pseudo-Riemannian Research Articles

Triharmonic hypersurfaces with constant mean curvature in pseudo-Riemannian space forms

In this paper, triharmonic hypersurfaces with constant mean curvature in pseudo-Riemannian space forms are studied. Under the assumption that the shape operator is diagonalizable, we first classify completely the nonminimal hypersurfaces with at most two distinct principal curvatures and give some examples of non-biharmonic triharmonic hypersurfaces. Then, we prove that the hypersurfaces with at most four distinct principal curvatures have constant scalar curvature. As a consequence, we obtain that such triharmonic hypersurfaces in pseudo-Euclidean spaces are minimal, which gives an affirmative partial solution to the generalized Chen's conjecture in [21].

Weyl-ambient geometries

Weyl geometry is a natural extension of conformal geometry with Weyl covariance mediated by a Weyl connection. We generalize the Fefferman-Graham (FG) ambient construction for conformal manifolds to a corresponding construction for Weyl manifolds. We first introduce the Weyl-ambient metric motivated by the Weyl-Fefferman-Graham (WFG) gauge. From a top-down perspective, we show that the Weyl-ambient space as a pseudo-Riemannian geometry induces a codimension-2 Weyl geometry. Then, from a bottom-up perspective, we start from promoting a conformal manifold into a Weyl manifold by assigning a Weyl connection to the principal R+-bundle realizing a Weyl structure. We show that the Weyl structure admits a well-defined initial value problem, which determines the Weyl-ambient metric. Through the Weyl-ambient construction, we also investigate Weyl-covariant tensors on the Weyl manifold and define extended Weyl-obstruction tensors explicitly.

Chen Inequalities for Isotropic Submanifolds in Pseudo-Riemannian Space Forms

The class of isotropic submanifolds in pseudo-Riemannian manifolds is a distinguished family of submanifolds; they have been studied by several authors. In this article we establish Chen inequalities for isotropic immersions. An example of an isotropic immersion for which the equality case in the Chen first inequality holds is given.

Inequalities on Isotropic Submanifolds in Pseudo-Riemannian Space Forms

Spacelike and timelike isotropic submanifolds of pseudo-Riemannian spaces have interesting properties, with important applications in Mathematics and Physics. The article presents inequalities for isotropic spacelike and timelike submanifolds of pseudo-Riemannian space forms and isotropic Lorentzian submanifolds are also considered.

A Note on Some Generalized Curvature Tensor

For any semi-Riemannian manifold (M, g) we define some generalized curvature tensor E as a linear combination of Kulkarni-Nomizu products formed by the metric tensor, the Ricci tensor and its square of given manifold. That tensor is closely related to quasi-Einstein spaces, Roter spaces and some Roter type spaces.

Professor Krishan L. Duggal: A Biographical Note

In this note, we present a biographical sketch of the life and academic contributions of late Professor Krishan L. Duggal. His contributions span from Riemannian and Lorentzian geometries of manifolds with various structural groups of the tangent bundle, Lightlike curves and submanifolds, Cauchy-Riemann geometry, Symmetries of semi-Riemannian manifolds, to Killing horizons. In particular, his approach to the study of lightlike submanifolds is remarkable and drawn considerable interest of many geometers.

On Isometric Immersions of Null Manifolds into Semi-Riemannian Space Forms of Arbitrary Index

A null manifold is a differentiable manifold M endowed with a degenerate metric tensor g. In this work we provide sufficient conditions for a null manifold to be isometrically immersed as a hypersurface into a simple connected semi-Riemannian manifold of constant sectional curvature c and index q

Null-Projectability of Levi-Civita Connections

We study the natural property of projectability of a torsion-free connection along a foliation on the underlying manifold, which leads to a projected torsion-free connection on a local leaf space, focusing on projectability of Levi-Civita connections of pseudo-Riemannian metrics along foliations tangent to null parallel distributions. For the neutral metric signature and mid-dimensional distributions, Afifi showed in 1954 that projectability of the Levi-Civita connection characterizes, locally, the case of Patterson and Walker’s Riemann extension metrics. We extend this correspondence to null parallel distributions of any dimension, introducing a suitable generalization of Riemann extensions.

A microscopic analogue of the BMS group

We consider a microscopic analogue of the BMS analysis of asymptotic symmetries by analysing universal geometric structures on infinitesimal tangent light cones. Thereby, two natural microscopic symmetry groups arise: a non-trivially represented Lorentz group and a BMS-like group. The latter has a rich mathematical structure, since it contains the former as a non-canonical subgroup, next to infinitely many other Lorentz subgroups. None of those Lorentz subgroups appears to be intrinsically preferred, and hence, the microscopic BMS-like group constitutes a natural symmetry group for infinitesimal tangent light cones. We compare our investigation with the classical BMS analysis and show, that the microscopic BMS-like group is a gauge group for the bundle of null vectors. Motivated by the various applications of the original BMS group, our findings could have interesting implications: they identify a geometric structure that could be suitable for a bulk analysis of gravitational waves, they suggest a possible enlargement of the fundamental gauge group of gravity and they motivate the possibility of an interrelation between the UV structure of gauge theories, gravitational memory effects and BMS-like symmetries. Also, our results imply, that BMS-like groups arise not only as macroscopic, asymptotic symmetry groups in cosmology, but describe also a fundamental and seemingly unknown microscopic symmetry of pseudo-Riemannian geometry.

An anisotropic inverse mean curvature flow for spacelike graphic hypersurfaces with boundary in the Lorentz manifold bmM^n×mathbbR

令$M^{n}\times\mathbb{R}$是$(n+1)$-维Lorentz流形, 其中$M^{n}$为Ricci曲率非负、曲率张量及其一阶协变导数有界的、且含有一个极点的$n$-维($n\geq 2$)完备黎曼流形，$\mathbb{R}$是1维欧氏空间，$\Omega$是$M^{n}\times\mathbb{R}$中的有界类空凸超曲面.本文考虑了$M^{n}\times\mathbb{R}$中定义在$\Omega$上的类空图超曲面，沿着具有退化Neumann边值条件的一类各向异性逆平均曲率流的演化过程，可证明该流的长时间存在性.此外，通过适当的伸缩变换后我们可以得到，当$t\rightarrow\infty$时，演化类空图超曲面光滑地收敛为定义在$\Omega$上的一个常值函数的图曲面.

Parallel spinor flows on three-dimensional Cauchy hypersurfaces

The three-dimensional parallel spinor flow is the evolution flow defined by a parallel spinor on a globally hyperbolic Lorentzian four-manifold. We prove that, despite the fact that Lorentzian metrics admitting parallel spinors are not necessarily Ricci flat, the parallel spinor flow preserves the vacuum momentum and Hamiltonian constraints and therefore the Einstein and parallel spinor flows coincide on common initial data. Using this result, we provide an initial data characterization of parallel spinors on Ricci flat Lorentzian four-manifolds, which in turn yields the first initial data characterization of Ricci-flat pp-waves. Furthermore, we explicitly solve the left-invariant parallel spinor flow on simply connected Lie groups, obtaining along the way necessary and sufficient conditions for the flow to be immortal. These are, to the best of our knowledge, the first non-trivial examples of evolution flows of parallel spinors. Finally, we use some of these examples to construct families of -Einstein cosymplectic structures and to produce solutions to the left-invariant Ricci flow in three dimensions. This suggests the intriguing possibility of using first-order hyperbolic spinorial flows to construct special solutions of curvature flows in Riemannian signature.

Families of Geodesic Orbit Spaces and Related Pseudo-Riemannian Manifolds

Families of Geodesic Orbit Spaces and Related Pseudo-Riemannian Manifolds

Complex actions and causality violations: applications to Lorentzian quantum cosmology

For the construction of the Lorentzian path integral for gravity one faces two main questions: firstly, what configurations to include, in particular whether to allow Lorentzian metrics that violate causality conditions. And secondly, how to evaluate a highly oscillatory path integral over unbounded domains. Relying on Picard–Lefschetz theory to address the second question for discrete Regge gravity, we will illustrate that it can also answer the first question. To this end we will define the Regge action for complexified variables and study its analytical continuation. Although there have been previously two different versions defined for the Lorentzian Regge action, we will show that the complex action is unique. More precisely, starting from the different definitions for the action one arrives at equivalent analytical extensions. The difference between the two Lorentzian versions is only realized along branch cuts which arise for a certain class of causality violating configurations. As an application we discuss the path integral describing a finite evolution step of the discretized de Sitter Universe. We will in particular consider an evolution from vanishing to finite scale factor, for which the path integral defines the no-boundary wave function.

Gravitational Decoupling as a Generator of Curvature and Matter Distribution

AbstractIn this work, we generate curved Lorentzian manifolds from a flat Minkowskian space–time, through the Gravitational Decoupling by the Minimal Geometric Deformation method interpreting the decoupling sector as a generator of curvature and matter distribution. In particular, we obtain some new wormhole geometries by imposing either physically relevant equations of state or certain physically–motivated geometric constraints. These solutions are analyzed in detail. For all these solutions, we study the associated energy conditions, the geometric behavior and construct their embedding diagrams.

Conformal η-Ricci soliton in Lorentzian para Kenmotsu manifolds

The objective of the present paper is to study conformal η-Ricci soliton on Lorentzian Para-Kenmotsu manifolds with some curvature conditions. We study Concircular curvature tensor, Quasi conformal curvature tensor, Codazi type of Ricci tensor and cyclic parallel Ricci tensor in Lorentzian Para-Kenmotsu manifolds. At last we give examples of such manifolds.

Pseudo-Kähler and pseudo-Sasaki structures on Einstein solvmanifolds

The aim of this paper is to construct left-invariant Einstein pseudo-Riemannian Sasaki metrics on solvable Lie groups. We consider the class of mathfrak {z}-standard Sasaki solvable Lie algebras of dimension 2n+3, which are in one-to-one correspondence with pseudo-Kähler nilpotent Lie algebras of dimension 2n endowed with a compatible derivation, in a suitable sense. We characterize the pseudo-Kähler structures and derivations giving rise to Sasaki–Einstein metrics. We classify mathfrak {z}-standard Sasaki solvable Lie algebras of dimension le 7 and those whose pseudo-Kähler reduction is an abelian Lie algebra. The Einstein metrics we obtain are standard, but not of pseudo-Iwasawa type.

Lightlike Hypersurfaces in Semi-Riemmanian Manifolds Admitting Affine Conformal Vector Fields

Lightlike hypersurfaces with integrable screen distributions are very important as far as lightlike geometry is concerned. They include, among others, screen conformal and screen totally umbilic ones. In this paper, we show that any lightlike hypersurface of a semi-Riemannian manifold admitting a certain closed affine conformal vector field has an integrable screen distribution. Several examples are furnished in support of the main results.

Almost η-Ricci Solitons on the Pseudosymmetric Lorentzian Para-Kenmotsu Manifolds

In this paper, we consider Lorentzian para-Kenmotsu manifold admitting almost $\eta-$Ricci solitons by virtue of some curvature tensors. Ricci pseudosymmetry concepts of Lorentzian para-Kenmotsu manifolds admitting $\eta-$Ricci soliton have introduced according to the choice of some curvature tensors such as Riemann, concircular, projective, $\mathcal{M-}$projective, $W_{1}$ and $W_{2}.$ After then, according to the choice of the curvature tensors, necessary conditions are given for Lorentzian para-Kenmotsu manifold admitting $\eta-$Ricci soliton to be Ricci semisymmetric. Then some characterizations are given and classifications have made under the some conditions.

Symmetric Ricci Flows of Semisymmetric Connections on Three-Dimensional Metrical Lie Groups: An Analysis

The study of Ricci flows, which describe the deformation of (pseudo) Riemannian metrics on a manifold, and their solutions, Ricci solitons, has garnered much attention from mathematicians. However, previous studies have typically focused on manifolds with Levi-Civita connections. This paper breaks new ground by considering manifolds with semisymmetric connections, which also include the Levi-Civita connection. Metric connections with vector torsion, or semisymmetric connections, were first studied by E. Cartan on (pseudo) Riemannian manifolds. Later, K. Yano and I. Agricola studied tensor fields and geodesic lines of such connections, while P.N. Klepikov,
 E.D. Rodionov, and O.P. Khromova considered the Einstein equation of semisymmetric connections on three-dimensional locally homogeneous (pseudo) Riemannian manifolds. Because the Ricci tensor of a semisymmetric connection is not symmetric in general, we focus on studying the symmetric and skew-symmetric parts of the Ricci tensor. Specifically, we investigate symmetric Ricci flows on three-dimensional Lie groups with J. Milnor's left-invariant (pseudo) Riemannian metric and E. Cartan's semisymmetric connection.

Curved Versions of the Ovsienko–Redou Operators

Abstract We construct a family of conformally covariant bidifferential operators on pseudo-Riemannian manifolds. Our construction is analogous to the construction of Graham–Jenne–Mason–Sparling of conformally covariant differential operators via tangential powers of the Laplacian in the Fefferman–Graham ambient space. In fact, we completely classify the tangential bidifferential operators on the ambient space, which are expressed purely in terms of the ambient Laplacian. This gives a curved analogue of the classification, due to Ovsienko–Redou and Clerc, of conformally invariant bidifferential operators on the sphere. As an application, we construct a large class of formally self-adjoint conformally invariant differential operators.