Journal of Finsler Geometry and its Applications
Volume:3 Issue: 2, Dec 2022

This paper mainly studies the volume comparison in Finsler geometry under the condition that the weighted Ricci curvature Ric∞ has a lower bound. By using the Laplacian comparison theorems of distance function, we characterize the growth ratio of the volume coefficients. Further, some volume comparison theorems of Bishop-Gromov type are obtained.

In this paper, we study generalized symmetric Finsler spaces with special (α , β ) -space. In fact, we study this spaces with square-root metric and we prove that generalized symmetric (α , β ) -spaces with square-root metric must be Riemannian.

Every Landsberg metric and every Landsbeg metric is a weakly Landsberg metric, but the converse is not true generally. Let (M, F) be a 3-dimensional Finsler manifold. In this paper, we find a condition under which the notions of weakly Landsberg metric and Landsberg metric are equivalent.

In this paper, we study conformally flat 5-th root (α, β)-metrics. We prove that everyconformally flat 5-th root (α, β)-metric with relatively isotropic mean Landsberg curvaturemust be either Riemannian metrics or locally Minkowski metrics.

In order to extend the sphere theorem for Finsler metrics, the concept of reversibil-ity introduced by H-B. Rademacher for a compact Finsler manifold. In this paper, weextend this notion to the general Finsler manifolds. Then we find an upper bound forthe reversibility of some important spherically symmetric Finsler metrics. Furthermore,we introduce the concept of sub-reversibility for a general Finsler manifold and obtain anon-zero lower bound for this new quantity.

In this paper, we give an explicit formula for the flag curvature of invariant square metric and Randers change of square metric.

In this paper, we construct a family of Finsler metrics, called square-type Finsler metrics. We obtain the  flag curvature of this metric. Then we  find a necessary and sufficient condition under which the  flag curvature of square-type Finsler metrics becomes zero.

In this paper, we find necessary and sufficient conditions underwhich the infinite series metric and Randers metric on a manifold M of dimension n >3 be projectively related.

General (α, β) metrics form a rich and important class of metrics. Many well-known Finsler metrics of constant flag curvature can be locally expressed as a general (α, β) metrics. In this paper, we study the general (α, β) metrics with constant Ricci curvature (tensor) and constant flag curvature. Moreover we study the vanishing non-Riemannian quantity χ-curvature.

In this paper, we study the Finsler warped product metric which is Einstein. We find equation that characterize Einstein Finsler warped product metrics with vanishing Douglas curvature. Moreover, we obtain the differential equation that characterizes Einstein Finsler warped product metrics of locally projectively flat.

The main focus of this paper is concern to the study on the point-wise Osserman structure on 4-dimensional Lorentzian Lie group. In this paper we study on the spectrum of the Jacobi operator and spectrum of the skew-symmetric curvature operator on the non-abelian 4-dimensional Lie group G, whenever G equipped with an orthonormal left invariant pseudo-Riemannian metric g of signature (-;+;+; +), i.e, Lorentzian metric, where e1 is a unit time-like vector. The Lie algebra structure in dimension four has key role in our investigation, also in this case we study on the classification of 1-Stein and mixed IP spaces. At the end we show that G does not admit any space form and Einstein structures.

In [12], authors introduced some geometric concepts such as (almost) product, para-complex, para-Hermitian and para-Kähler structures for hom-Lie algebras and they presented an example of a 4-dimensional hom-Lie algebra, which contains these concepts. In this paper, we classify two-dimensional hom-Lie algebras containing these structures. In particular, we show that there doesn't exist para-Kahler proper hom-Lie algebra of dimension 2.