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

The theory of connections is an important field of research in differential geometry. It was initially developed to solve pure geometrical problems. In the Riemannian contex, M. M. Tripathi introduced a new linear connection on a Riemannian manifold, which generalizes many Riemannian connections such as symmetric, semi-symmetric, qurter-symmetric; Ricci qurter-symmetric; metric, non-metric and recurrent connections. In this paper, we extend the work of M. M. Tripath from Riemannian geometry to Finsler geometry, precisely, we investigate a new linear Finsler connection, which unifies the well known linear connections and provides new connections in Finsler geometry. This connection will be named general linear Finsler (GF-) connection. The existence and uniqueness of such a connection is proved. The curvature and torsion tensors are computed. A general reformulation for Cartan, Berwald, Chern and Hashiguchi connections is obtained. Various special cases and connections are studied and introduced. Moreover, some examples of this connection are studied.

In Finsler Geometry, all special spaces are investigated locally (using local coordinates) by many authors. On the other hand, the global (or intrinsic, free from local coordinates) investigation of such spaces is very rare in the literature. The aim of the present paper is to provide an intrinsic investigation of two special Finsler spaces whose defining properties are related to Berwald connection, namely, Finsler space of scalar curvature and of constant curvature. Some characterizations of a Finsler space of scalar curvature are proved. Necessary and sufficient conditions under which a Finsler space of scalar curvature reduces to a Finsler space of constant curvature are investigated. It should finally be noted that the present work is formulated in a prospective modern coordinate-free form. Moreover, the outcome of this work is twofold. Firstly, the local expressions of the obtained results, when calculated, coincide with the existing local results. Secondly, new coordinates-free proofs have been established.

One of the most important concepts in geometry is of geodesics. Geodesic in a manifold is the generalization of notion of a straight line in an Euclidean space. A geodesic in a homogeneous Finsler space (G/H, F) is called homogeneous geodesic if it is an orbit of a one-parameter subgroup of G. Homogeneous geodesics on homogeneous Riemannian manifolds have been studied by many authors. Latifi has extended the concept of homogeneous geodesics in homogeneous Finsler spaces. He has given a criterion for characterization of geodesic vectors. Latifi and Razavi have studied homogeneous geodesics in a 3-dimensional connected Lie group with a left invariant Randers metric and show that all the geodesics on spaces equipped with such metrics are homogeneous.  In this paper, first we give basic definitions required to define a honogeneous Finsler space. Next, we study geodesics and geodesic vectors for homogeneous Finsler space with infinite series (α, β)-metric. Next, we give a lemma in which the existence of invariant vector field corresponding to 1-form β for a homogeneous Finsler space with infinite series metric is proved. Further, we find necessary and sufficient condition for a non-zero vector in this homogeneous space to be a geodesic vector.

In this paper, we study the H-curvature, an important non-Riemannian quantity, for a rich and important class of Finsler metrics called Finsler warped product metrics. We find an equation that characterizes the metrics of almost vanishing H-curvature. Further, we show that, if F is a Finsler warped product metric, then the H-curvature vanishes if and only if the Χ-curvature vanishes.

In this paper, we study generalized symmetric Finsler spaces with Matsumoto metric, infinite series metric and exponential metric.The definition of generalized symmetric Finsler spaces is a natural generalization of the definition of Riemannian generalized symmetric spaces. We prove that generalized symmetric (α, β)−spaces with Matsumoto metric, infinite series metric and exponential metric are Riemannian. We also prove that if (M, F) be a generalized symmetric Matsumoto space with F defined by the Riemannian metric a~ and the vector field X, Then the regular s−structure {sx} of (M, F) is also a regular s−structure of the Riemannian manifold (M, ã) and if (M, ã) be a generalized symmetric Riemannian space and Also suppose that F is a Matsumoto metric introduced by ã and a vector field X, Then the regular s−structure {sx} of (M, ã) is also a regular s−structure of (M, F) if and only if X is sx−invariant for all x in M.

In this paper, we prove that every generalized cubic Finsler metric with vanishing Landsberg curvature is a Berwald metric. This yields an extension of Matsumoto theorem for the cubic metric. Then, we show that every generalized 4-th root Finsler metric with vanishing Landsberg curvature is a Berwald metric.

In this paper, we study a special class of Finsler metrics F = F(x, y) in Rn that satisfy F(−x, y) = F(x, y). We show the induced distance function of F satisfies dF (p, q) = dF (−q,−p) for all p, q in Rn. The geodesics of these metrics have special property and many well-known Finsler metrics belong to this class. We prove that these metrics with constant S-curvature satisfy S = 0.

Z. Shen proved that Finsler manifold with unbounded Cartan torsion can not be isometrically imbedded into any Minkowski space. This shows that the norm of Cartan torsion of Finsler metrics has an essential role for studying of immersion theory in Finsler geometry. In this paper, we study the norm of Cartan torsion of Ingarden-Tàmassy and Arctangent Finsler metrics that are special (α, β)-metrics.

In this paper, we are going to consider a class of (α, β)-metrics which introduced by Matsumoto. We find a condition under which a Matsumoto metric of almost vanishing H-curvature reduces to a Berwald metric.

We consider the Lorentzian pr-waves three-manifolds with recurrect curvature. We obtain a full classification of the Killing and homothetic vector fields of these spaces.

In this paper, we study homogeneous geodesics in homogeneous Randers spaces. we give a four dimensional example and we obtain homogeneous geodesics of this space in some special cases.

In this paper, we characterize locally projectively flat left-invariant Randers metrics on simply connected three dimensional Lie groups.