arXiv research

The paper classifies 3D Lorentzian $Ein(2)$ Lie groups.

Study left-invariant Codazzi tensors and harmonic curvature on Lorentzian Lie groups.

The paper classifies Ricci solitons on specific Lorentzian Lie groups.

Study classifies Riemann solitons on specific 3D Lorentzian groups.

Classifies Ricci collineations on specific 3D Lorentzian Lie groups.

Complete description of flat Lorentzian Lie groups solved.

Classifies Ricci solitons on specific Lorentzian Lie groups.

Study left invariant Lorentzian metrics on 3D non-unimodular Lie groups.

A Lorentzian flat Lie group is a Lie group $G$ with a flat left invariant metric $μ$ with signature $(1,n-1)=(-,+,\ldots,+)$. The Lie algebra $\mathfrak{g}=T_eG$ of $G$ endowed with $\langle\;,\;\rangle=μ(e)$ is called flat Lorentzian Lie algebra. It is known that the metric of a flat Lorentzian Lie group is geodesical…

The paper classifies Lorentzian Lie groups based on Codazzi tensors and quasi-statistical structures.

Study finds all conformal Ricci collineations on specific 3D Lorentzian groups.

Classifies a specific type of Lie groups related to Einstein geometry.

The study classifies special flows on specific geometric groups.

We call a connected Lie group endowed with a left-invariant Lorentzian flat metric Lorentzian flat Lie group. In this Note, we determine all Lorentzian flat Lie groups admitting a timelike left-invariant Killing vector field. We show that these Lie groups are 2-solvable and unimodular and hence geodesically complete. M…

In this paper, we study Lorentzian left invariant Einstein metrics on nilpotent Lie groups. We show that if the center of such Lie groups is degenerate then they are Ricci-flat and their Lie algebras can be obtained by the double extension process from an abelian Euclidean Lie algebra. We show that all nilpotent Lie gr…

Study of special Lorentzian Lie groups with 4D isometry group, finding all are non-gradient expanding Ricci solitons.

Study finds all Ricci collineations for specific connections on 3D Lorentzian groups.

Study on special Lie groups with Lorentzian metrics.

Defines and classifies algebraic Schouten solitons in 3D Lorentzian Lie groups.

Study explores solvable Lie groups' actions on closed Lorentzian manifolds.

We call the Lie algebra of a Lie group with a left invariant pseudo-Riemannian flat metric pseudo-Riemannian flat Lie algebra. We give a new proof of a classical result of Milnor on Riemannian flat Lie algebras. We reduce the study of Lorentzian flat Lie algebras to those with trivial center or those with degenerate ce…

New conformally Einstein metrics on Heisenberg group found.

In this paper, we compute the Wanas tensor associated to canonical connections on three-dimensional Lorentzian Lie groups with some product structure. We define algebraic Wanas solitons associated to canonical connections. We classify algebraic Wanas solitons associated to canonical connections on three-dimensional Lor…

We classify Algebraic Ricci Solitons of three-dimensional Lorentzian Lie groups. All algebraic Ricci solitons that we obtain are sol-solitons. In particular, we prove that, contrary to the Riemannian case, Lorentzian Ricci solitons need not to be algebraic Ricci solitons. We classify Algebraic Ricci Solitons of three-d…

New types of Ricci solitons found in 4D Lorentzian geometry.

In this paper, we compute canonical connections and Kobayashi-Nomizu connections and their curvature on three-dimensional Lorentzian Lie groups with some product structure. We define algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections. We classify algebraic Ricci solitons assoc…

Paper generalizes results from nilpotent Lie algebras to broader types.

We determine, for all three-dimensional non-unimodular Lie groups equipped with a Lorentzian metric, the set of homogeneous geodesics through a point. Together with the results of [C] and [CM2], this leads to the full classification of three-dimensional Lorentzian g.o. spaces and naturally reductive spaces.

We consider Lie groups equipped with a left-invariant cyclic Lorentzian metric. As in the Riemannian case, in terms of homogeneous structures, such metrics can be considered as different as possible from bi-invariant metrics. We show that several results concerning cyclic Riemannian metrics do not extend to their Loren…

Study homogeneous Lorentzian manifolds under reductive Lie groups, reducing descriptions to semisimple groups.

In this paper we will show the existence and uniqueness of the solution of the Björling problem for minimal surfaces in a 3-dimensional Lorentzian Lie group.

The paper constructs a new Ricci-flat metric on almost abelian Lie groups.

Study of symmetry distributions in Lorentzian naturally reductive nilmanifolds.

We determine all Ricci flat left invariant Lorentzian metrics on simply connected 2-step nilpotent Lie groups. We show that the $2k+1$-dimensional Heisenberg Lie group $H_{2k+1}$ carries a Ricci flat left invariant Lorentzian metric if and only if $k=1$. We show also that for any $2\leq q\leq k$, $H_{2k+1}$ carries a R…

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

There are five unimodular simply connected three dimensional unimodular non abelian Lie groups: the nilpotent Lie group $\mathrm{Nil}$, the special unitary group $\mathrm{SU}(2)$, the universal covering group $\widetilde{\mathrm{PSL}}(2,\mathbb{R})$ of the special linear group, the solvable Lie group $\mathrm{Sol}$ and…

Study on completeness of metrics on specific Lie groups.

The paper examines geodesic completeness in Lie groups with specific vector fields.

In this work it is shown that a necessary condition for the completeness of the geodesics of left invariant pseudo-Riemannian metrics on Lie groups is also sufficient in the case of 3-dimensional unimodular Lie groups, and not sufficient for 3-dimensional non unimodular Lie groups. As a consequence it is possible to id…

Non-lorentzian geometry reviewed, including Lie algebras and Klein geometries.

Researchers found the longest arcs for specific sub-Lorentzian structures.

Research explores Lorentzian distances on a specific geometric plane.

We describe the structure of $d$-dimensional homogeneous Lorentzian $G$-manifolds $M=G/H$ of a semisimple Lie group $G$. Due to a result by N. Kowalsky, it is sufficient to consider the case when the group $G$ acts properly, that is the stabilizer $H$ is compact. Then any homogeneous space $G/\bar H$ with a smaller gro…

Researchers found sub-Lorentzian geodesics on a specific Lie subgroup.

Study isometric immersions in 3D Lie groups, proving new characterizations and classifications.

Researchers classify harmful structures on unimodular Lie groups.

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…

The Oscillator Groups,$\G_λ,$ are the only solvable, non commutative, simply connected Lie groups to admit a Lorentzian bi-invariant metric. For these groups, we give sufficient conditions for a left-invariant pseudo-Riemannian metric to be complete, we determine the group of isometries, we exhibit a left-invariant aff…