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_<jk>=1/2g^<im>∂mg_<jk>. 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.

公開日:2010年1月24日で登録したコンテンツは、国立情報学研究所において電子化したものです。