An Algebraic QFT Approach to the Wetterich Equation on Lorentzian Manifolds

We discuss the scaling of the effective action for the interacting scalar quantum field theory on generic spacetimes with Lorentzian signature and in a generic state (including vacuum and thermal states, if they exist). This is done constructing a flow equation, which is very close to the renown Wetterich equation, by means of techniques recently developed in the realm of perturbative Algebraic Quantum Field theory (pAQFT). The key ingredient that allows one to obtain an equation which is meaningful on generic Lorentzian backgrounds is the use of a local regulator, which keeps the theory covariant. As a proof of concept, the developed methods are used to show that non-trivial fixed points arise in quantum field theories in a thermal state and in the case of quantum fields in the Bunch–Davies state on the de Sitter spacetime.

Global Lipschitz stability for inverse problems of wave equations on Lorentzian manifolds

We prove global Lipschitz stability for an inverse source problem of a system of wave equations on a Lorentzian manifold. The method used in this paper is widely known as the Bukhgeim–Klibanov method. However, the conventional method is not sufficient for the application to the hyperbolic partial differential equation with time-dependent coefficients to obtain the Lipschitz stability, and further innovations are needed. In this paper, we present an improved global Carleman estimate and an energy estimate to obtain the Lipschitz stability.

Effect of [formula omitted]-curvature tensor in spacetimes and [formula omitted]-gravity

First, we prove that a Lorentzian manifold is of constant curvature if and only if W2-curvature tensor vanishes. We investigate W2-flat perfect fluid spacetime as a solution of fG-gravity theory and we obtain a relation among deceleration, jerk, and snap parameters in a W2-flat spacetime using Friedmann–Robertson–Walker metric. Several energy conditions in terms of Ricci scalar are investigated with the models fG=a1Gn+b1a2Gn+b2 and fG=log(αGn+βe−Gα). For these models, the weak, null and dominant energy conditions are satisfied, while the strong energy condition is violated, which is good agreement with the recent observational studies and this reveals that the current Universe is in accelerating phase.

The Linear CS/WZW Bulk/Boundary System in AQFT

This paper constructs in the framework of algebraic quantum field theory (AQFT) the linear Chern–Simons/Wess–Zumino–Witten system on a class of 3-manifolds M whose boundary ∂M\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial M$$\\end{document} is endowed with a Lorentzian metric. It is proven that this AQFT is equivalent to a dimensionally reduced AQFT on a 2-dimensional manifold B, whose restriction to the 1-dimensional boundary ∂B\\documentclass[12pt]{minimal} \\usepackage{amsmath} \\usepackage{wasysym} \\usepackage{amsfonts} \\usepackage{amssymb} \\usepackage{amsbsy} \\usepackage{mathrsfs} \\usepackage{upgreek} \\setlength{\\oddsidemargin}{-69pt} \\begin{document}$$\\partial B$$\\end{document} is weakly equivalent to a chiral free boson.

Paracausal deformations of Lorentzian metrics and Møller isomorphisms in algebraic quantum field theory

Given a pair of normally hyperbolic operators over (possibnly different) globally hyperbolic spacetimes on a given smooth manifold, the existence of a geometric isomorphism, called Møller operator, between the space of solutions is studied. This is achieved by exploiting a new equivalence relation in the space of globally hyperbolic metrics, called paracausal relation. In particular, it is shown that the Møller operator associated to a pair of paracausally related metrics and normally hyperbolic operators also intertwines the respective causal propagators of the normally hyperbolic operators and it preserves the natural symplectic forms on the space of (smooth) initial data. Finally, the Møller map is lifted to a *-isomorphism between (generally off-shell) CCR-algebras. It is shown that the Wave Front set of a Hadamard bidistribution (and of a Hadamard state in particular) is preserved by the pull-back action of this *-isomorphism.

Einstein’s equations and the pseudo-entropy of pseudo-Riemannian information manifolds

Einstein’s equations and the pseudo-entropy of pseudo-Riemannian information manifolds

On Lightlike Submanifolds of Poly-Norden Semi-Riemannian Manifolds

In this paper, we introduce and examine invariant and semi-invariant lightlike submanifolds of a poly-Norden semi-Riemannian manifold. Also, we obtain some examples of such types submanifolds and study the conditions for both integrability and totally geodesic foliation description of distributions.

On Quasi Hemi-Slant Submersions

The paper deals with the notion of quasi hemi-slant submersions from Lorentzian para Sasakian manifolds onto Riemannian manifolds. These submersions are generalization of hemi-slant submersions and semi-slant submersions. In this paper, we also study the geometry of leaves of distributions which are involved in the definition of the submersion. Further, we obtain the conditions for such distributions to be integrable and totally geodesic. Moreover, we also give the characterization theorems for proper quasi hemi-slant submersions and provide some examples of it.

Pairs of Associated Yamabe Almost Solitons with Vertical Potential on Almost Contact Complex Riemannian Manifolds

Almost contact complex Riemannian manifolds, also known as almost contact B-metric manifolds, are, in principle, equipped with a pair of mutually associated pseudo-Riemannian metrics. Each of these metrics is specialized as a Yamabe almost soliton with a potential collinear to the Reeb vector field. The resulting manifolds are then investigated in two important cases with geometric significance. The first is when the manifold is of Sasaki-like type, i.e., its complex cone is a holomorphic complex Riemannian manifold (also called a Kähler–Norden manifold). The second case is when the soliton potential is torse-forming, i.e., it satisfies a certain recurrence condition for its covariant derivative with respect to the Levi–Civita connection of the corresponding metric. The studied solitons are characterized. In the three-dimensional case, an explicit example is constructed, and the properties obtained in the theoretical part are confirmed.

Resonances and residue operators for pseudo-Riemannian hyperbolic spaces

For any pseudo-Riemannian hyperbolic space X over R,C,H or O, we show that the resolvent R(z)=(□−zId)−1 of the Laplace–Beltrami operator −□ on X can be extended meromorphically across the spectrum of □ as a family of operators Cc∞(X)→D′(X). Its poles are called resonances and we determine them explicitly in all cases. For each resonance, the image of the corresponding residue operator in D′(X) forms a representation of the isometry group of X, which we identify with a subrepresentation of a degenerate principal series. Our study includes in particular the case of even functions on de Sitter and Anti-de Sitter spaces.For Riemannian symmetric spaces analogous results were obtained by Miatello–Will and Hilgert–Pasquale. The main qualitative differences between the Riemannian and the non-Riemannian setting are that for non-Riemannian spaces the resolvent can have poles of order two, it can have a pole at the branching point of the covering to which R(z) extends, and the residue representations can be infinite-dimensional.

Lorentzian manifolds properly isometrically embeddable in Minkowski spacetime

I characterize the Lorentzian manifolds properly isometrically embeddable in Minkowski spacetime (i.e., the Lorentzian submanifolds of Minkowski spacetime that are also closed subsets). Moreover, I prove that the Lorentzian manifolds that can be properly conformally embedded in Minkowski spacetime coincide with the globally hyperbolic spacetimes. Finally, by taking advantage of the embedding, I obtain an infinitesimal version of the distance formula.

Probing time orientability of spacetime

In general relativity, cosmology and quantum field theory, spacetime is assumed to be an orientable manifold endowed with a Lorentz metric that makes it spatially and temporally orientable. The question as to whether the laws of physics require these orientability assumptions is ultimately of observational or experimental nature, or the answer might come from a fundamental theory of physics. The possibility that spacetime is time non-orientable lacks investigation, and so should not be dismissed straightaway. In this paper, we argue that it is possible to locally access a putative time non-orientability of Minkowski empty spacetime by physical effects involving quantum vacuum electromagnetic fluctuations. We set ourselves to study the influence of time non-orientability on the stochastic motions of a charged particle subject to these electromagnetic fluctuations in Minkowski spacetime equipped with a time non-orientable topology and with its time orientable counterpart. To this end, we introduce and derive analytic expressions for a statistical time orientability indicator. Then we show that it is possible to pinpoint the time non-orientable topology through an inversion pattern displayed by the corresponding orientability indicator, which is absent when the underlying manifold is time orientable.

Decomposable (5, 6)-solutions in eleven-dimensional supergravity

We present decomposable (5, 6)-solutions M̃1,4×M6 in eleven-dimensional supergravity by solving the bosonic supergravity equations for a variety of non-trivial flux forms. Many of the bosonic backgrounds presented here are induced by various types of null flux forms on products of certain totally Ricci-isotropic Lorentzian Walker manifolds and Ricci-flat Riemannian manifolds. These constructions provide an analogy of the work performed by Chrysikos and Galaev [Classical Quantum Gravity 37, 125004 (2020)], who made similar computations for decomposable (6, 5)-solutions. We also present bosonic backgrounds that are products of Lorentzian Einstein manifolds with a negative Einstein constant (in the “mostly plus” convention) and Riemannian Kähler–Einstein manifolds with a positive Einstein constant. This conclusion generalizes a result of Pope and van Nieuwenhuizen [Commun. Math. Phys. 122, 281–292 (1989)] concerning the appearance of six-dimensional Kähler–Einstein manifolds in eleven-dimensional supergravity. In this setting, we construct infinitely many non-symmetric decomposable (5, 6)-supergravity backgrounds by using the infinitely many Lorentzian Einstein–Sasakian structures with a negative Einstein constant on the 5-sphere, known from the work of Boyer et al. [Commun. Math. Phys. 262, 177–208 (2006)].

Characterization of Ricci Almost Soliton on Lorentzian Manifolds

Ricci solitons (RS) have an extensive background in modern physics and are extensively used in cosmology and general relativity. The focus of this work is to investigate Ricci almost solitons (RAS) on Lorentzian manifolds with a special metric connection called a semi-symmetric metric u-connection (SSM-connection). First, we show that any quasi-Einstein Lorentzian manifold having a SSM-connection, whose metric is RS, is Einstein manifold. A similar conclusion also holds for a Lorentzian manifold with SSM-connection admitting RS whose soliton vector Z is parallel to the vector u. Finally, we examine the gradient Ricci almost soliton (GRAS) on Lorentzian manifold admitting SSM-connection.

η-Ricci solitons on Lorentzian β-Kenmotsu manifold

휂-Ricci solitons on Lorentzian 훽-Kenmotsu manifold are considered an manifolds satisfying certain curvature Conditions, R(ξ,X).S=0, S(ξ,X).R=0, W2(ξ,X).S=0,S(ξ,X).W2=0 We proved that in Lorentzian β-Kenmotsu manifold (M,φ,ξ,η,푔). Then the existence of an 휂-Ricci solitons implies that M is Einstein manifold and if the Ricci curvature tensor satisfies, S(ξ,X).R=0, then Ricci solitons M is steady. If the condition 휇=0, then 휆=0, which shows that 휆is steady

Almost Robinson geometries

We investigate the geometry of almost Robinson manifolds, Lorentzian analogues of almost Hermitian manifolds, defined by Nurowski and Trautman as Lorentzian manifolds of even dimension equipped with a totally null complex distribution of maximal rank. Associated to such a structure, there is a congruence of null curves, which, in dimension four, is geodesic and non-shearing if and only if the complex distribution is involutive. Under suitable conditions, the distribution gives rise to an almost Cauchy–Riemann structure on the leaf space of the congruence. We give a comprehensive classification of such manifolds on the basis of their intrinsic torsion. This includes an investigation of the relation between an almost Robinson structure and the geometric properties of the leaf space of its congruence. We also obtain conformally invariant properties of such a structure, and we finally study an analogue of so-called generalised optical geometries as introduced by Robinson and Trautman.