Abstract
We present a first structure theorem for compact simply connected positively curved manifolds with arbitrarily small pinching constants: For each n∈N and 0<δ≤1, there exists a positive number V = V(n,δ) such that if (M,g) is a compact simply connected n-dimensional Riemannian manifold with sectional curvature 0<δ≤K≤1 whose volume is less than V, then there is a smooth locally free S1-action on M, and the base space of the induced Seifert fibration is a simply connected Riemannian orbifold which carries itself a metric of positive curvature.
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