On the Curvature of Biquotients

As a means to better understanding manifolds with positive curvature, there has been much recent interest in the study of nonnegatively curved manifolds which contain either a point or an open dense set of points at which all 2-planes have positive curvature. We study infinite families of biquotients defined by Eschenburg and Bazaikin from this viewpoint, together with torus quotients of S × S. There exist many examples of (compact) manifolds with non-negative curvature. All homogeneous spaces G/H and all biquotients G/U inherit non-negative curvature from the bi-invariant metric on G. Additionally, it is shown in [GZ] that all cohomogeneity-one manifolds, namely manifolds admitting an isometric group action with one-dimensional orbit space, with singular orbits of codimension ≤ 2 admit metrics with non-negative curvature. On the other hand, the known examples with positive curvature are very sparse (see [Zi1] for a survey). Other than the rank-one symmetric spaces there are isolated examples in dimensions 6, 7, 12, 13 and 24 due to Wallach [Wa] and Berger [Ber], and two infinite families, one in dimension 7 (Eschenburg spaces; see [AW], [E1], [E2]) and the other in dimension 13 (Bazaikin spaces; see [Ba]). In recent developments, two distinct metrics with positive curvature on a particular cohomogeneity-one manifold have been proposed ([GVZ], [D]), while in [PW2] the authors propose that the Gromoll-Meyer exotic 7-sphere admits positive curvature, which would be the first exotic sphere known to exhibit this property. Unfortunately, for a simply connected manifold which admits a metric of non-negative curvature there are no known obstructions to admitting positive curvature. In this paper we are interested in the study of manifolds which lie “between” those with non-negative and those with positive sectional curvature. It is hoped that the study of such manifolds will yield a better understanding of the differences between these two classes. Recall that a Riemannian manifold (M, 〈 , 〉) is said to have quasi-positive curvature (resp. almost positive curvature) if (M, 〈 , 〉) has non-negative sectional curvature and there is a point (resp. an open dense set of points) at which all 2-planes have positive sectional curvature. Our main result is: Date: May 12, 2009. 1