Non-negative curvature obstructions in cohomogeneity one and the Kervaire spheres

Abstract

In contrast to the homogeneous case, we show that there are compact cohomogeneity one manifolds that do not support invariant metrics of nonnegative sectional curvature. In fact we exhibit infinite families of such manifolds including the exotic Kervaire spheres. Such examples exist for any codimension of the singular orbits except for the case when both are equal to two, where existence of non-negatively curved metrics is known. Mathematics Subject Classification (2000): 53C20. Most examples of manifolds with non-negative sectional curvature are obtained via product and quotient constructions, starting from compact Lie groups with biinvariant metrics. In [GZ1] a new large class of non-negatively curved compact manifolds was constructed by using Lie group actions whose quotients are onedimensional, so called cohomogeneity one actions. It was shown that if the action has two singular orbits of codimension two, then the manifold carries an invariant metric with non-negative sectional curvature. This in particular gives rise to non-negatively curved metrics on all sphere bundles over S4, which include the Milnor exotic spheres. In [GZ1] it was also conjectured that any cohomogeneity one manifold should carry an invariant metric with non-negative sectional curvature. The most interesting application of this conjecture would be the Kervaire spheres, which can be presented as particular 2n − 1 dimensional Brieskorn varieties M2n−1 d ⊂ Cn+1 defined by the equations zd 0 + z2 1 + · · · z2 n = 0 , |z0| + · · · |zn| = 1. The first named author was supported in part by the Danish Research Council, the second named author was supported in part by GNSAGA and the last named author by the Francis J. Carey Term Chair and the Clay Institute. The first and the last two authors were also supported by grants from the National Science Foundation. Received February 2, 2006; accepted April 12, 2006.