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We determine the structure of the fundamental group of the regular leaves of a closed singular Riemannian foliation on a compact, simply connected Riemannian manifold. We also study closed singular Riemannian foliations whose leaves are homeomorphic to aspherical or to Bieberbach manifolds. These foliations, which we call  A‐foliations  and  B‐foliations , respectively, generalize isometric torus actions on Riemannian manifolds. We apply our results to the classification problem of compact, simply connected Riemannian   4  ‐ and   5  ‐manifolds with positive or non‐negative sectional curvature.

Singular Riemannian foliations and applications to positive and non‐negative curvature