Cohomogeneity one manifolds with positive Ricci curvature
A connected manifold \(M\) is said to have cohomogeneity one if \(M\) supports a smooth action by a compact Lie group \(G\) such that the orbit space \(M/G\) is one-dimensional. In this paper, the authors prove the following theorem A: A compact cohomogeneity one manifold admits an invariant metric with positive Ricci curvature if and only if its fundamental group is finite. A special case of Theorem A (the case of Kervaire spheres) was treated earlier by J. Cheeger (1973) and by A. Back and W. Y. Hsiang (1987). The authors obtain also the following result, which in fact was the starting point for Theorem A. Theorem B: Any cohomogeneity one manifold admits a complete invariant metric with nonnegative Ricci curvature. Theorem B establishes a special case of a conjecture which was made by the authors in their paper [Ann. Math. (2) 152, 331--367 (2000; Zbl 0991.53016)].
scientific article; zbMATH DE number 1965434