Singular Riemannian foliations and applications to positive and nonnegative curvature

Abstract

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 nonnegative sectional curvature.

23 pages, 1 figure. Final version. Accepted for publication by the Journal of Topology

References in corpus (4)

• Examples of Riemannian Manifolds with non-negative sectional curvature
• A knot characterization and 1-connected nonnegatively curved 4-manifolds with circle symmetry
• Low dimensional Singular Riemannian Foliations in spheres
• Differentiable classification of 4-manifolds with singular Riemannian foliations

Cited by in corpus (10)

• Structure of Submetries
• Isoparametric hypersurfaces with four principal curvatures, IV
• Positive curvature and torus symmetry in small dimensions, I -- Dimensions 10, 12, 14, and 16
• Differentiable classification of 4-manifolds with singular Riemannian foliations
• Riemannian foliation with exotic tori as leaves
• Core reduction for singular Riemannian foliations in positive curvature
• Yamabe problem in the presence of singular Riemannian Foliations
• A-Foliations of codimension two on compact simply-connected manifolds
• A Weyl's Law for Singular Riemannian Foliations with Applications to Invariant Theory
• Singular Riemannian flows and characteristic numbers