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Singular Riemannian foliations and collapse

Diego Corro

TL;DR

This work surveys the interface between symmetry in Riemannian geometry and collapse phenomena, focusing on singular Riemannian foliations (SRFs) and their relation to Lie groupoid actions. It extends classical deformation techniques, including Cheeger deformations and bounded-curvature collapse via F- and N-structures, to the singular setting, and develops a Cheeger-like deformation for proper Lie groupoid actions. A core theme is the link between closed SRFs and Lie groupoid representations, where holonomy and infinitesimal foliations encode local transverse symmetry and enable global deformation analysis under controlled curvature. The results shed light on when SRFs behave like homogeneous actions (orbit-like) and provide a framework for constructing metrics with controlled curvature along collapsing directions, with potential implications for the Grove symmetry program and metrics of positive or nonnegative curvature.

Abstract

In this survey we present classical results on methods to use group actions to collapse manifolds to the orbit spaces while keeping some control on the curvature, and recent extensions of these constructions to the setting of singular Riemannian foliations.

Singular Riemannian foliations and collapse

TL;DR

This work surveys the interface between symmetry in Riemannian geometry and collapse phenomena, focusing on singular Riemannian foliations (SRFs) and their relation to Lie groupoid actions. It extends classical deformation techniques, including Cheeger deformations and bounded-curvature collapse via F- and N-structures, to the singular setting, and develops a Cheeger-like deformation for proper Lie groupoid actions. A core theme is the link between closed SRFs and Lie groupoid representations, where holonomy and infinitesimal foliations encode local transverse symmetry and enable global deformation analysis under controlled curvature. The results shed light on when SRFs behave like homogeneous actions (orbit-like) and provide a framework for constructing metrics with controlled curvature along collapsing directions, with potential implications for the Grove symmetry program and metrics of positive or nonnegative curvature.

Abstract

In this survey we present classical results on methods to use group actions to collapse manifolds to the orbit spaces while keeping some control on the curvature, and recent extensions of these constructions to the setting of singular Riemannian foliations.

Paper Structure

This paper contains 18 sections, 41 theorems, 58 equations.

Table of Contents

  1. Introduction
  2. Classical deformations
  3. Gromov-Hausdorff distance and collapse
  4. Riemannian submersions and Cheeger deformations
  5. Collapse with bounded curvature F-structures
  6. Singular Riemannian foliations
  7. Infinitesimally foliation
  8. Collapse of singular Riemannian foliations with flat leaves
  9. Lie groupoid action
  10. Normal representation
  11. Riemannian Lie groupoids and invariant actions
  12. Hausdorff proper Lie groupoids and existence of 2-metr-ics
  13. Cheeger-like deformation for Lie groupoid actions
  14. Horizontal distribution
  15. Cheeger tensor
  16. ...and 3 more sections

Key Result

Theorem 1

þ Consider $(M, \mathcal{F}, \mathtt{g})$ a regular closed non-trivial foliation such that for $L\in \mathcal{F}$ we have $(L,\mathtt{g}|_{L})$ is a flat manifold. For $0<\delta\leq 1$ and $p\in M$ given we consider $T_p M= T_pL_p \oplus \nu_p (L_p)$ and set Then $(M,\mathtt{g}_\delta)$ collapses with uniformly bounded curvature and uniformly bounded diameter.

Theorems & Definitions (84)

  • Theorem 1: Theorem B in Corro2024
  • Theorem 2
  • Corollary 3
  • Remark 1.1
  • Theorem 4
  • Conjecture 1.3: GonzalezAlvaroGuijarro2023Grove2002
  • Proposition 2.1: Proposition 7.4.11 in BuragoBuragoIvanov
  • Remark 2.2
  • Theorem 2.3: O'Neill's formula ONeill1966, see Section 1.5 in GromollWalschap
  • Theorem 2.4: Section 3.d in Mueter
  • ...and 74 more