Riemannian submersions of compact simple Lie groups with connected totally geodesic fibres

Abstract

In [1, 2] and [5] Riemannian submersions of spheres and complex projective spaces with totally geodesic fibres have been classified. It is natural to investigate similar submersions from other symmetric spaces. In this paper, we address ourselves to the classification problem of Riemannian submersions with totally geodesic fibres of compact simple Lie groups with biinvariant metrics. The most natural examples of such Riemannian submersions in the case of compact Lie groups are given by closed subgroups H and the quotient maps n: G->G/H (or ~: G~H\G) . Notice that the submersions given by left cosets are equivalent to those given by right cosets via the isometry i: G~G given by i(g)=g -1. The obvious question therefore is to ask whether the submersions with totally geodesic fibres are all obtained in the above manner or not. In this paper we give a partially affirmative answer to this question. More precisely, we have