Curvature properties of the positively curved Eschenburg spaces

In this thesis we show that all of the Eschenburg spaces of positive curvature have their pinching bounded above by [special characters omitted]in a one parameter family of positively curved metrics. Since the Aloff-Wallach spaces are the homogeneous Eschenburg spaces and since Püttmann has calculated the pinching of [special characters omitted] in the U(2) biinvariant metric to be exactly [special characters omitted] this upper bound is sharp. It is also shown that the only Eschenburg space with pinching exactly [special characters omitted] is W1,1. In addition, using a Cheeger construction, we enlarge the known metrics of positive curvature on the Eschenburg spaces to a simple explicit four parameter family. In order to calculate exact pinching constants, we designed a self-checking numerical algorithm for computing extreme sectional curvatures which applies to any Riemannian manifold. We applied this method to the calculation of the maximal pinching constants of the cohomogeneity one Eschenburg spaces where it performed well and calculated the pinching to six digits of accuracy for many of these spaces