On the Finsler Connection Associated with a Linear Connection Satisfying P^h_<ikj>=0

In a Finsler manifold provided with a linear connection Γ^i_(x), a Finsler connection such as Γ^*={Γ^i_(x), Γ^i_(x)y^l, C^i_} can be considered, where C^i_=1/2g^ ∂_mg_. The connection Γ^* is called the Finsler connection associated with Γ^i_(x). In this case, the A-covariant derivative ▽k, v-covariant derivative ▽k and the hv-curvature tensor P^h_, which have been defined by Matsumoto [8], can be also considered

Kaehlerian Finsler Manifolds

In the present paper, continued the preceding paper [10], we are mainly concerned with a Kaehlerian Finsler manifold (M, f, g). First, in the Kaehlerian Finsler manifold, we define a generalized Finsler metric g^^～ by g^^～=(g+^tfgf)/2. We investigate the relation between the Finsler metric g, the generalized Finsler metric g^^～, the complex structure f and several Finsler connections derived from g and g^^～. In consequence of it, we obtain that the Kaehlerian Finsler manifold is a Landsberg space and the generalized Finsler metric g^^～ can be regarded as a real representation of a complex Finsler metric in a sense. Finally we find a necessary and sufficient condition for an Hermitian structure on the tangent bundle over a Kaehlerian Finsler manifold to be a Kaehler structure

Conformally Flat Finsler Structures

In the present paper, we consider the conformal theory of Finsler manifolds. We find, under a certain condition, a conformally invariant Finsler connection and several conformally invariant tensors of a Finsler metric. Finally we come to show, in terms of the conformally invariant tensors, the necessary and sufficient condition for a Finsler manifold to be conformally flat

Finsler Manifolds with a Linear Connection

In the previous paper [5], the present author has treated Finsler manifolds with such a property that the tangent spaces at arbitrary points are congruent (isometrically linearly isomorphic) to a single Minkowski space, and introduced, as a typical example of such a space, the notion of {V, H}-manifolds. At the same time, it has been shown that the {V, H}-manifolds are generalized Berwald spaces defined by Hashiguchi [3] and Wagner [9]. Now, the present paper has two main purposes. One is to consider the converse of the above-mentioned result. After some preparation, it will be proved, in §3, that if a generalized Berwald space is connected, it is actually a {V, H}-manifold. Next, in a Minkowski space is presented a Riemann metric, which is different from the Minkowski norm. Therefore, geodesic lines with respect to this Riemann metric can be introduced in the Minkowski space, which we call C-geodesics. A connected Finsler manifold M with a linear connection Γ^i_(x) is to be considered. With regard to arbitrary two points p and q in M and any piecewise differentiable curve C joining p and q, we can define a linear isomorphic mapping σ between the tangent Minkowski spaces Tp(M] and Tq(M) by parallel displacement with respect to Γ^i_(x) along the curve C. Now, the other main purpose of the present paper is to find a necessary and sufficient condition for a to map any C-geodesic in Tp(M) to a C-geodesic in Tq(M). It will be shown, in the last section, that the condition is C^i_=0 or equivalently P^i_=0 with respect to the Finser connection associated with the linear connection Γ^i_(x)

On the Conditions for a {V, H}-manifold to be Locally Minkowskian or Conformally Flat

The present paper is the continuation of the serial papers concerning the Finsler manifold modeled on a Minkowski space ([5], [6], [7]). A Finsler manifold whose tangent spaces at arbitrary points are congruent to a unique Minkowski space is called a Finsler manifold modeled on a Minkowski space. As an example of it, the notion of the {V, H}-manifold has been introduced in the paper [5]. On the other hand, M. Hashiguchi has defined a notion of a generalized Berwald space [2]. Following his definition, it is a Finsler manifold admitting a linear connection Γ(x) with respect to which ▽g=0 holds, where ▽ denotes the covariant derivative with respect to the Finsler connection (r'jk(x), rimk(x)ym). It has been shown, in the paper [5], that the [V, H}-manifold is a generalized Berwald space. In the paper [6] it has been proved that a standard generalized Berwald space is a {V, H}-manifold. And also it has been found, in the paper [7], that a Finsler manifold with a linear connection Γ(x) with respect to which ▽C=0 holds good becomes a {V, H}-manifold under some condition, where C is the tensor given by C^i_=1/2g^∂mg_. Now, the main purpose of the present paper is to find the following two: The one is the condition for the {V, H}-manifold to be locally Minkowskian and the another is the condition for the {V, H}-manifold to be locally conformal to a Minkowski space. These will be shown in § 1 and § 2 using the terminology of the theory of G-structures. In section 3, examples of these manifolds will be shown especially in the case where the manifolds admit a Randers metric. The last section is devoted to consideration on the case that a Finsler manifold is globally conformal to a locally Minkowskian manifold

On Holonomy Mappings Associated with a Non-linear Connection

Let us consider a manifold provided with a non-linear connection. In the preceding paper [6] we have defined a notion of a holonomy mapping and found a gemetrical significance for the condition that the hv-curvature tensor of a Finsler connection vanishes. In the present paper, we investigate the holonomy mappings in detail and consider, in connection with the holonomy mappings, the character of Landsberg spaces, Berwald spaces, generalized Berwald spaces and almost Hermitian Finsler spaces

General Projective Connections and Finsler Metric

This paper is the continuation of the paper formerly written by one of the authors (Ichijyo). In the former paper, the general projective connections on the tangent bundle over a C^∞-manifold were discussed. But, in that case, it was necessary to choose canonical parameters independently. In this paper, we first consider a vector bundle having R^, the real number space of (n+1)-dimensions, as the standard fibre and a subgroup of GL(n+1; R) as the structural group. This vector bundle was introduced by T. Otsuki for studying his restricted projective connection and was named a projective vector bundle. Now, our intention is on the generalization of the former case to the projective vector bundle. In §§1 and 2, we define the projective vector bundle and the general projective connection on it, and discuss some properties of them. Then, a projectively invariant distribution p is defined. The integrability condition for p is discussed in §3. §4 is devoted to the study of the holonomy group of the general projective connection, especially the case in which the holonomy group leaves a certain hypercone invariant is studied. In the last section we try to extend some known results on holonomy groups to the case in which the base manifold of the projective vector bundle is assumed to have a Finsler metric. As for the references, we wish to refer the former paper