Curvature properties of the positively curved Eschenburg spaces

Using a Cheeger construction, we enlarge the known one parameter family of metrics of positive curvature on the Eschenburg spaces to a simple explicit four parameter family. Then we show in a smaller class of metrics that all of the Eschenburg spaces of positive curvature have their pinching bounded above by 1/37. Since the Aloff–Wallach spaces are the homogeneous Eschenburg spaces and since Püttmann has calculated the pinching of W1,1=SU(3)/S^1 1,1 in the U(2) biinvariant metric to be exactly 1/37 this upper bound is sharp. It is also shown that the only Eschenburg space with pinching exactly 1/37 is W1,1.