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^<n+1>, 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.

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。