Three-dimensional Lie Groups Research Articles

Classification of harmonic homomorphisms between Riemannian three-dimensional unimodular Lie groups

PurposeThe purpose of this study is to classify harmonic homomorphisms ϕ : (G, g) → (H, h), where G, H are connected and simply connected three-dimensional unimodular Lie groups and g, h are left-invariant Riemannian metrics.Design/methodology/approachThis study aims the classification up to conjugation by automorphism of Lie groups of harmonic homomorphism, between twodifferent non-abelian connected and simply connected three-dimensional unimodular Lie groups (G, g) and (H, h), where g and h are two left-invariant Riemannian metrics on G and H, respectively.FindingsThis study managed to classify some homomorphisms between two different non-abelian connected and simply connected three-dimensional uni-modular Lie groups.Originality/valueThe theory of harmonic maps into Lie groups has been extensively studied related homomorphism in compact Lie groups by many mathematicians, harmonic maps into Lie group and harmonics inner automorphisms of compact connected semi-simple Lie groups and intensively study harmonic and biharmonic homomorphisms between Riemannian Lie groups equipped with a left-invariant Riemannian metric.

On Parametric Surfaces with Constant Mean Curvature Along Given Smarandache Curves in Lie Group

This paper finds sufficient conditions to determine a surface whose mean curvature along a given Smarandache curve is constant in a three-dimensional Lie group. This is accomplished by using the Frenet frames of the specified curve to express surfaces that span the TN, NB, and TB Smarandache curves parametrically. In terms of the curvatures of given Smarandache curves, marching scale functions, and their partial derivatives, the mean curvatures of these surfaces along the given TN, NB, and TB Smarandache curves are determined. Sufficient conditions are found to maintain the provided mean curvatures of the resulting surfaces at a constant value. Finally, some examples are provided.

Framed curves in three-dimensional Lie groups and a Berry phase model

In this study, we introduce framed curves which can have singular points in three-dimensional Lie groups. We give Frenet-Serret type formulas of framed curves with the help of a Lie curvature in three-dimensional Lie groups. Then, we define adapted frames of framed curves in three-dimensional Lie groups such that Frenet-Serret type frame and Bishop frame. Finally, as a physical application, we give Berry phase model of polarized light wave along an optical fiber in Lie groups.

On the Symmetric Einstein Equation for Three-Dimensional Lie Groups with Left-Invariant Riemannian Metric and Semi-Symmetric Connection

Riemannian manifolds with a Levi-Civita connection and constant Ricci curvature, or Einstein manifolds, were studied in the works of many mathematicians. This question has been most studied in the homogeneous Riemannian case. In this direction, the most famous ones are the results of works by D.V. Alekseevsky, M. Wang, V. Ziller, G. Jensen, H.Laure, Y.G. Nikonorov, E.D. Rodionov and other mathematicians. At the same time, the question of studying Einstein manifolds has been little studied for the case of an arbitrary metric connection. This is primarily due to the fact that the Ricci tensor of metric connection is not, generally speaking, symmetric.
 In this paper, we consider semisymmetric connections on 3-dimensional Lie groups with a left-invariant Riemannian metric. For the symmetric part of the Ricci tensor, the Einstein equation is studied. As a result of the research carried out, a classification of 3-dimensional metric Lie groups and the corresponding semisymmetric connections in the case of the Einstein symmetric equation has been obtained.
 Earlier in the works of P.N. Klepikov, E.D. Rodionov and O.P. Khromova, the classical Einstein equation was studied, and it was proved that if the classical.
 The Einstein equation holds for a 3-dimensional Lie group with left-invariant (pseudo) Riemannian metric and semi-symmetric connection. Then, either the connection is a Levi-Civita connection, or the curvature tensor of the connection is equal to zero.

Constructions of helicoidal minimal surfaces and minimal annuli in $$\widetilde{E(2)}$$

In this article, we construct two one-parameter families of properly embedded minimal surfaces in a three-dimensional Lie group \(\widetilde{E(2)}\), which is the universal covering of the group of rigid motions of Euclidean plane endowed with a left-invariant Riemannian metric. The first one can be seen as a family of helicoids, while the second one is a family of catenoidal minimal surfaces. The main tool that we use for the construction of these surfaces is a Weierstrass-type representation introduced by Meeks, Mira, Pérez and Ros for minimal surfaces in Lie groups of dimension three. In the end, we study the limit of the catenoidal minimal surfaces. As an application of this limit case, we get a new proof of a half-space theorem for minimal surfaces in \(\widetilde{E(2)}\).

Classification of Lorentzian Lie Groups Based on Codazzi Tensors Associated with Yano Connections

In this paper, we derive the expressions of Codazzi tensors associated with Yano connections in seven Lorentzian Lie groups. Furthermore, we complete the classification of three-dimensional Lorentzian Lie groups in which Ricci tensors associated with Yano connections are Codazzi tensors. The main results are listed in a table, and indicate that G1 and G7 do not have Codazzi tensors associated with Yano connections, G2, G3, G4, G5 and G6 have Codazzi tensors associated with Yano connections.

Generalized Ricci Solitons of Three-Dimensional Lorentzian Lie Groups Associated Canonical Connections and Kobayashi-Nomizu Connections

In this paper, we study the affine generalized Ricci solitons on three-dimensional Lorentzian Lie groups associated canonical connections and Kobayashi-Nomizu connections and we classifying these left-invariant affine generalized Ricci solitons with some product structure.

Non-Abelian T-duality of AdS_{dle 3} families by Poisson-Lie T-duality

We proceed to investigate the non-Abelian T-duality of AdS_{2}, AdS_{2}times S^1 and AdS_{3} physical backgrounds, as well as the metric of the analytic continuation of AdS_{2} from the point of view of Poisson-Lie (PL) T-duality. To this end, we reconstruct these metrics of the AdS families as backgrounds of non-linear sigma -models on two- and three-dimensional Lie groups. By considering the Killing vectors of these metrics and by taking into account the fact that the subgroups of isometry Lie group of the metrics can be taken as one of the subgroups of the Drinfeld double (with Abelian duals) we look up the PL T-duality. To construct the dualizable metrics by the PL T-duality we find all subalgebras of Killing vectors that generate subgroup of isometries which acts freely and transitively on the manifolds defined by aforementioned AdS families. We then obtain the dual backgrounds for these families of AdS in such a way that we apply the usual rules of PL T-duality without further corrections. We have also investigated the conformal invariance conditions of the original backgrounds (AdS families) and their dual counterparts. Finally, by using the T-duality rules proposed by Kaloper and Meissner (KM) we calculate the Abelian T-duals of BTZ black hole up to two-loop by dualizing on the coordinates varphi and t. When the dualizing is implemented by the shift of direction varphi , we show that the horizons and singularity of the dual spacetime are the same as in charged black string derived by Horne and Horowitz without alpha '-corrections, whereas in dualizing on the coordinate t we find a new three-dimensional black string whose structure and asymptotic nature are clearly determined. For this case, we show that the T-duality transformation changes the asymptotic behavior from AdS_3 to flat.

Canonical Connections and Algebraic Ricci Solitons of Three-dimensional Lorentzian Lie Groups

In this paper, the author computes canonical connections and Kobayashi-Nomizu connections and their curvature on three-dimensional Lorentzian Lie groups with some product structure. He defines algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections. He classifies algebraic Ricci solitons associated to canonical connections and Kobayashi-Nomizu connections on three-dimensional Lorentzian Lie groups with some product structure.

Three-Dimensional Nonunimodular Lie Groups with a Riemannian Metric of an Invariant Ricci Soliton and a Semisymmetric Metric Connection

Three-Dimensional Nonunimodular Lie Groups with a Riemannian Metric of an Invariant Ricci Soliton and a Semisymmetric Metric Connection

Gauss-Bonnet Theorem in the Universal Covering Group of Euclidean Motion Group E(2) with the General Left-Invariant Metric

The universal covering group of Euclidean motion group E(2) with the general left-invariant metric is denoted by (widetilde{E(2)},g_L(lambda _1,lambda _2)), where lambda _1ge lambda _2>0. It is one of three-dimensional unimodular Lie groups which are classified by Milnor. In this paper, we compute sub-Riemannian limits of Gaussian curvature for a Euclidean C^2-smooth surface in (widetilde{E(2)},g_L(lambda _1,lambda _2)) away from characteristic points and signed geodesic curvature for Euclidean C^2-smooth curves on surfaces. Based on these results, we get a Gauss-Bonnet theorem in the universal covering group of Euclidean motion group E(2) with the general left-invariant metric.

Sweeping Surfaces in theThree-Dimensional Lie Group

This paper investigated the rotation minimizing frames that are related to the space curves and the sweeping surfaces that are traced by these frames in the three-dimensional Lie group. Then, the sufficient and necessary conditions for the sweeping surface to be a developable ruled surface were obtained. In particular, we mostly focused on the study of the resulting developable surface is a cylinder, cone, or tangent surface. Meanwhile, to support the results in the paper, some illustrative examples are presented.

On the Cerbo Conjecture on Lie Groups with the Left-Invariant Lorentzian Metric

Manifolds with constraints on tensor fields include Einstein manifolds, Einstein-like manifolds, conformally flat manifolds, and a number of other important classes of manifolds. The work of many mathematicians is devoted to the study of such manifolds, which is reflected in the monographs of A. Besse, M. Berger, M.-D. Cao, M. Wang.
 Ricci solitons are one of the natural generalizations of Einstein's metrics. If a Riemannian manifold is a Lie group, one speaks of invariant Ricci solitons.
 Invariant Ricci solitons were studied in most detail in the case of unimodular Lie groups with left-invariant Riemannian metrics and the case of low dimension. Thus, L. Cerbo proved that all invariant Ricci solitons are trivial on unimodular Lie groups with left-invariant Riemannian metric and Levi-Civita connection.A similar result up to dimension four was obtained by P.N. Klepikov and D.N. Oskorbin for the non-unimodular case.
 We study invariant Ricci solitons on three-dimensional unimodular Lie groups with the Lorentzian metric.The study results show that unimodular Lie groups with left-invariant Lorentzian metric admit invariant Ricci solitons other than trivial ones. In this paper, a complete classification of invariant Ricci solitons on three-dimensional unimodular Lie groups with leftinvariant Lorentzian metric is obtained.

Isometry groups of three-dimensional Lie groups

We compute the full isometry group of any left-invariant metric on a simply connected, non-unimodular Lie group of dimension three. As an application, we determine the index of symmetry of such metrics and prove that the singularities of the moduli space of left-invariant metrics, up to isometric automorphism are contained in the subspace of classes of metrics with maximal index of symmetry.

Topological and homological properties of the orbit space of a simple three-dimensional compact linear Lie group

Topological and homological properties of the orbit space of a simple three-dimensional compact linear Lie group

Mathematical modeling in the study of semisymmetric connections on three-dimensional Lie groups with the metric of the Ricci soliton

Semisymmetric connections were first discovered by E. Cartan and are a natural generalization of the Levi-Civita connection. The properties of the parallel transfer of such connections and the basic tensor fields were investigated by I. Agrikola, K. Yano and other mathematicians. In this paper, a mathematical model is constructed for studying semisymmetric connections on three-dimensional Lie groups with the metric of an invariant Ricci soliton. A classification of these connections on three-dimensional unimodular Lie groups with left-invariant Riemannian metric of the Ricci soliton is obtained. It is proved that in this case there are nontrivial invariant semisimetric connections. Previously, the authors carried out similar studies in the class of Einstein metrics.

Affine Ricci solitons associated to the Bott connection on three-dimensional Lorentzian Lie groups

In this paper, we compute the Bott connection and their curvature on three-dimensional Lorentzian Lie groups with three different distributions, then we classify affine Ricci solitons associated to the Bott connection on the spaces under study.

On Invariant Semisymmetric Connections on Three-Dimensional Non-Unimodular Lie Groups with the Metric of the Ricci Soliton

Metric connections with vector torsion, or semisymmetric connections, were first discovered by E. Cartan. They are a natural generalization of the Levi-Civita connection. The properties of such connections and the basic tensor fields were investigated by I. Agrikola, K. Yano, and other mathematicians.
 Ricci solitons are the solution to the Ricci flow and a natural generalization of Einstein's metrics. In the general case, they were investigated by many mathematicians, which was reflected in the reviews by H.-D. Cao, R.M. Aroyo — R. Lafuente. This question is best studied in the case of trivial Ricci solitons, or Einstein metrics, as well as the homogeneous Riemannian case.
 This paper investigates semisymmetric connections on three-dimensional Lie groups with the metric of an invariant Ricci soliton. A classification of these connections on three-dimensional non-unimodularLie groups with the left-invariant Riemannian metric of the Ricci soliton is obtained. It is proved that there are nontrivial invariant semisymmetric connections in this case. In addition, it is shown that there are nontrivial invariant Ricci solitons.

Codazzi Tensors and the Quasi-Statistical Structure Associated with Affine Connections on Three-Dimensional Lorentzian Lie Groups

In this paper, we classify three-dimensional Lorentzian Lie groups on which Ricci tensors associated with Bott connections, canonical connections and Kobayashi–Nomizu connections are Codazzi tensors associated with these connections. We also classify three-dimensional Lorentzian Lie groups with the quasi-statistical structure associated with Bott connections, canonical connections and Kobayashi–Nomizu connections.

Natural and conjugate mates of Frenet curves in three-dimensional Lie group

In this study, we introduce the natural mate and conjugate mate of a Frenet curve in a three dimensional Lie group $ \mathbb{G} $ with bi-invariant metric. Also, we give some relationships between a Frenet curve and its natural mate or its conjugate mate in $ \mathbb{G} $. Especially, we obtain some results for the natural mate and the conjugate mate of a Frenet curve in $ \mathbb{G} $ when the Frenet curve is a general helix, a slant helix, a spherical curve, a rectifying curve, a Salkowski (constant curvature and non-constant torsion), anti-Salkowski (non-constant curvature and constant torsion), Bertrand curve. Finally, we give nice graphics with numeric solution in Euclidean 3-space as a commutative Lie group.