The well-known Li-Crampin theorem states that on weakly Berwald manifolds, a Finsler metric is a Landsberg metric if and only if it is a Berwald metric. In this paper, we extend this result to the class of (alpha, /3)-metrics. More precisely, we prove that an (alpha, /3)-metric with isotropic mean Berwald curvature is a generalized Landsberg metric if and only if it is a Berwald metric. As a consequence, every generalized Berwald C-reducible metric is a generalized Landsberg metric if and only if it is a Berwald metric. Furthermore, we show that any connected, positively (or negatively) complete generalized Berwald surface is a generalized Landsberg surface if and only if it is either Riemannian or locally Minkowskian, generalizing Vincze's result for Landsberg surfaces. Additionally, we construct a special pair consisting of a Riemannian metric and a 1-form, which can be used to generate infinitely many pure generalized Berwald metrics. We then investigate certain Riemannian and non-Riemannian curvature properties of the Randers metric derived from this pair. Finally, by providing an illustrative example, we demonstrate that the 2-dimensional Bartelme ss-Lang rigidity theorem does not extend to higher dimensions. (c) 2026 Elsevier B.V. All rights are reserved, including those for text and data mining, AI training, and similar technologies.

In this paper, we study the class of two-dimensional Douglas-recurrent Finsler metrics, which we refer to as D-recurrent Finsler surfaces for simplicity. First, we derive the necessary and sufficient condition under which a Finsler surface is D-recurrent. Then, we show that a Kropina surface is D-recurrent if and only if it is a Douglas surface

The class of spherically symmetric Finsler metrics forms a rich and important class of Finsler metrics. In this paper, for an arbitrary spherically symmetric Finsler metric, we construct four classes of Finsler metrics which are projectively related to it.

In this paper, we study some non-Riemannian curvature properties of general spherically symmetric Finsler metrics. First, we prove that every general spherically symmetric Finsler metric is semi-C-reducible. Then, we find the necessary and sufficient condition under which a general spherically symmetric Finsler metric has vanishing weakly stretch curvature.

In this paper, we study the conformal transformation of recent defined non-Riemannian curvature in Finsler Geometry, namely $\Xi$-curvature. Indeed, we obtain the necessary and sufficient condition under which the conformal transformation preserves the $\Xi$-curvature.

In this paper, we study some non-Riemannian curvature properties of general spherically symmetric Finsler metrics. First, we find the necessary and sufficient condition under which a general spherically symmetric Finsler metric be weakly Berwaldian. Then we find the necessary and sufficient condition under which a general spherically symmetric Finsler metric be of isotropic Berwald curvature. Finally, we find the necessary and sufficient condition under which a general spherically symmetric Finsler metric be Landsbergian.

The class of generalized Berwald metrics contains the class of Berwald metrics as a special case. Let F= αΦ(s), s=β/α, be a generalized Berwald (α, β)-metric on manifold M. We show that F has vanishing S-curvature S=0 and is of relatively isotropic Landsberg curvature L+ cFC=0 if and only if B=0, where c=c(x) is a scalar function on M.

In this paper, we are going to study the Landsberg curvature of 4-dimensional Finsler manifolds. First, we characterize 4-dimensional Landsberg manifolds. Then, we study the mean Landsberg curvature of 4-dimensional Finsler manifolds and give the necessary and sufficient condition under which these manifolds are weakly Landsberg. Finally, by using the obtained results, we find the necessary and sufficient condition under which a 4-dimensional Finsler manifold is a Landsberg manifold.

Every Berwald metric is a special generalized Berwald metric. In this paper, we study the class of projectively flat generalized Berwald (alpha, beta)-metrics of isotropic S-curvature. We find some conditions under which this class of Finsler metrics reduces to the class of Berwald metrics.

There are several non-Riemannian curvatures in Finsler geometry which show the complexity of Finsler geometry with respect to Riemannian geometry. Amon these quantities, the Cartan and mean Cartan torsion have very important and brilliant positions. In this paper, we find the necessary and sufficient condition under which a 4-dimensional Finsler manifold is C-reducible. Also, we find the necessary and sufficient condition under which a 4-dimensional Finsler manifold has vanishing ${\bar I}$-curvature.

We show that every Finsler surface with isotropic main scalar and isotropic flag curvature is Riemannian or relatively constant Landsberg metric. Using it, we prove that every homogeneous Finsler surface with isotropic flag curvature and isotropic main scalar is Riemannian or locally Minkowskian.

Let F = alpha phi(s), s := beta/alpha, be a generalized Berwald (alpha, beta)-metric on a 2-dimensional manifold. Suppose that F has vanishing S-curvature and phi '(0) not equal 0. We show that if F is regular, then it is a locally Minkowskian metric. If F is an almost regular and non-locally Minkowskian metric, then we explicitly determine the function phi which results in generalized Berwald metrics not belonging to the classes of Berwald, Landsb erg or Douglas metrics. Furthermore, we prove that a left-invariant Finsler metric on a 2-dimensional Lie group has vanishing S-curvature if and only if it is a Riemannian metric of constant Gaussian curvature. Finally, we construct a family of Randers-type generalized Berwald metrics on an arbitrary odd-dimensional manifold.

In this paper, we study some important Remannian and non-Riemannian curvature properties of spherically symmetric Finsler metrics. Under a condition on the geodesic coefficient, we find the necessary and sufficient conditions under which spherically symmetric metrics are of scalar flag curvature, [Formula: see text]-quadratic or projectively Ricci-flat. For spherically symmetric metrics of relatively isotropic Landsberg curvature, we find the necessary and sufficient conditions under which these metrics are of constant flag curvature or Ricci-quadratic. Finally, we prove a rigidity result that every spherically symmetric metric of relatively isotropic Landsberg curvature is Ricci-quadratic if and only if it is a Berwald metric. Moreover, if the Finsler metric is negatively complete then it reduces to a Riemannian metric.

In this paper, we study some important non-Riemannian curvature properties of the new class of (α ,β ) -metrics introduced by Pişcoran–Mishra in Finsler geometry. We prove that this class of Finsler metrics are Landsbergian if and only if they are weakly Landsbergian if and only if they are Berwaldian. Then, we show that this class of Finsler metrics has vanishing Ξ -curvature if and only if they have vanishing S-curvature S=0 . Finally, we show that this class of Finsler metrics has almost vanishing H-curvature if and only if H=0 .

Let (M,F) be a Finsler surface with the isotropic main scalar I=I(x). The well-known Berwald’s theorem states that F is a Berwald metric if and only if it has a constant main scalar I=constant. This ensures a kind of equality of two non-Riemannian quantities for Finsler surfaces. In this paper, we consider a positively curved Finsler surface and show that H=0 if and only if I=0. This provides an extension of Berwald’s theorem. It follows that F has an isotropic scalar flag curvature if and only if it is Riemannian. Our results yield an infrastructural development of some equalities for two-dimensional Finsler manifolds.

In this paper, we study the Landsberg curvature of the class of spherically symmetric Finsler metrics. We find the necessary and sufficient condition under which a spherically symmetric Finsler metric of dimension n >= 3 has relatively isotropic Landsberg curvature. This yields an extension of Mo-Zhou and Elgendi's results that proved for the spherically symmetric Finsler metric with vanishing Landsberg curvature. Finally, we characterize spherically symmetric Finsler metrics with vanishing stretch curvature.

This paper is divided into two main parts. In the first part, we study left-invariant Randers metrics on Lie groups. We characterize the class of left-invariant Randers metrics with isotropic mean Berwald and isotropic Berwald curvatures on Lie groups. This yields an extension of Deng's well-known theorem for left-invariant Randers metrics with isotropic S-curvature. In the second part, we consider the left-invariant Randers metrics on tangent Lie groups. Let G be a Lie group equipped with a left-invariant Randers metric F. Suppose that F-v and F-c denote the vertical and complete lift of F on TG, respectively. First, we find the necessary and sufficient condition under which these metrics are weakly Berwaldian. Then, we prove that these lifting Randers metrics are isotropic Berwald metrics if and only if F reduces to a Berwald metric. Finally, we give the necessary and sufficient conditions under which these metrics are of Douglas-type metrics.

In this paper, we study a long existing open problem on Landsberg metrics in Finsler geometry. For this aim, we study the Landsberg curvature of three-dimensional homogeneous Finsler manifolds. First, we express the second Matsumoto torsion of three-dimensional Finsler manifolds, explicitly. Then, we show that the mean Lands berg curvature of three-dimensional homogeneous Finsler manifolds satisfy an ODE. Finally, we prove that every homogeneous 3-dimensional L-reducible Finsler manifold has constant relatively isotropic mean Landsberg curvature if and only if it is a Lands berg metric or a Randers metric of Berwald-type.

In this paper, we have considered conformal-Matsumoto change of the class of m-th root Finsler metrics. We have established the necessary and sufficient condition for the transformed metric to be projectively flat or locally dually flat. Further, we have proved the non-existence of the concerned metric which is projectively flat with nonzero flag curvature.

The theory of mth root Finsler metrics has been applied to Biology, Ecology, Gravitation, Seismic ray theory, etc. It is regarded as a direct generalization of Riemannian metric in a sense, namely, the second root metric is a Riemannian metric. On the other hand, the Riemannian curvature faithfully reveals the local geometric properties of a Riemann–Finsler metric. The reversibility of Riemannian and Ricci curvatures of Finsler metrics is an essential concept in Finsler geometry. Here, we study the Riemannian curvature of the class of third and fourth root [Formula: see text]-metrics. Then, we find the necessary and sufficient condition under which a cubic and fourth root [Formula: see text]-metric be Einstein-reversible.