Metrics of positive Ricci curvature on quotient spaces

Abstract

One of the classical problems in differential geometry is the investigation of closed manifolds which admit Riemannian metrics with given lower bounds for the sectional or the Ricci curvature and the study of relations between the existence of such metrics and the topology and geometry of the underlying manifold. Despite many efforts during the past decades, this problem is still far from being understood. For example, so far the only obstructions to the existence of a metric with positive Ricci curvature come from the obstructions to the existence of metrics with positive scalar curvature ([Li], [Hi], [Ro], [SchY], [Ta]) and the Bonnet-Myers theorem which implies that the fundamental group of a closed manifold with positive Ricci curvature must be finite. Fruitful constructions of metrics with positive Ricci curvature on closed manifolds have so far been established by techniques that include deformation of metrics ([Au], [Eh], [We]), Kahler geometry ([Yau1], [Yau2]), bundles and warping ([Po], [Na], [BB], [GPT]), special kinds of surgery ([SY], [Wr]), metrical glueing ([GZ2]), and Sasakian geometry ([BGN]). Particularly large classes of examples of manifolds with positive Ricci curvature are given by all compact homogeneous spaces with finite fundamental group ([Na]) and all closed cohomogeneity one manifolds with finite fundamental group ([GZ2]). In this article we present several new classes of manifolds which admit metrics of positive or nonnegative Ricci curvature. The idea is to consider quotients of manifolds (M,g) of positive or nonnegative Ricci curvature by a free isometric action. While taking such a quotient non-decreases the sectional curvature, it may well happen in general that the Ricci curvature of the quotient is inferior to the one of M . Our first results, however, state that the quotient of M does admit metrics of positive Ricci curvature if M belongs to one of the aforementioned classes of homogeneous spaces or spaces with a cohomogeneity one action. More precisely, we prove: