G-ベクトル束と不変ベクトル場について

その他

The main purpose of this note is to exhibit an isomorphism of semi-groups between the equivalence classes of G-vector bundles over a G-manifold with one orbit type and. some classes of vector bundles over the orbit space. The article is a continuation of the author's preceding paper [4]. In which the author has proposed a too restrictive condition, i. e. the normalizer of the isotropy subgroup is the direct product, (C_2) in §2 of [4]. For example, in §4 of Chapter 4, [2], SO(n), SU(n)-actions have been mvestigated. In these cases, the normalizers are semi-direct products, which are shown in §1 of this note. In this note we attain to some kind of vector bundles over orbit spaces, called local H-vector bundles, which behave in a rather different manner than the usual H-vector bundles. We treat in this note only G-manifolds with one orbit type for a simplicity. We could reformulate the theorem 2 in [4] in a semi-direct product case, but the verification is too long, and so we will omit it. Thus this note is a theory concerning fiber bundles with Lie group actions of one orbit type.
In §2, we reconstruct the characterization of G-vector bundles along the line of Part 1, [6]. A pair of transition functions is obtained.
§3 contains a proof of the contonuoty of them, and the main theorem is given.
In §4, we calculate Grothendieck group of local H-vector bundles over spheres.
As in [4], the invariant fields problem is treated in §5. Tangent bundles over G-manifolds are typical exa㎜ples of G-vector bundles. The structure of them as coordinate bundles is analized, and applied to the investigation of invariant fields. The Stiefel manifold is a suitable example for a concrete calculation. In this section we discuss about the total space of a Stiefel manifold bundle over a Stiefel manifold.