pseudo-Riemannian Research Articles

Golden slant lightlike submanifolds of golden semi-Riemannian manifolds

Golden slant lightlike submanifolds of golden semi-Riemannian manifolds

Fundamental theorems of quasi-geodesic mappings of generalized-recurrent-parabolic spaces

In previous papers we studied mappings of pseudo-Riemannian spaces being mutually quasi-geodesic and almost geodesic of the 2nd type. As a result, we arrived at the quasi-geodesic mapping f: (Vn, gij, Fih) → (Vn, gij, Fih) of spaces with an affine structure, which was called generalized-recurrent. Quasi-geodesic mappings are divided into two types: general and canonical. In this article, the fundamental issues of the theory of quasi-geodesic mappings of generalized-recurrent-parabolic spaces are considered. First, the fundamental equations of quasi-geodesic mappings are reduced to a form that allows effective investigation. Then, using a new form of the fundamental equations, we prove theorems that allow for any generalized-recurrent-parabolic space (Vn, gij, Fih) or to find all spaces (Vn, gij, Fih) onto which Vn admits a quasi-geodesic mapping of the general form, or prove that there are no such spaces.

Conformal flatness of compact three-dimensional Cotton-parallel manifolds

A three-dimensional pseudo-Riemannian manifold is called essentially conformally symmetric (ECS) if its Cotton tensor is parallel but nowhere-vanishing. In this note we prove that three-dimensional ECS manifolds must be noncompact or, equivalently, that every compact three-dimensional Cotton-parallel pseudo-Riemannian manifold must be conformally flat.

Some Results on Semi-Symmetric Spaces

We give some properties of semi-symmetric pseudo-Riemannian manifolds as an indecomposable irreducible Ricci pseudo-Riemannian manifold (i.e. the minimal polynomial of its Ricci operator is irreducible) is semi symmetric if and only if it is locally symmetric. We also show that any semi-symmetric pseudo-Riemannian manifold will be foliated. Moreover, if the metric is Lorentzian, the Ricci operator has only real eigenvalues and more precisely, on each leaf, it is diagonalizable with at most a single non zero eigenvalue or isotropic.

On Prescribed Values of the Operator of Sectional Curvature on Three-Dimensional Locally Homogeneous Lorentzian Manifolds

On Prescribed Values of the Operator of Sectional Curvature on Three-Dimensional Locally Homogeneous Lorentzian Manifolds

On Semi-Riemannian Manifolds Satisfying Some Generalized Einstein Metric Conditions

The derivation-commutator $R \cdot C - C \cdot R$ of a semi-Riemannian manifold $(M,g)$, $\dim M \geq 4$, formed by its Riemann-Christoffel curvature tensor $R$ and the Weyl conformal curvature tensor $C$, under some assumptions, can be expressed as a linear combination of $(0,6)$-Tachibana tensors $Q(A,T)$, where $A$ is a symmetric $(0,2)$-tensor and $T$ a generalized curvature tensor. These conditions form a family of generalized Einstein metric conditions. In this survey paper we present recent results on manifolds and submanifolds, and in particular hypersurfaces, satisfying such conditions.

Some curvature properties of perfect fluid spacetimes

In this paper we assume that a perfect fluid is the source of the gravitational field while analyzing the solutions to the Einstein field equations. With this new and innovative approach, we investigate some curvature properties of perfect fluid spacetimes. Firstly, we provide a necessary condition for a Lorentzian manifold to be a perfect fluid spacetime. We prove that a conformally flat perfect fluid space-time is a spacetime of quasi constant sectional curvature. Also, we acquire that a conharmonically flat perfect fluid spacetime represents the radiation era. Next, we show that if a stiff matter fluid obeys Yang’s equations, then the vorticity of the fluid vanishes. Moreover, we find that if the Ricci tensor is Killing, then the perfect fluid spacetime is (i) expansion-free and shear-free, and its flow is geodesic, however, not necessarily vorticity-free, and (ii) σ and p are constant. Besides, we establish that Ricci semi-symmetric or Ricci symmetric perfect fluid spacetimes represent either dark matter era, or phantom era. Finally, we acquire that if a perfect fluid is Ricci symmetric, then either the spacetime represents a dark matter era, or a static spacetime.

Generalized Minkowski Type Integral Formulas for Compact Hypersurfaces in Pseudo-Riemannian Manifolds

We obtain some generalized Minkowski type integral formulas for compact Riemannian (resp., spacelike) hypersurfaces in Riemannian (resp., Lorentzian) manifolds in the presence of an arbitrary vector field that we assume to be timelike in the case where the ambient space is Lorentzian. Some of these formulas generalize existing formulas in the case of conformal and Killing vector fields. We apply these integral formulas to obtain interesting results concerning the characterization of such hypersurfaces in some particular cases such as when the ambient space is Einstein admitting an arbitrary (in particular, conformal or Killing) vector field, and when the hypersurface has a constant mean curvature.

A classification of triharmonic hypersurfaces in a pseudo-Riemannian space form

The purpose of this article is to investigate triharmonic hypersurfaces Mtn in a pseudo-Riemannian space form Nt1n+1(c). Assuming that the shape operator is diagonalizable, we show that every CMC triharmonic hypersurface either has constant scalar curvature or it is minimal if cε>0 and every CMC triharmonic hypersurface is minimal if cε<0, where ε is the inner product of the unit normal vector of Mtn. Furthermore, we prove that any CMC triharmonic hypersurface with diagonalizable shape operator in a pseudo-Euclidean space Et1n+1 is actually minimal provided that zero is a principal curvature of multiplicity at most one.

A “magic” approach to octonionic Rosenfeld spaces

In his study on the geometry of Lie groups, Rosenfeld postulated a strict relation between all real forms of exceptional Lie groups and the isometries of projective and hyperbolic spaces over the (rank-2) tensor product of Hurwitz algebras taken with appropriate conjugations. Unfortunately, the procedure carried out by Rosenfeld was not rigorous, since many of the theorems he had been using do not actually hold true in the case of algebras that are not alternative nor power-associative. A more rigorous approach to the definition of all the planes presented more than thirty years ago by Rosenfeld in terms of their isometry group, can be considered within the theory of coset manifolds, which we exploit in this work, by making use of all real forms of Magic Squares of order three and two over Hurwitz normed division algebras and their split versions. Within our analysis, we find seven pseudo-Riemannian symmetric coset manifolds which seemingly cannot have any interpretation within Rosenfeld’s framework. We carry out a similar analysis for Rosenfeld lines, obtaining that there are a number of pseudo-Riemannian symmetric cosets which do not have any interpretation á la Rosenfeld.

Pseudo-Conformal actions of the Möbius group

We study compact connected pseudo-Riemannian manifolds (M,g) on which the conformal group Conf(M,g) acts essentially and transitively. We prove, in particular, that if the non-compact semi-simple part of Conf(M,g) is the Möbius group, then (M,g) is conformally flat.

Conelike radiant structures

Analogues of the classical affine-projective correspondence are developed in the context of statistical manifolds compatible with a radiant vector field. These utilize a formulation of Einstein equations for special statistical structures that generalizes the usual Einstein equations for pseudo-Riemannian metrics and is of independent interest. A conelike radiant structure is a not necessarily flat affine connection equipped with a family of surfaces that behave like the intersections of the planes through the origin with a convex cone in a real vector space. A radiant structure is a torsion-free affine connection and a vector field whose covariant derivative is the identity endomorphism. A radiant structure is conelike if for every point and every two-dimensional subspace containing the radiant vector field there is a totally geodesic surface passing through the point and tangent to the subspace. Such structures exist on the total space of any principal bundle with one-dimensional fiber and on any Lie group with a quadratic structure on its Lie algebra. The affine connection of a conelike radiant structure can be normalized in a canonical way to have antisymmetric Ricci tensor. Applied to a conelike radiant structure on the total space of a principal bundle with one-dimensional fiber this yields a generalization of the classical Thomas connection of a projective structure. The compatibility of radiant and conelike structures with metrics is investigated and yields a construction of connections for which the symmetrized Ricci curvature is a constant multiple of a compatible metric that generalizes well-known constructions of Riemannian and Lorentzian Einstein–Weyl structures over Kähler–Einstein manifolds having nonzero scalar curvature. A formulation of Einstein equations for special statistical manifolds is given that generalizes the Einstein–Weyl equations and encompasses these more general examples. There are constructed left-invariant conelike radiant structures on a Lie group endowed with a left-invariant nondegenerate bilinear form, and the case of three-dimensional unimodular Lie groups is described in detail.

A remark on convex functions and isometry groups of Lorentzian manifolds

A remark on convex functions and isometry groups of Lorentzian manifolds

On the Geometry of the Null Tangent Bundle of a Pseudo-Riemannian Manifold

When we consider a non-definite pseudo-Riemannian manifold, we obtain lightlike tangent vectors that constitute the null tangent bundle, whose fibers are lightlike cones in the corresponding tangent spaces. In this paper, we define and study a class of “g-natural” metrics on the tangent bundle of a pseudo-Riemannian manifold and we investigate the geometry of the null tangent bundle as a lightlike hypersurface equipped with an induced g-natural metric.

Weakly-Einstein conditions over four-dimensional homogenous pseudo-Riemannian manifolds

A curvature identity is studied through homogenous pseudo-Riemannian four-manifolds with nontrivial isotropy. Then, special classes of the considered spaces i.e. weakly-Einstein examples are completely classified.

On an Equation in the Ricci Solitons Theory with a Semisymmetric Connection

The study of Ricci solitons and invariant Ricci solitons with connections of various types has garnered much attention from many mathematicians. 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.&#x0D; In the previous paper, the authors studied invariant Ricci solitons with a semisymmetric connection. They are an important subclass of the class of homogeneous Ricci solitons. We obtained the classification of invariant Ricci solitons on three-dimensional Lie groups with a left-invariant Riemannian metric and a semisymmetric connection different from the Levi-Civita connection. Also, the existence of invariant Ricci solitons with a non-conformal Killing vector field was proved for the such case. Moreover, a part of the proofs was obtained using the analytical calculation software packages.&#x0D; In this paper, we investigate invariant Ricci solitons on three-dimensional nonunimodular Lie groups with a left-invariant Riemannian metric and a semisymmet-ric connection. Analytical proofs of all theorems completing the classification of such solitons are presented.

Four-dimensional compact Clifford–Klein forms of pseudo-Riemannian symmetric spaces with signature (2, 2)

We give a classification of reducible and indecomposable pseudo-Riemannian symmetric spaces with signature [Formula: see text] which admit compact Clifford–Klein forms. For this, we develop a method that applies to particular 1-connected solvable symmetric spaces. We also examine a “solvable analogue” of Kobayashi’s conjecture for reductive type and find an evidence that the assumption “reductive type” in the conjecture is crucial.

The Lorentzian Lichnerowicz conjecture for real-analytic, three-dimensional manifolds

Abstract We prove that, for a compact, 3-dimensional, real-analytic, Lorentzian manifold, if the group of conformal transformations does not preserve any metric in the conformal class, then the metric is conformally flat.

Hyperbolic cones for casual diffeomorphisms

This paper begins with introducing a new class of diffeomorphisms on pseudo-Riemannian manifolds which are called casual diffeomorphisms. We deduce a new pseudo-Riemannian metric via a casual diffeomorphism. We introduce a class of cones for a diffeomorphism, and we prove the existence of such cones for each p in a hyperbolic set of a casual diffeomorphism. As an example we find a hyperbolic casual diffeomorphism on a Schwarzschild pseudo-Riemannian manifold, and show that the whole manifold is a hyperbolic set for it.

Projective Collineations in Warped Product Manifolds and (PRS)n Manifolds

The current work first explores projective collineations on pseudo-Riemannian manifolds. Projective collineations, curvature collineations, and Ricci curvature collineations are examined in relation to one another. On warped product manifolds, the projective collineations of the form ζ=ζ1+ζ2 are investigated. We scrutinize various inheritance aspects in projective collineations from warped product manifolds to its factor manifolds. This provides, for example, a partially negative solution to Besse’s problem regarding the existence of Einstein warped product manifolds. Finally, Pseudo-Ricci symmetric space-times admitting projective collineations are investigated.