Positively Curved Cohomogeneity One Manifolds and 3-Sasakian Geometry

Since the round sphere of constant positive (sectional) curvature is the simplest and most symmetric topologically non-trivial Riemannian manifold, it is only natural that manifolds with positive curvature always will have a special appeal, and play an important role in Riemannian geometry. Yet, the general knowledge and understanding of these objects is still rather limited. In particular, although only a few obstructions are known, examples are notoriously hard to come by. The additional structure provided by the presence of a large isometry group has had a significant impact on the subject (for a survey see [Gr]). Aside from classification and structure theorems in this context (as in [HK], [GS1], [GS2], [GK], [Wi2], [Wi3] and [Ro], [FR2], [FR3]), such investigations also provide a natural framework for a systematic search for new examples. In retrospect, the classification of simply connected homogeneous manifolds of positive curvature ([Be],[Wa],[AW],[BB]) is a prime example. It is noteworthy, that in dimensions above 24, only the rank one symmetric spaces, i.e., spheres and projective spaces appear in this classification. The only further known examples of positively curved manifolds are all biquotients [E1, E2, Ba], and so far occur only in dimension 13 and below. A natural measure for the size of a symmetry group is provided by the so-called cohomogeneity, i.e. the dimension of its orbit space. It was recently shown in [Wi3], that the lack of positively curved homogeneous manifolds in higher dimensions in the following sense carries over to any cohomogeneity: If a simply connected positively curved manifold with cohomogeneity k ≥ 1 has dimension at least 18(k + 1)2, then it is homotopy equivalent to a rank one symmetric space.