Nonnegatively curved five-manifolds with non-abelian symmetry

Abstract

We classify compact simply-connected 5-dimensional manifolds which admit a metric of nonnegative curvature with a connected non-abelian group acting by isometries. We show that they are diffeomorphic to either \(\hbox {S}^{5}, \hbox {S}^{3} \times \hbox {S}^{2}\), the nontrivial \(\hbox {S}^{3}\)-bundle over \(\hbox {S}^{2}\) or the Wu-manifold, SU(3)/SO(3). This result is a consequence of our equivariant classification of all SO(3) and SU(2)-actions on compact simply-connected five-manifolds. In the case of positive curvature we obtain a partial classification.