Non-negatively Curved Cohomogeneity One Manifolds

Non-negatively Curved Cohomogeneity One Manifolds Chenxu He Prof. Wolfgang Ziller, Advisor A Riemannian manifold M is called cohomogeneity one if it admits an isometric action by a compact Lie group G and the orbit space is one dimension. Many new examples of non-negatively curved manifolds were discovered recently in this category. However not every cohomogeneity one manifold carries an invariant metric with non-negative sectional curvature. We show a large family of cohomogeneity one manifolds is obstructed to have a non-negatively curved metric. It generalizes the first examples obtained by K. Grove, B. Wilking, L. Veridiani and W. Ziller. The cohomogeneity one manifolds which have a small family of invariant metrics are also studied. If the principal isotropy action has three or less summands and G is a simple Lie group, then the manifold is equivariantly diffeomorphic to a double or a symmetric space.