Positively curved cohomogeneity one manifolds and 3-Sasakian geometry

Abstract

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. This paper deals with manifolds of cohomogeneity one. Recall that in [GZ] a wealth of new nonnegatively curved examples were found among such manifolds. Our ultimate goal is to classify positively curved (simply connected) cohomogeneity one manifolds. The spheres and projective spaces admit an abundance of such actions (cf. [HL, St, Iw1, Iw1], and [Uc]). In [Se], however, it was shown that in dimensions at most six, these are in fact the only ones. In [PV2] it was shown that this is also true in dimension 7, as long as the symmetry group is not locally isomorphic to S 3 × S 3 . Recently Verdiani completed the classification in even dimensions (see [PV1, V1, V2]) :Theorem (Verdiani). An even dimensional simply connected cohomogeneity one manifold with an invariant metric of positive sectional curvature is equivariantly diffeomorphic to a compact rank one symmetric space with a linear action.The same conclusion is false in odd dimensions. There are three normal homogeneous manifolds of positive curvature which admit cohomogeneity one actions: The Berger space B 7 =

Cohomogeneity one manifolds with positive Euler characteristic

“…Vice versa, assuming M being simply connected, one arrives at the following properties of the groups involved: Proposition 1.4 (Grove et al [2008]). Let M be a simply connected cohomogeneity one G-manifold.…”

“…Most of the claims are easily checked, the more involved examples can be found in Grove et al [2008].…”

“…As a motivation for studying primitive G-manifolds, note that if N has non-negative curvature, so has G × L N. The technical value of primitivity lies mostly in the following lemma, taken from Grove et al [2008]: Lemma 1.3. Assume the primitive cohomogeneity one action of G on M is effective (almost effective).…”

“…If M 1 , M 2 carry a cohomogeneity one action of G, we will call M 1 and M 2 G-equivariantly diffeomorphic if there is a diffeomorphism ψ : M 1 → M 2 such that ψ(gp) = gψ(p) for all g ∈ G, p ∈ M 1 . This will determine the group diagram of a cohomogeneity one G-manifold M up to the following operations (see Grove et al [2008]):…”

Ancient solutions of Ricci flow on spheres and generalized Hopf fibrations

“…It is also not known if one can extend Fateev's construction of type-II ancient solutions. As a modest classification question we propose that for the homogenous ancient solutions as well as ones of the cohomogeneity one in view of the recent progresses in this direction on positively curved manifolds [23]. A more specific question is to classify the cohomogeneity one three dimensional ancient solutions.…”