Some New Examples with Quasi-positive Curvature

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 a point at which all 2-planes have positive curvature. We show that there are generalisations of the well-known Eschenburg spaces together with quotients of S×S which admit metrics with this property. It is an unfortunate fact that for a simply connected manifold which admits a metric of non-negative curvature there are no known obstructions to admitting positive curvature. While there exist many examples of manifolds with non-negative curvature, the known examples with positive curvature are very sparse (see [Zi] for a comprehensive survey of both situations). 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. 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: Theorem A. (i) Let Lp,q ⊂ U(n+ 1)× U(n+ 1), n ≥ 2, be defined by Lp,q = {(diag(z1 , . . . , zn+1),diag(z1, z2 , A)) | z ∈ S, A ∈ U(n− 1)}, Date: October 24, 2008.