arXiv research

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

Study finds index of symmetry for solvable 3D Lie groups with left-invariant metrics.

The paper classifies structures on specific Lie groups.

The paper classifies Ricci solitons on specific Lorentzian Lie groups.

The paper classifies solitons on specific Lie groups.

Study classifies Riemann solitons on specific 3D Lorentzian groups.

In this paper, we study spinor Frenet equations in three dimensional Lie Groups with a bi-invariant metric. Also, we obtain spinor Frenet equations for special cases of three dimensional Lie groups.

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

The study classifies special flows on specific geometric groups.

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

We study the subelliptic heat kernels of the CR three dimensional solvable Lie groups. We first classify all left-invariant sub-Riemannian structures on three dimensional solvable Lie groups and obtain representations of these groups. We give expressions for the heat kernels on these groups and obtain heat semigroup gr…

The paper classifies Ricci solitons on specific Lorentzian Lie groups.

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

Classifies Ricci collineations on specific 3D Lorentzian Lie groups.

In this paper we classify those three-dimensional Riemannian Lie groups which admit harmonic morphisms to surfaces.

Classifies Ricci solitons on specific Lorentzian Lie groups.

We prove existence and uniqueness of the solution of the Björling problem for minimal surfaces in a three-dimensional Lie group.

The main purpose of this paper is to investigate the Schouten-Weyl tensor on the three-dimensional Lie groups with left-invariant Lorenzian metrics. The left-invariant Lorentzian metrics on the three-dimensional Lie groups with squared length zero Schouten-Weyl tensor are studied. Moreover, the three-dimensional metric…

Study classifies biharmonic and harmonic homomorphisms between specific Lie groups.

We give a complete list of those left invariant unit vector fields on three-dimensional Lie groups with the left-invariant metric that generate a totally geodesic submanifold in the unit tangent bundle of a group with the Sasaki metric. As a result, each class of three-dimensional Lie groups admits the totally geodesic…

Study on harmonic spinors on specific Lie groups.

In this paper we describe the geodesics of a left-invariant sub-Riemannian metric on the three-dimensional solvable Lie group $SOLV^-$.

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

Study finds abnormal paths on specific Lie groups using algebraic structures.

In this paper, we define slant helices in three dimensional Lie Groups with a bi-invariant metric and obtain a characterization of slant helices. Moreover, we give some relations between slant helices and their involutes, spherical images.

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

In this paper, we shall use a method based on the theory of extensions of left-symmetric algebras to classify complete left-invariant affine real structures on solvable non-unimodular three-dimensional Lie groups.

In this paper, we investigate the relationship between algebraic soliton metrics and soliton metrics for geometric evolution equations on Lie groups. After discussing the general relationship between algebraic soliton metrics and soliton metrics, we investigate the cross curvature flow and the second order renormalizat…

In this paper, we define the corresponding submanifolds to left-invariant Riemannian metrics on Lie groups, and study the following question: does a distinguished left-invariant Riemannian metric on a Lie group correspond to a distinguished submanifold? As a result, we prove that the solvsolitons on three-dimensional s…

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…

Study on curvatures of surfaces in specific Lie groups.

We derive the Weierstrass (or spinor) representation for surfaces in three-dimensional Lie groups Nil, \tilde{SL}_2, and Sol with Thurston's geometries and establish the generating equations for minimal surfaces in these groups. By using the spectral properties of the corresponding Dirac operators we find analogs of th…

This is a survey of results on surfaces in noncommutative three-dimensional Lie groups obtained by using the Weierstrass (spinor) representation of surfaces. It is based on the talk given at the conference "Geometry related to the theory of integrable systems" (RIMS, Kyoto, September 2007).

Motivated by a number of recent investigations, we define and investigate the various properties of the ruled surfaces depend on three dimensional Lie groups with a bi-variant metric. We give useful results involving the characterizations of these ruled surfaces. Some special ruled surfaces such as normal surface, bino…

Study proves all left-invariant contact structures on 3D Lie groups are tight.

Classifies left invariant Kundt structures on 3D Lie groups.

We study three-dimensional generalized Ricci solitons, both in Riemannian and Lorentzian settings. We shall determine their homogeneous models, classifying left-invariant generalized Ricci solitons on three-dimensional Lie groups.

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.

Cocalibrated G_2-structures are structures naturally induced on hypersurfaces in Spin(7)-manifolds. Conversely, one may start with a seven-dimensional manifold M endowed with a cocalibrated G_2-structure and construct via the Hitchin flow a Spin(7)-manifold which contains M as a hypersurface. In this article, we consid…

Study uses Lie group subgroups to identify special subspaces in calibrations.

We classify six-dimensional Lie groups which admit a left-invariant half-flat SU(3)-structure and which split in a direct product of three-dimensional factors. Moreover, a complete list of those direct products is obtained which admit a left-invariant half-flat SU(3)-structure such that the three-dimensional factors ar…

Classifies invariant generalised Killing spinors on Lie groups.

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

In this paper, we give the defination of harmonic curvature function some special curves such as helix, slant curves, Mannheim curves and Bertrand curves. Then, we recall the characterizations of helices [8], slant curves (see [19]) and Mannheim curves (see [12]) in three dimensional Lie groups using their harmonic cur…

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 connects Lie groups to specific Riemannian manifolds.

Geometrically constructs Virasoro-Bott group from circle diffeomorphisms.

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.