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Cheeger Deformations for Lie groupoid actions

Diego Corro

TL;DR

This work extends Cheeger deformation techniques to singular Riemannian foliations arising from proper Lie groupoid actions, providing a one-parameter family of metrics that collapses along orbits while preserving transverse geometry. The construction leverages a transversely invariant base metric and a 1-metric from a 2-metric on the groupoid, yielding a deformation whose curvature is controlled by an O'Neill-type decomposition involving the base foliation, the groupoid metric, the $A$-tensor, and the second fundamental form. It also clarifies limitations in the global deformation problem for general foliations and recovers the classical Cheeger deformation in the special case of a pair groupoid from a group action. The results enable explicit curvature computations for the deformed metrics and connect the limiting behavior to the metric on the leaf space, suggesting potential applications in constructing metrics with prescribed curvature lower bounds in foliated or groupoid-symmetric settings.

Abstract

We give an extension of Cheeger's deformation techniques for smooth Lie group actions on manifolds to the setting of singular Riemannian foliations induced by Lie groupoids actions. We give an explicit description of the sectional curvature of our generalized Cheeger deformation.

Cheeger Deformations for Lie groupoid actions

TL;DR

This work extends Cheeger deformation techniques to singular Riemannian foliations arising from proper Lie groupoid actions, providing a one-parameter family of metrics that collapses along orbits while preserving transverse geometry. The construction leverages a transversely invariant base metric and a 1-metric from a 2-metric on the groupoid, yielding a deformation whose curvature is controlled by an O'Neill-type decomposition involving the base foliation, the groupoid metric, the -tensor, and the second fundamental form. It also clarifies limitations in the global deformation problem for general foliations and recovers the classical Cheeger deformation in the special case of a pair groupoid from a group action. The results enable explicit curvature computations for the deformed metrics and connect the limiting behavior to the metric on the leaf space, suggesting potential applications in constructing metrics with prescribed curvature lower bounds in foliated or groupoid-symmetric settings.

Abstract

We give an extension of Cheeger's deformation techniques for smooth Lie group actions on manifolds to the setting of singular Riemannian foliations induced by Lie groupoids actions. We give an explicit description of the sectional curvature of our generalized Cheeger deformation.

Paper Structure

This paper contains 15 sections, 27 theorems, 72 equations.

Table of Contents

  1. Introduction
  2. Preliminaries
  3. Riemannian submersions and curvature
  4. Cheeger Deformations
  5. Lie groupoid actions
  6. Left groupoid action
  7. Singular Riemannian foliations
  8. General smooth singular foliations
  9. Normal representation
  10. Riemannian Lie groupoids and invariant actions
  11. Hausdorff proper Lie groupoids and existence of 2-metr-ics
  12. Cheeger-like deformation for Lie groupoid actions
  13. Horizontal distribution
  14. Cheeger tensor
  15. Curvature

Key Result

Theorem 1

þ Let $\mathcal{G}\rightrightarrows M$ be a proper Lie groupoid acting along $\alpha\colon P\to M$ given by $\mu$, where $\mathcal{G}$, $M$ and $P$ are compact. Denote by $\mathcal{F}$ the singular foliation on $P$ induced by the orbits of the action. Then for any $g$ transversely $\mu$-invariant Ri

Theorems & Definitions (68)

  • Theorem 1
  • Theorem 2
  • Theorem 2.1: O'Neill's formula ONeill1966, see Section 1.5 in GromollWalschap
  • Theorem 2.2: Section 3.d in Mueter
  • Remark 2.3
  • Example 3.1: Lie groups
  • Example 3.2: Manifolds
  • Example 3.3: Pair groupoid
  • Example 3.4: Submersion groupoid
  • Example 3.5: Restriction groupoid
  • ...and 58 more