Singular Riemannian foliations and applications to positive and nonnegative curvature

TL;DR

This paper investigates the structure of singular Riemannian foliations on compact manifolds, classifies certain types, and applies findings to classify low-dimensional positively curved manifolds.

Contribution

It characterizes fundamental groups of regular leaves and introduces A- and B-foliations, generalizing torus actions, with applications to curvature classification.

Findings

Fundamental group structure of regular leaves determined.

Classification of A- and B-foliations on specific manifolds.

Applications to classifying 4- and 5-manifolds with positive or 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.