球面上のベクトル束へのS^1-作用

In the previous papers [6], [7] and [8], we studied some group actions on sphere bundles over spheres and proved some non existence theorems. In this paper, we shall study S^1 -vector bundle structures on vector bundles over sphercs. We shall fix a representation of S^1 on the fiber over the north pole.
Section 1 provides some preliminaries and we shall prove some classification theorems for S^1-vector bundles with a designated action on the fibre over the north pole (Theorem 1 and Theorem 2). As a corollary, we shall obtain a non existence theorem.
In seotion 2, we shall construct two kinds of lifting actions. One of them is a lifting of a linear action on a sphere and the other is a lifting of a quasi linear action.
In the last section, we shall show that some action on a sphere bundle over a sphere can not be derived from an S^1-vector bundle with a specified action on the fibre over the north pole.