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Collapsing regular Riemannian foliations with flat leaves

Diego Corro

TL;DR

This work develops a geometric collapse framework for manifolds with closed flat regular Riemannian foliations of positive leaf dimension, preserving uniform curvature bounds. By constructing explicit collapsing metrics and then approximating them with $N$-structures, the authors show that, on compact simply connected manifolds, such foliations arise from torus actions. The results yield a precise link between collapse with bounded curvature and torus-symmetric foliations, providing a rigidity-type characterization of aspherical regular foliations. The approach combines foliations, F- and N-structure theory, and equivariant convergence to bridge geometric collapse and group actions, with potential implications for understanding symmetry in Riemannian geometry.

Abstract

In this manuscript we present how to collapse a manifold equipped with a closed flat regular Riemannian foliation with leaves of positive dimension, while keeping the sectional curvature uniformly bounded from above and below. From this deformation, we show that in the case when the manifold is compact and simply connected the foliation is given by torus actions. This gives a geometric characterization of aspherical regular Riemannian foliations given by torus actions

Collapsing regular Riemannian foliations with flat leaves

TL;DR

This work develops a geometric collapse framework for manifolds with closed flat regular Riemannian foliations of positive leaf dimension, preserving uniform curvature bounds. By constructing explicit collapsing metrics and then approximating them with -structures, the authors show that, on compact simply connected manifolds, such foliations arise from torus actions. The results yield a precise link between collapse with bounded curvature and torus-symmetric foliations, providing a rigidity-type characterization of aspherical regular foliations. The approach combines foliations, F- and N-structure theory, and equivariant convergence to bridge geometric collapse and group actions, with potential implications for understanding symmetry in Riemannian geometry.

Abstract

In this manuscript we present how to collapse a manifold equipped with a closed flat regular Riemannian foliation with leaves of positive dimension, while keeping the sectional curvature uniformly bounded from above and below. From this deformation, we show that in the case when the manifold is compact and simply connected the foliation is given by torus actions. This gives a geometric characterization of aspherical regular Riemannian foliations given by torus actions
Paper Structure (13 sections, 26 theorems, 39 equations)

This paper contains 13 sections, 26 theorems, 39 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Singular Riemannian foliations
  4. Holonomy
  5. A- and B-foliations
  6. Compact flat metric spaces
  7. Alexandrov spaces
  8. Equivariant convergence
  9. N- and F-structures
  10. Proofs of Main Theorems
  11. Proof of Theorem \ref{['MT: Collapse of flat foliations']}
  12. Proof of Theorem \ref{['MT: Approximating flat foliations by N-structures']}
  13. Proof of Theorem \ref{['MT: flat foliations are given by group actions']}

Key Result

Theorem 1

þ Consider $(M,\mathcal{F},g)$ a regular $B$-foliation on a compact simply-connected Riemannian manifold. Then the foliation $\mathcal{F}$ is induced by an isometric torus action.

Theorems & Definitions (44)

  • Theorem 1
  • Theorem 2
  • Theorem 3
  • Remark 1.1
  • Corollary 4
  • Proposition 2.1: Proposition 2.17 in Radeschi2017
  • Theorem 2.2: Corollary 1.5 in MendesRadeschi2019
  • Theorem 2.3: Theorem B in GalazGarciaRadeschi2015
  • Remark 2.4
  • Theorem 2.5: Corollary 3.2 and Proposition 3.3 in Corro2023
  • ...and 34 more