In this paper we study Generalized symmetric Berwald spaces. We show that if a Berwald space (M, F) admits a parallel s-structure then it is locally symmetric. For a complete Berwald space which admits a parallel s-structure we show that if the flag curvature of (M, F) is everywhere nonzero, then F is Riemannian.

Recently we introduced a general class of Finsler connections which lead to a smart representation of connection theory in Finsler geometry and yielded a classification of Finsler connections into the three classes. Here, the properties of one of these classes, namely the Berwald-type connection which contains Berwald and Chern (Rund) connections as special cases are studied. It is proved, among other results, that the hv-curvature of these connections vanishes if and only if the Finsler space is a Berwald one. Some applications of this connection is discussed.

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.

The notion of Generalized Berwald manifolds goes back to V. Wagner [60]. They are Finsler manifolds admitting linear connections on the base manifold such that the parallel transports preserve the Finslerian length of tangent vectors (compatibility condition). Presenting a panoramic view of the general theory we are going to summarize some special problems and results. Spaces of special metrics are of special interest in the Generalized Berwald manifold theory. We discuss the case of Generalized Berwald Randers metrics, Finsler surfaces and Finsler manifolds of dimension three. To provide the unicity of the compatible linear connection we are looking for, we introduce the notion of the extremal compatible linear connection minimizing the norm of the torsion tensor point by point. The mathematical formulation is given in terms of a conditional extremum problem for checking the existence of compatible linear connections in general. Explicite computations are presented in the special case of Generalized Berwald Randers metrics.

By using Berwald connection, we show that there are linear connections Ñ which are projectively equivalent and belong to the same projective structure on TM. We found a condition for the geodesics of the Berwald connection under which Ñ is complete.

A Cartan manifold is a smooth manifold M whose slit cotangent bundle T* M0 is endowed with a regular Hamiltonian K which is positively homogeneous of degree 2 in momenta. The Hamiltonian K defines a (pseudo) -Riemannian metric Gij in the vertical bundle over T* M0 and using it, a Sasaki type metric on T* M0 is constructed. A natural almost complex structure is also defined by K on T* M0 in such a way that pairing it with the Sasaki type metric an almost Kahler structure is obtained. In this paper we deform Gij to a pseudo-Riemannian metric Gij and we define a corresponding almost complex Kahler structure. We determine the Levi-Civita connection of G and compute all the components of its curvature. Then we prove that if the structure (T* M0, G, J)is Kahler-Einstein, then the Cartan structure given by K reduces to a Riemannian one.

It is well-known that M. Matsumoto and R. Miron constructed an orthonormal frame for an n-dimensional Finsler space and the frame was called ’Miron frame’. Then, T. N. Pandey and D. K. Diwedi and P. N. Pandey and Manish Gupta studied four dimensional Finsler spaces in terms of scalars. P. N. Pandey and Manish Gupta also studied four dimensional Berwald space with vanishing h-connection vector. Gauree Shankar, G C. Chaubey and Vinay Pandey studied the main scalar of a five-dimensional Finsler space. In the present paper, we study a five dimensional Berwald space with vanishing hconnection vector hi, Ji, ki.

In this paper, we study Generalized Douglas-Weyl Finsler metrics. We find some conditions under which the class of Generalized Douglas-Weyl (a, b)-metric with vanishing S-curvature reduce to the class of Berwald metrics.

This paper introduces new non-Riemannian quantities and classes of Finsler metrics. The study focuses on the class of Generalized Douglas Weyl GDW-metrics, which is contained in the class of Finsler metrics. The paper constructs the new sub-classes of GDW-metrics and presents illustrative examples.