On the structure of compact simply connected manifolds of positive sectional curvature (Q1366283)

The author obtains a first structure theorem for compact simply connected positively curved manifolds with arbitrarily small pinching constants. Indeed, he proves the following. For each \(n\in \mathbb{N}\) and \(0< \delta\leq 1\) there exists a positive number \(V=V(n, \delta)\) such that if \((M,g)\) is a compact simply connected Riemannian manifold of dimension \(n\) with sectional curvature satisfying \(0<\delta <K\leq 1\) and volume \(\text{vol} M<V\), then there is a smooth locally free \(S^1\)-action on \(M\) such that the base space of the induced Seifert fibration is a simply connected Riemannian orbifold which carries itself a metric of positive curvature.