arXiv subjects

Hamid Reza Salimi Moghaddam

Publications and source records attributed to Hamid Reza Salimi Moghaddam.

At least 19 recordsLinked to original sources

arXiv2026

Ricci Solitons of Left-Invariant Metrics on Lie Groups with Derived Algebra of Dimension at Most Two

In this article, we investigate when a left-invariant Riemannian metric on a Lie group is a Ricci soliton, under the assumption that the derived algebra has dimension at most two. We establish computable necessary and sufficient conditions for a given left-invariant Riemannian metric to be a Ricci soliton. As applications, we obtain several examples of Ricci nilsolitons and apply our results to indecomposable Lie groups of dimension at least five with two-dimensional derived algebra.

math.DG↗

arXiv2026

Hermitian and Kähler Structures of Two Types on Lie Groups with Two-Dimensional Derived Algebras

We study left-invariant Hermitian structures on Lie groups whose derived algebra is two-dimensional. We consider two natural types of almost Hermitian structures, according to the position of the derived algebra with respect to the almost complex structure. For each type, we obtain explicit necessary and sufficient conditions for integrability. In particular, for the second type, integrability is equivalent to the abelianity of the complex structure. We then characterize the Kähler structures of both types and show that, in each case, the Kähler condition imposes strong restrictions on the Lie bracket. Moreover, we compute the corresponding Bismut connections and torsion forms explicitly. As applications, we construct examples of Kähler, WKT, and SKT structures in dimensions four, six, and in all even dimensions greater than six. In particular, we obtain a family of abelian SKT structures on $\mathfrak r_2\oplus\mathfrak h_3\oplus\mathbb R^{2n-5}$ for $n\geq3$.

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arXiv2026

On Naturally Reductive $\boldsymbol{(α_1,α_2)}$-Metrics

In this paper, we investigate the converse of the Tan-Xu theorem, which states that the naturally reductive property of a Riemannian metric is inherited by a naturally reductive $(α_1,α_2)$-metric, and we show that, under certain conditions, the converse also holds. We also examine the relationship between geodesic vector fields on homogeneous Riemannian spaces and homogeneous $(α_1,α_2)$-spaces. Finally, we construct left-invariant $(α_1,α_2)$-metrics on the tangent bundle of Lie groups using left-invariant Randers metrics on the base Lie group, and study their geometric relations.

math.GM↗

arXiv2026

On the Riemann-Finsler Geometry of Tangent Bundle of Lie Groups with Two-Dimensional Commutator Subgroup

We begin by studying the Riemannian geometry of the tangent Lie group $TG$ associated with a Lie group $G$ whose commutator subgroup is two-dimensional, equipped with the lift of a left-invariant Riemannian metric on $G$. We establish the relationship between the sectional curvatures of $G$ and those of $TG$. Next, we define a Randers metric on $G$ from a left-invariant Riemannian metric and a left-invariant vector field, and lift it vertically and completely to $TG$. We investigate the conditions under which this Randers metric is of Berwald and Douglas type, respectively, and compute the flag curvatures in the Berwald case. In an addendum, we discuss geodesic vectors and bi-invariant Riemannian metrics on these Lie groups, highlighting the special unimodularity conditions. Finally, we provide explicit formulas for the Riemannian curvature tensor on the tangent bundle of such a Lie group.

math.DG↗

arXiv2024

New Randers metrics defined by the other Randers metrics

In this short article, using a left-invariant Randers metric $F$, we define a new left-invariant Randers metric $\tilde{F}$. We show that $F$ is of Berwald (Douglas) type if and only if $\tilde{F}$ is of Berwald (Douglas) type. In the case of Berwaldian metrics, we give the relation between their flag curvatures. Also, we have studied the relations between their base Riemannian metrics. Finally, as examples, the results are studied in the Heisenberg group and almost Abelian Lie groups.

math.DG↗

arXiv2024

Two-step homogeneous geodesics in some homogeneous Finsler manifolds

A natural extension of a homogeneous geodesic in homogeneous Riemannian spaces $G/H$, known as a two-step homogeneous geodesic, can be expressed of the form $γ(t)=π(\exp(tx)\exp(ty))$, where $x$ and $y$ are elements of the Lie algebra of $G$. This paper aims to expand this concept to homogeneous Finsler spaces. We provide certain sufficient conditions for $(α,β)$ spaces and decomposable cubic spaces to possess a one-parameter family of invariant Finsler metrics that can be classified as two-step Finsler geodesic orbit spaces. Additionally, we present some illustrative examples of these spaces.

math.DG↗

arXiv2024

f-left-invariant Riemannian metrics on Lie groups

With a f-left-invariant Riemannian metric on a Lie group $G$, we mean a Riemannian metric which is conformally equivalent to a left-invariant Riemannian metric, with the conformal factor $f$. In this article, we study the geometry of such metrics and give a necessary and sufficient condition for an f-left-invariant Riemannian metric to be a Ricci soliton. Using this result, for any expansion constant $λ$, we obtain a flat gradient Ricci soliton on some two and three-dimensional non-abelian Lie groups. We give an example of a non-flat steady gradient Ricci soliton and construct some examples of non-flat shrinking, steady, and expanding non-gradient Ricci solitons on the non-abelian Lie group $\Bbb{R}\rtimes\Bbb{R}^{+}$. Finally, we study f-left-invariant Riemannian metrics on the Heisenberg group.

math.DG↗

arXiv2023

Riemannian geometry of tangent Lie groups using two left invariant Riemannian metrics

In this paper, we consider a Lie group $G$ equipped with two left-invariant Riemannian metrics $g^1$ and $g^2$. Using these two left-invariant Riemannian metrics we define a left-invariant Riemannian metric $\tilde{g}$ on the tangent Lie group $TG$. The Levi-Civita connection, tensor curvature, and sectional curvature of $(TG,\tilde{g})$ in terms of $g^1$ and $g^2$ are given. Also, we give a sufficient condition for $\tilde{g}$ to be bi-invariant. Finally, motivated by the recent work of D. N. Pham, using symplectic forms $ω_1$ and $ω_2$ on $G$ we define a symplectic form $\tildeω$ on $TG$.

math.DG↗

arXiv2022

Conformally related invariant $(α,β)$-metrics on homogeneous spaces

In this paper, we give the flag curvature formula of general $(α,β)$-metrics of Berwald type. We study conformally related $(α,β)$-metrics, especially general $(α,β)$-metrics that are conformally related to invariant $(α,β)$-metrics. Also, a necessary and sufficient condition for a Finsler metric conformally related to an $(α,β)$-metric is given, and conformally related Douglas Randers metrics are studied. Finally, we present some examples of conformally related $(α,β)$-metrics.

math.GM↗

arXiv2021

On the classification of five-dimensional nilsolitons

In 2002, using a variational method, Lauret classified five-dimensional nilsolitons. In this work, using the algebraic Ricci soliton equation, we obtain the same classification. We show that, among ten classes of five-dimensional nilmanifolds, seven classes admit Ricci soliton structure. In any case, the derivation which satisfies the algebraic Ricci soliton equation is computed.

math.DG↗

arXiv2021

Invariant Einstein Kropina metrics on Lie groups and homogeneous spaces

In this article, we study Einstein Kropina metrics on Lie groups and homogeneous spaces. We give a method to construct Einstein Kropina metrics on Lie groups. As an example of this method, a family of non-Riemannian Einstein Kropina metrics on the special orthogonal group $SO(n)$ is given. Then, we classify all left invariant Einstein Kropina metrics on simply connected $3$-dimensional real Lie groups. We provide a procedure to build Einstein Kropina metrics on homogeneous spaces. Using this technique, we study invariant Einstein Kropina metrics on spheres. Finally, we show that projective spaces do not admit any homogeneous non-Riemannian Einstein Kropina metrics.

math.DG↗

arXiv2019

Classification of Douglas $(α,β)$-metrics on five dimensional nilpotent Lie groups

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.

math.DG↗

arXiv2019

Randers Ricci soliton homogeneous nilmanifolds

Let $F$ be a left invariant Randers metric on a simply connected nilpotent Lie group $N$, induced by a left invariant Riemannian metric ${\hat{\textbf{\textit{a}}}}$ and a vector field $X$ which is $I_{\hat{\textbf{\textit{a}}}}(M)$-invariant. 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.

math.DG↗

arXiv2019

The Relation Between Automorphism Group and Isometry Group of Left Invariant $ (α,β)$-metrics

This work generalizes the results of an earlier paper by the second author, from Randers metrics to $(α,β)$-metrics. Let $F$ be an $(α,β)$-metric which is defined by a left invariant vector field and a left invariant Riemannian metric on a simply connected real Lie group $G$. We consider the automorphism and isometry groups of the Finsler manifold $(G,F)$ and their intersection. We prove that for an arbitrary left invariant vector field $X$ and any compact subgroup $K$ of automorphisms which $X$ is invariant under them, there exists an $(α,β)$-metric such that $K$ is a subgroup of its isometry group.

math.DG↗

arXiv2019

Left invariant lifted $(α,β)$-metrics of Douglas type on tangent Lie groups

In this paper we study lifted left invariant $(α,β)$-metrics of Douglas type on tangent Lie groups. Let $G$ be a Lie group equipped with a left invariant $(α,β)$-metric of Douglas type $F$, induced by a left invariant Riemannian metric $g$. Using vertical and complete lifts, we construct the vertical and complete lifted $(α,β)$-metrics $F^v$ and $F^c$ on the tangent Lie group $TG$ 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 $F^v$ and $F^c$ in the cases of Randers metrics of Douglas type, and Kropina and Matsumoto metrics of Berwald type are given.

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arXiv2019

On the Existence of Homogeneous Geodesics in Homogeneous Kropina Spaces

Recently, it is shown that each regular homogeneous Finsler space $M$ admits at least one homogeneous geodesic through any point $o\in M$. 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. Also, 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 $3$-dimensional non-unimodular real Lie groups equipped with a left invariant Randers metric of Douglas type are investigated.

math.DG↗

arXiv2016

On the left invariant $(α,β)$-metrics on some Lie groups

We give the explicit formulas of the flag curvatures of left invariant Matsumoto and Kropina metrics of Berwald type. We can see these formulas are different from previous results given recently. Using these formulas, we prove that at any point of an arbitrary connected non-commutative nilpotent Lie group, the flag curvature of any left invariant Matsumoto and Kropina metrics of Berwald type admits zero, positive and negative values, this is a generalization of Wolf's theorem. Then we study $(α,β)$-metrics of Berwald type and also Randers metrics of Douglas type on two interesting families of Lie groups considered by Milnor and Kaiser, containing Heisenberg Lie groups. On these spaces, we present some necessary and sufficient conditions for $(α,β)$-metrics to be of Berwald type and also some necessary and sufficient conditions for Randers metrics to be of Douglas type. All left invariant non-Berwaldian Randers metrics of Douglas type are given and the flag curvatures are computed.

math.DG↗

arXiv2016

Naturally reductive homogeneous $(α,β)$-metric spaces

In the present paper we study naturally reductive homogeneous $(α,β)$-metric spaces. Under some conditions, we give some necessary and sufficient conditions for a homogeneous $(α,β)$-metric space to be naturally reductive. Then we show that for such spaces the two definitions of naturally reductive homogeneous Finsler space, given in literature, are equivalent. Finally we compute the flag curvature of naturally reductive homogeneous $(α,β)$-metric spaces.

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