Torus Bundles over a Torus

1. If a compact Lie group P acts on a completely regular topological space E then E is said to be a principal P-bundle if whenever the relation px — x holds for pÇzP, x£E it follows that p = e, the identity of P. The orbit space X = E/P is called the base space and the map tt: E—>X carrying y into its orbit P-y is called the projection. Suppose now that G is a Lie group, 77i a closed subgroup of G and 77 a closed, normal subgroup of 77i such that P — Hi/H is compact. Then E = G/H becomes a principal P-bundle with base space X = G/H\ and projection g77—->g77i under the action p(gII)=gp~1H. Such a principal bundle will be called canonical. The purpose of this note is to show that if X is a torus of dimension w and P a torus of dimension m then every principal P-bundle over X is canonical and further that the group G lies in an extremely narrow class. Roughly speaking, our method is to try to lift the action of euclidean w-space up to the total space of the bundle and observe what obstructs this effort. The torus of dimension k will be denoted by Tk, the corresponding euclidean space by Rk. We view Rk as acting transitively on Tk with the lattice of integral points in Rk acting ineffectively. We denote by p: Rk-+Tk the usual covering map. Without loss of generality we assume the Pm-bundles over T" are differentiable. %(X) will denote the Lie algebra of infinitely differentiable vector fields on X. Exactly how narrow the class of Lie groups that G lies in will be left to Theorem 2. For the present we prove: