arXiv research

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

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

Study classifies Riemann solitons on specific 3D Lorentzian groups.

Classifies Ricci collineations on specific 3D Lorentzian Lie groups.

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

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

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

Study on completeness of metrics on specific Lie groups.

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

Study classifies special metrics for which the isometry group of 3D Lie groups is larger than expected.

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

The paper classifies Ricci solitons on specific Lorentzian Lie groups.

Complete description of flat Lorentzian Lie groups solved.

Classifies Ricci solitons on specific Lorentzian Lie groups.

The paper classifies Ricci solitons on specific Lorentzian 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 solitons on specific Lie groups.

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

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…

Proves Lorentzian manifold properties for analytic 3D spaces.

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 on special Lie groups with Lorentzian metrics.

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.

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.

The paper classifies isometries on specific Lie groups.

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…

The study finds Ricci solitons on a specific type of 3D Lorentzian Walker manifold.

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

Prove an isoperimetric inequality for compact bodies in 3D contact non-unimodular Lie 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…

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.