In this paper, we shall construct compact Lie group actions on total spaces of orientable sphere bundles over spheres. All actions considered in this paper preserve bundle structures, that is, each element of groups gives a bundle map. The author intends to construct actions on all sphere bundles over S^n for n≦8. For n > 8, actions on S^k-bundlles over S^n are given for the case of k ≧ n and other particular n, k. Thus we can conclude that these bundle spaces have positive degrees of symmetry.
In section 1, we construct actions on S^3-bundles over S^4 and S^7-bundles over S^8. By means of reductions of structure groups, we can give S^1-actions on S^k-bundles over S^n for k≧n. Using well-knowm results from the homotopy theory of spheres and rotation groups, we construct actions on S^k-bundles over S^n for k＜n≦8 in section 2. In the last section, we construct actions on S^<4s-1>_bundles over S^<4s> of types B_<l,o>, B_<o,l> and B,εk m k, where ε=1 if s is odd, ε=2 if s is even, m-(2s-1)!/2 and k is an integer.
The technique used in this paper is quite homotopical and actually elementary. Our results are essentially due to the computatioms of M. A. Kervaire, [3] and we shall use it frequently in this paper.