Some new examples with almost positive curvature

TL;DR

This paper explores new examples of manifolds with almost positive curvature, extending known spaces like Eschenburg spaces and certain quotients, to better understand manifolds with near-positive curvature properties.

Contribution

It introduces generalized manifolds that admit metrics with almost positive curvature, expanding the class of known examples in differential geometry.

Findings

Existence of generalized Eschenburg spaces with almost positive curvature

Construction of quotients of S^7 x S^7 with similar properties

New insights into manifolds with near-positive curvature

Abstract

As a means to better understanding manifolds with positive curvature, there has been much recent interest in the study of non-negatively curved manifolds which contain points at which all 2-planes have positive curvature. We show that there are generalisations of the well-known Eschenburg spaces and quotients of $\sph^7 \x \sph^7$ which admit metrics with this property.