Journal of Finsler Geometry and its Applications
Volume:1 Issue: 2, Dec 2020

In a Finsler spaces, we consider a special (A, B )-metric L satisfying L^2(A, B) = c_1A^2 + 2c_2AB + c_3B^2, where c_i are constant. In this paper, the existence of invariant vector elds on a special homogeneous (A;B )-space with L metric is proved. Then we study geodesic vectors and investigate the set of all homogeneous geodesics of invariant (A;B )-metric L on homogeneous spaces and simply connected 4-dimensional real Lie groups admitting invariant hypercomplex structure.

In this paper, we give a classification of left-invariant Douglas and Berwald $(alpha,beta)$-metrics on simply connected four-dimensional nilpotent Lie groups. We show that there are not any bi-invariant Randers metrics on four-dimensional nilpotent Lie groups. Then, we explicitly give the flag curvature formulas and geodesic vectors of these spaces. Finally, we give the formula of $S$-curvature of left-invariant Randers metrics of Douglas type.

Let ∇ be a symmetric connection on an n-dimensional manifold M n and T ∗ M n its cotangent bundle. In this paper, firstly, we determine the infinitesimal fiber-preserving projective(IFP) transformations on T ∗ M n with respect to the Riemannian connection of the modified Riemannian extension ˜ g ∇,c where c is a symmetric (0,2)-tensor field on M n . Then we prove that, if (T ∗ M n , ˜ g ∇,c ) admits a non-affine infinitesimal fiber- preserving projective transformation, then M n is locally flat, where ∇ is the Levi-Civita connection of a Riemannian metric g on M n . Finally, the infinitesimal complete lift, horizontal and vertical lift projective trans- formations on (T ∗ M n , ˜ g ∇,c ) are studied.

Inspired by the notion of projectively related spherically symmetric metrics, we study the class of Finsler metrics whose geodesics have the same shape with a difference in rotation or reflection of their graphs. This class of metrics contains the class of projectively related Finsler metrics. First, we characterize the class of Randers metrics, ( α, β )-metrics and spherically symmetric metrics in this class of metrics

‎In this paper‎, ‎we study a special class of Finsler metrics $F= breve{alpha}phi(r‎, ‎s)$ called warped product metrics where $breve{alpha}$ is a Riemannian metric‎, ‎$r= u^{1}$ and $ s= frac{v^{1}}{breve{alpha}}$‎. ‎We show that every projectively flat Finsler warped product metrics with isotropic $E$-curvature is a Randers metric‎.

In this paper, we study the class of 3-dimensional Finsler manifolds. We find the necessary and sufficient condition under which a 3-dimensional weakly Landsberg metric reduces to a Landsberg metric.

In this paper, we consider the class of L-reducible Finsler metrics which contains the class of C-reducible metrics and the class of Landsberg metrics. Let (M,F) be a 3-dimensional L-reducible Finsler manifold. Suppose that F has a relatively isotropic mean Landsberg curvature. We find a condition on the main scalars of F under which it reduces to a Randers metric or a Landsberg metric.

In this paper, we study the projective vector fields on two special (α,β)-metrics, namely Kropina and Matsumoto metrics. First, we consider the Kropina metrics, and show that if a Kropina metric F = α^2/β admits a projective vector field, then this is a conformal vector field with respect to Riemannian metric a or F has vanishing S-curvature. Then we study the Matsumoto metric F = α^2/(α−β) and prove that if the Matsumoto metric F = α^2/β admits a projective vector field, then this is a conformal vector field with respect to Riemannian metric a or F has vanishing S-curvature.

The class of semi-P-reducible Finsler metrics is a rich and basic class of Finsler metrics that contains the class of $L$-reducible metrics, $C$-reducible metrics, and Landsberg metrics. In this paper, we prove that every semi-P-reducible manifold with isotropic Landsberg curvature reduces to semi-$C$-reducible manifolds. Also, we prove that a semi-P-reducible Finsler metric of relatively isotropic mean Landsberg curvature has relatively isotropic Landsberg curvature if and only if it is a semi-$C$-reducible Finsler metric.

In this article, we study the algebraic Ricci solitons of three-dimensional Lie group $mathbb{H}^{2}timesmathbb{R}$, endowed with a left-invariant Riemannian metric. Also, we examine the existence of sol-solitons on the three-dimensional Lie group $Sol_{3}$, endowed with a left-invariant Riemannian metric.

It is proved that every locally flat Finsler manifold is a locally flat Riemannian manifold. Some low dimensional locally Finsler manifolds are classified. It is also proved that in a categorical sense, there is a correspondence between locally flat Finsler manifolds and locally hessian Riemannian manifolds

In this paper, we study the class of semi-C-reducible Finsler manifolds. Under a condition, we prove that every semi-C-reducible Finsler spaces with a semi-P-reducible metric has constant characteristic scalar along Finslerian geodesics or reduces to a Landsberg metric. By this fact, we characterize the class of semi-P-reducible spaces equipped with an (α, β)-metric. More precisely, we proved that such metrics are Berwaldian B= 0,or have vanishing S-curvature S= 0 or satisfy a well-known ODE. This yields an extension of Tayebi-Najafi’s classification for 3-dimensional (α, β)-metric of Landsberg-type.