arXiv research

Study properties of semi-symmetric Lorentzian spaces, foliated manifolds.

Causal properties of Lorentzian symmetric spaces are investigated in the paper. The global hyperbolicity of the Cahen--Wallach Lorentzian symmetric spaces is proved.

In this paper, we address the problem of determining a function in terms of its orbital integrals on Lorentzian symmetric spaces. It has been solved by S. Helgason for even-dimensional isotropic Lorentzian symmetric spaces via a limit formula involving the Laplace-Beltrami operator. The result has been extended by J. O…

Third-order symmetric Lorentzian manifolds, i.e. Lorentzian manifold with zero third derivative of the curvature tensor, are classified. These manifolds are exhausted by a special type of pp-waves, they generalize Cahen-Wallach spaces and second-order symmetric Lorentzian spaces.

We solve the twistor equation on all indecomposable Lorentzian symmetric spaces explicity.

We give a complete description of semi-symmetric algebraic curvature tensors on a four-dimensional Lorentzian vector space and we use this description to determine all four-dimensional homogeneous semi-symmetric Lorentzian manifolds.

Study classifies Einstein-Yang-Mills spaces in 4D symmetric spaces.

Geodesic completeness proven for certain Lorentzian spaces.

The notion of $Γ$-symmetric space is a natural generalization of the classical notion of symmetric space based on $\z_2$-grading of Lie algebras. In our case, we consider homogeneous spaces $G/H$ such that the Lie algebra $\g$ of $G$ admits a $Γ$-grading where $Γ$ is a finite abelian group. In this work we study Rieman…

In this work, we are interested in a non symmetric homogeneous space, namely $SO(2m)/Sp(m)$. We show that this space admits a structure of $Z_2^2$-symmetric space. We describe all the non degenerated metrics and classify the Riemannian and Lorentzian ones.

We classify two-symmetric Lorentzian manifolds using methods of the theory of holonomy groups. These manifolds are exhausted by a special type of pp-waves and, like the symmetric Cahen-Wallach spaces, they have commutative holonomy.

This paper explores Lorentzian manifolds with specific connections and their symmetries.

Spaces of polynomials are shown to be Euclidean balls.

Identifies Lorentzian locally symmetric spaces where Calabi operator suffices to determine Killing operator range.

I will discuss the emergence of lorentzian symmetric spaces as supersymmetric supergravity backgrounds. I will focus on supergravity theories in dimension 11, 10, and 6, and will concentrate on the determination of the so-called maximally supersymmetric backgrounds, for which a classification exists up to local isometr…

We show that a Lorentzian homogeneous space admitting a homogeneous structure of type T1 + T3 is either a (locally) symmetric space or a singular homogeneous plane wave.

The notion of $Γ$-symmetric space is a natural generalization of the classical notion of symmetric space based on $\Z_2$-grading of Lie algebras. In our case, we consider homogeneous spaces $G/H$ such that the Lie algebra $\g$ of $G$ admits a $Γ$-grading where $Γ$ is a finite abelian group. In this work we study Rieman…

Second-order symmetric Lorentzian spaces, that is to say, Lorentzian manifolds with vanishing second derivative of the curvature tensor R, are characterized by several geometric properties, and explicitly presented. Locally, they are a product M=M_1 x M_2 where each factor is uniquely determined as follows: M_2 is a Ri…

In this work we study riemannian metrics on flag manifolds adapted to the symmetries of these homogeneous nonsymmetric spaces. We first introduce the notion of riemannian $Γ$-symmetric space when $Γ$ is a general abelian finite group, the symmetric case corresponding to $Γ=\Z_2$. We describe and study all the riemannia…

The study examines a semi-symmetric metric connection in perfect fluid space-time and phantom barriers.

We prove that each special Lorentzian holonomy group (with the exception of those including the isotropy groups of Kähler symmetric spaces with rank greater than one) can be realized as the holonomy group of a globally hyperbolic Lorentzian manifold.

We study curvature properties of four-dimensional Lorentzian manifold with two-symmetry property. We then consider Einstein-like metrics, Ricci solitons and homogeneity over these spaces.

Study how algebraic conditions on isotropy group affect Lorentzian homogeneous space geometry.

We present an approach to solvable pseudo-Riemannian symmetric spaces based on papers of M.Cahen, M.Parker and N.Wallach. Thereby we reproduce the classification of solvable symmetric triples of Lorentzian signature $(1,n-1)$ and complete the case of signature $(2,n-2)$. Moreover we discuss the topology of non-simply-c…

It is shown how one can apply the classification of the holonomy algebras of Lorentzian manifolds to solve some problems. In particular, a new proof to the classification of Lorentzian manifolds with recurrent curvature tensor is given; the classification of two-symmetric Lorentzian manifolds is explained; conformally …

The paper explores Lorentzian connections with parallel skew torsion.

In this paper we study homogeneous backgrounds of type IIB supergravity where the underlying geometry is that of a symmetric space. We determine which ten-dimensional lorentzian symmetric spaces (up to local isometry) admit such backgrounds and in about two thirds of the cases we determine fully their moduli space.

New connections found with specific torsion properties.

Since the work of Henri Cartan finite dimensional Riemannian symmetric spaces are an important subject of mathematical interest. They are related in a natural way to semisimple Lie groups. In this work we introduce and study their infinite dimensional generalization: Affine Kac-Moody symmetric spaces. Affine Kac-Moody …

Characterizes Lorentzian manifolds with semi-symmetric metric connections.

The paper explores conjugate points in Lorentzian spaces, comparing different definitions and proving related theorems.

Study on surfaces in Heisenberg group with constant mean curvature.

Departing from the observation that the Penrose limit of AdS_3 x S^3 is a group contraction in the sense of Inonu and Wigner, we explore the relation between the symmetric D-branes of AdS_3 x S^3 and those of its Penrose limit, a six-dimensional symmetric plane wave analogous to the four-dimensional Nappi--Witten space…

We classify symmetric backgrounds of eleven-dimensional supergravity up to local isometry. In other words, we classify triples (M,g,F), where (M,g) is an eleven-dimensional lorentzian locally symmetric space and F is an invariant 4-form, satisfying the equations of motion of eleven-dimensional supergravity. The possibl…

We give a summary of recent results on the explicit local form of the second-order symmetric Lorentzian manifolds in arbitrary dimension, and its global version. These spacetimes turn out to be essentially a specific subclass of plane waves.

Having developed a description of indefinite extrinsic symmetric spaces by corresponding infinitesimal objects in the preceding paper we now study the classification problem for these algebraic objects. In most cases the transvection group of an indefinite extrinsic symmetric space is not semisimple, which makes the cl…

Let $M^n$ be an $n$-dimensional umbilic-free hypersurface in the $(n+1)$-dimensional Lorentzian space form $M^{n+1}_1(c)$. Three basic invariants of $M^n$ under the conformal transformation group of $M^{n+1}_1(c)$ are a $1$-form $C$, called conformal $1$-form, a symmetric $(0,2)$ tensor $B$, called conformal second fun…

The n-dimensional Lorentzian manifolds with vanishing second covariant derivative of the Riemann tensor (2-symmetric spacetimes) are characterized and classified. The main result is that either they are locally symmetric or they have a covariantly constant null vector field, in this case defining a subfamily of Brinkma…

In this paper, a survey of the recent results about the classification of the connected holonomy groups of the Lorentzian manifolds is given. A simplification of the construction of the Lorentzian metrics with all possible connected holonomy groups is obtained. As the applications, the Einstein equation, Lorentzian man…

Indecomposable symmetric Lorentzian manifolds of non-constant curvature are called Cahen-Wallach spaces. Their isometry classes are described by continuous families of real parameters. We derive necessary and sufficient conditions for the existence of compact quotients of Cahen-Wallach spaces in terms of these paramete…

The present study initially identify the generalized symmetric connections of type $(α,β)$, which can be regarded as more generalized forms of quarter and semi-symmetric connections. The quarter and semi-symmetric connections are obtained respectively when $(α,β)=(1,0)$ and $(α,β)=(0,1)$. Taking that into account, a ne…

We study the full holonomy group of Lorentzian manifolds with a parallel null line bundle. We prove several results that are based on the classification of the restricted holonomy groups of such manifolds and provide a construction method for manifolds with disconnected holonomy which starts from a Riemannian manifold …

Lorentzian manifolds with vanishing second covariant derivative of the Riemann tensor are studied. Their existence, classification and explicit local expression are considered. Related issues and open questions are briefly commented.

We obtain Harnack estimates for a class of curvature flows in Riemannian manifolds of constant non-negative sectional curvature as well as in the Lorentzian Minkowski and de Sitter spaces. Furthermore, we prove a Harnack estimate with a bonus term for mean curvature flow in locally symmetric Riemannian Einstein manifol…

The paper classifies metrics on specific Lie groups and finds unique properties of these metrics.

We show that Lorentzian manifolds whose isometry group is of dimension at least $\frac{1}{2}n(n-1)+1$ are expanding, steady and shrinking Ricci solitons and steady gradient Ricci solitons. This provides examples of complete locally conformally flat and symmetric Lorentzian Ricci solitons which are not rigid.

Together with spaces of constant sectional curvature and products of a real line with a manifold of constant curvature, the socalled Egorov spaces and $\varepsilon$-spaces exhaust the class of $n$-dimensional Lorentzian manifolds admitting a group of isometries of dimension at least ${1/2} n(n-1)+1$, for almost all val…

Study of a series of Lorentzian structures on SL(2,R) with SO(1,1) symmetry.