Journal of Mathematical Physics, Analysis, Geometry (18175805)17(2)pp. 201-215
In the paper, lifted left invariant (α, β)-metrics of Douglas type on tangent Lie groups are studied. Suppose that g is a left invariant Riemannian metric on a Lie group G, and F is a left invariant (α, β)-metric of Douglas type induced by g. Using vertical and complete lifts, we construct the vertical and complete lifted (α, β)-metrics Fv and Fc on the tangent bundle T G and give necessary and sufficient conditions for them to be of Douglas type. Then the flag curvature of these metrics are studied. Finally, as some special cases, the flag curvatures of Fv and Fc are given for Randers metrics of Douglas type and Kropina and Matsumoto metrics of Berwald type. © Masumeh Nejadahm and Hamid Reza Salimi Moghaddam, 2021.
Archivum Mathematicum (00448753)57(1)pp. 1-11
In the present paper we study naturally reductive homogeneous (α, β)-metric spaces. We show that for homogeneous (α, β)-metric spaces, under a mild condition, the two definitions of naturally reductive homogeneous Finsler space, given in the literature, are equivalent. Then, we compute the flag curvature of naturally reductive homogeneous (α, β)-metric spaces. © 2021, Masaryk University. All rights reserved.
Mathematica Scandinavica (00255521)127(1)pp. 100-110
Let F be a left-invariant Randers metric on a simply connected nilpotent Lie group N, induced by a left-invariant Riemannian metric â and a vector field X which is Iâ (M)-invariant. We show that if the Ricci flow equation has a unique solution then, (N, F) is a Ricci soliton if and only if (N, F) is a semialgebraic Ricci soliton. © 2021 Mathematica Scandinavica. All rights reserved.
International Journal of Geometric Methods in Modern Physics (17936977)17(8)
In this paper, we classify all simply connected five-dimensional nilpotent Lie groups which admit (α,β)-metrics of Berwald and Douglas type defined by a left invariant Riemannian metric and a left invariant vector field. During this classification, we give the geodesic vectors, Levi-Civita connection, curvature tensor, sectional curvature and S-curvature. © 2020 World Scientific Publishing Company.
Bulletin Of The Iranian Mathematical Society (10186301)46(2)pp. 457-469
Recently, it is shown that each regular homogeneous Finsler space M admits at least one homogeneous geodesic through any point o∈ M. The purpose of this article is to study the existence of homogeneous geodesics on singular homogeneous (α, β) -spaces, specially, homogeneous Kropina spaces. We show that any homogeneous Kropina space admits at least one homogeneous geodesic through any point. It is shown that, under some conditions, the same result is true for any (α, β) -homogeneous space. In addition, in the case of homogeneous Kropina space of Douglas type, a necessary and sufficient condition for a vector to be a geodesic vector is given. Finally, as an example, homogeneous geodesics of three-dimensional non-unimodular real Lie groups equipped with a left invariant Randers metric of Douglas type are investigated. © 2019, Iranian Mathematical Society.
Journal of Lie Theory (09495932)29(4)pp. 957-968
We give a necessary and sufficient condition for an arbitrary real Lie group, to admit an algebraic Ricci soliton. As an application, we classify all algebraic Ricci solitons on three-dimensional real Lie groups, up to automorphism. This classification shows that, in dimension three, there exist a solvable Lie group and a simple Lie group such that they do not admit any algebraic Ricci soliton. Also it is shown that there exist three-dimensional unimodular and non-unimodular Lie groups with left invariant Ricci solitons. Finally, for a unimodular solvable Lie group, the solution of the Ricci soliton equation is given, explicitly. © 2019 Heldermann Verlag
Houston Journal of Mathematics (03621588)45(4)pp. 1071-1088
We give the explicit formulas of the flag curvature of (α, β)metrics of Berwald type, and correct an error of the first and third authors in a previous article. Then we prove that at any point of a connected non-commutative nilpotent Lie group, the flag curvature of any left invariant (α, β)-metric of Douglas type admits zero, positive and negative values, generalizing a theorem of Wolf. Moreover, we study left invariant (α, β)-metrics of Douglas type on two interesting families of Lie groups considered by Milnor and Kaiser, including Heisenberg Lie groups. On these spaces, we present some necessary and sufficient conditions for (α, β)-metrics to be of Berwald type, as well as some necessary and sufficient conditions for Randers metrics to be of Douglas type. We show that every left invariant (α, β)-metric of Douglas type on G ∈ G1, the family which is defined by Milnor, is a locally projectively flat Randers metric. We also give the explicit formulas of the flag curvature of left invariant Randers metrics of Douglas type on these spaces and show that, under a condition, the flag curvature is negative. © 2019 University of Houston
Iranian Journal of Science and Technology, Transaction A: Science (10286276)43(3)pp. 1197-1202
In this paper, we study left invariant (α, β) -metrics on four-dimensional real Lie groups equipped with left invariant Einstein Riemannian metrics. We classify all left invariant (α, β) -metrics of Berwald type induced by a left invariant Einstein Riemannian metric and a left invariant vector field and show that all of them are locally Minkowskian. All left invariant Randers metrics of Douglas type, and all Einstein Kropina metrics induced by a left invariant Riemannian metric and a left invariant vector field, are classified. Finally, the flag curvatures of these spaces are investigated and in a special case the geodesics are computed. © 2018, Shiraz University.
International Electronic Journal of Geometry (13075624)12(2)pp. 218-222
In this paper we study the Riemann-Finsler geometry of the Lie groups H(p,r) which are a generalization of the Heisenberg Lie groups. For a certain Riemannian metric 〈•, •〉, the Levi-Civita connection and the sectional curvature are given. We classify all left invariant Randers metrics of Douglas type induced by 〈•, •〉, compute their flag curvatures and show that all of them are non-Berwaldian. © 2019
International Journal of Geometric Methods in Modern Physics (17936977)15(1)
Let G be a Lie group equipped with a left invariant Randers metric of Berward type F, with underlying left invariant Riemannian metric g. Suppose that F and g are lifted Randers and Riemannian metrics arising from F and g on the tangent Lie group TG by vertical and complete lifts. In this paper, we study the relations between the flag curvature of the Randers manifold (TG,F) and the sectional curvature of the Riemannian manifold (G,g) when F is of Berwald type. Then we give all simply connected three-dimensional Lie groups such that their tangent bundles admit Randers metrics of Berwarld type and their geodesics vectors. © 2018 World Scientific Publishing Company.
Rendiconti del Circolo Matematico di Palermo (0009725X)67(3)pp. 539-545
In this paper we classify all non-Berwaldian Randers metrics of Douglas type arising from invariant hyper-Hermitian metrics on simply connected four-dimensional real Lie groups. Also, the formulas of the flag curvature are given and it is shown that, in some directions, the flag curvature of the Randers metrics and the sectional curvature of the hyper-Hermitian metrics have the same sign. © 2018, Springer-Verlag Italia S.r.l., part of Springer Nature.
Rendiconti del Circolo Matematico di Palermo (0009725X)67(2)pp. 185-195
In the present article we consider a Lie group G equipped with a left invariant Riemannian metric g. Then, by using complete and vertical lifts of left invariant vector fields we induce a left invariant Riemannian metric g~ on the tangent Lie group TG. The Levi-Civita connection and sectional curvature of (TG, g~) are given, in terms of Levi-Civita connection and sectional curvature of (G, g). Then, we present Levi-Civita connection, sectional curvature and Ricci tensor formulas of (TG, g~) in terms of structure constants of the Lie algebra of G. Finally, some examples of tangent Lie groups of strictly negative and non-negative Ricci curvatures are given. © 2017, Springer-Verlag Italia.
Bulletin Of The Iranian Mathematical Society (10186301)44(1)pp. 193-203
Using vertical and complete lifts, any left invariant Riemannian metric on a Lie group induces a left invariant Riemannian metric on the tangent Lie group. In the present article, we study the Riemannian geometry of tangent bundle of two families of real Lie groups. The first one is the family of special Lie groups considered by J. Milnor and the second one is the class of Lie groups with one-dimensional commutator groups. The Levi–Civita connection, sectional and Ricci curvatures have been investigated. © 2018, Iranian Mathematical Society.
Monatshefte fur Mathematik (14365081)177(4)pp. 649-658
In this paper we identify all simply connected 3-dimensional real Lie groups which admit Randers or Matsumoto metrics of Berwald type with a certain underlying left invariant Riemannian metric. Then we give their flag curvatures formulas explicitly. © 2015, Springer-Verlag Wien.
Osaka Journal of Mathematics (00306126)51(1)pp. 39-45
In this paper we consider invariant Matsumoto metrics which are induced by invariant Riemannian metrics and invariant vector fields on homogeneous spaces, and then we give the flag curvature formula of them. Also we study the special cases of naturally reductive spaces and bi-invariant metrics. We end the article by giving some examples of geodesically complete Matsumoto spaces.
Results in Mathematics (14209012)61(1-2)pp. 137-142
In this paper we consider simply connected Lie groups equipped with left invariant Randers metrics which arise from left invariant Riemannian metrics and left invariant vector fields. Then we study the intersection between automorphism and isometry groups of these spaces. Finally it has shown that for any left invariant vector field, in a special case, the Lie group admits a left invariant Randers metric such that this intersection is a maximal compact subgroup of the group of automorphisms with respect to which the considered vector field is invariant. © 2010 Springer Basel AG.
International Journal of Geometric Methods in Modern Physics (17936977)8(3)pp. 501-510
In this paper we study the geometry of simply connected two-step nilpotent Lie groups of dimension five. We give the Levi-Civita connection, curvature tensor, sectional and scalar curvatures of these spaces and show that they have constant negative scalar curvature. Also we show that the only space which admits left-invariant Randers metric of Berwald type has three-dimensional center. In this case the explicit formula for computing flag curvature is obtained and it is shown that flag curvature and sectional curvature have the same sign. © 2011 World Scientific Publishing Company.
International Journal of Geometric Methods in Modern Physics (17936977)6(4)pp. 619-624
In this paper we study sectional curvature of invariant hyper-Hermitian metrics on simply connected 4-dimensional real Lie groups admitting invariant hypercomplex structure. We give the Levi-Civita connections and explicit formulas for computing sectional curvatures of these metrics and show that all these spaces have constant scalar curvature. We also show that they are flat or they have only non-negative or non-positive sectional curvature. © 2009 World Scientific Publishing Company.
Archivum Mathematicum (00448753)45(3)pp. 159-170
In this paper, firstly we study some left invariant Riemannian metrics on para-hypercomplex 4-dimensional Lie groups. In each Lie group, the Levi-Civita connection and sectional curvature have been given explicitly. We also show these spaces have constant negative scalar curvatures. Then by using left invariant Riemannian metrics introduced in the first part, we construct some left invariant Randers metrics of Berwald type. The explicit formulas for computing flag curvature have been obtained in all cases. Some of these Finsler Lie groups are of non-positive flag curvature.
Journal of Physics A: Mathematical and Theoretical (17518113)42(9)
In the present paper we study Randers metrics of Berwald type on simply connected four-dimensional real Lie groups admitting invariant hypercomplex structure. On these spaces, the Randers metrics arising from invariant hyper-Hermitian metrics are considered. Then we give explicit formulae for computing the flag curvature of these metrics. By this study we construct two four-dimensional Berwald spaces, one of them with a non-negative flag curvature and the other one with a non-positive flag curvature. © 2009 IOP Publishing Ltd.
Journal of Geometry and Physics (03930440)59(7)pp. 969-975
In this paper, by using left invariant Riemannian metrics on some three-dimensional Lie groups, we construct some complete non-Riemannian Berwald spaces of non-positive flag curvature and several families of geodesically complete locally Minkowskian spaces of zero constant flag curvature. © 2009 Elsevier B.V. All rights reserved.
Mathematical Physics Analysis and Geometry (13850172)11(1)pp. 1-9
In the present paper, the flag curvature of invariant Randers metrics on homogeneous spaces and Lie groups is studied. We first give an explicit formula for the flag curvature of invariant Randers metrics arising from invariant Riemannian metrics on homogeneous spaces and, in special case, Lie groups. We then study Randers metrics of constant positive flag curvature and complete underlying Riemannian metric on Lie groups. Finally we give some properties of those Lie groups which admit a left invariant non-Riemannian Randers metric of Berwald type arising from a left invariant Riemannian metric and a left invariant vector field. © 2008 Springer Science+Business Media B.V.
Journal of Physics A: Mathematical and Theoretical (17518113)41(27)
In this paper, we study the flag curvature of invariant (α, β)-metrics of the form on homogeneous spaces and Lie groups. We give a formula for the flag curvature of invariant metrics of the form such that α is induced by an invariant Riemannian metric g on the homogeneous space and the Chern connection of F coincides to the Levi-Civita connection of g. Then some conclusions in the cases of naturally reductive homogeneous spaces and Lie groups are given. © 2008 IOP Publishing Ltd.
Balkan Journal of Geometry and its Applications (12242780)11(2)pp. 73-79
In this paper we show that every invariant Finsler metric on Lie group G, induces an invariant Finsler metric on quotient group G/H in the natural way, where H is a closed normal Lie subgroup of G. © Balkan Society of Geometers, Geometry Balkan Press 2006.
