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A curvature approach to fatness

Leonardo F. Cavenaghi, Lino Grama

TL;DR

The paper investigates fat Riemannian submersions under non-negative sectional curvature, using Cheeger deformations and associated bundle geometry to derive strong structural results. It proves that fibers must be rank-one symmetric spaces (dimensions $1,3,7$) under curvature constraints, and that the total space decomposes as an associated bundle $M \cong P \times_H (H/K)$, with the horizontal distribution encoding a transitive holonomy action. It further shows that dual foliations are rigid: in many nonnegative-curvature settings the dual foliation collapses to a single leaf, and fat foliations on Lie groups are isometric to coset foliations with holonomy limited to $SO(3)$ or $S^3$ when the fiber is higher dimensional. Collectively, these results sharply restrict the possible fibers and holonomy groups in fat submersions, aligning them with rank-one symmetric spaces and rank-one coset structures in compact Lie group settings.

Abstract

This paper delves into the concept of ``fat bundles'' within Riemannian submersions. One explores the structural implications of fat Riemannian submersions, particularly focusing on those with non-negative sectional curvature. The main results include the classification of fibers as symmetric spaces, the isometric correspondence of fat foliations with coset foliations on Lie groups, and the rigidity of dual foliations associated with fat Riemannian submersions.

A curvature approach to fatness

TL;DR

The paper investigates fat Riemannian submersions under non-negative sectional curvature, using Cheeger deformations and associated bundle geometry to derive strong structural results. It proves that fibers must be rank-one symmetric spaces (dimensions ) under curvature constraints, and that the total space decomposes as an associated bundle , with the horizontal distribution encoding a transitive holonomy action. It further shows that dual foliations are rigid: in many nonnegative-curvature settings the dual foliation collapses to a single leaf, and fat foliations on Lie groups are isometric to coset foliations with holonomy limited to or when the fiber is higher dimensional. Collectively, these results sharply restrict the possible fibers and holonomy groups in fat submersions, aligning them with rank-one symmetric spaces and rank-one coset structures in compact Lie group settings.

Abstract

This paper delves into the concept of ``fat bundles'' within Riemannian submersions. One explores the structural implications of fat Riemannian submersions, particularly focusing on those with non-negative sectional curvature. The main results include the classification of fibers as symmetric spaces, the isometric correspondence of fat foliations with coset foliations on Lie groups, and the rigidity of dual foliations associated with fat Riemannian submersions.
Paper Structure (12 sections, 23 theorems, 38 equations, 2 tables)

This paper contains 12 sections, 23 theorems, 38 equations, 2 tables.

Table of Contents

  1. Introduction
  2. Preliminaries in fat bundles and Cheeger deformations
  3. Preliminary aspects of fat submersions
  4. Preliminary aspects of Cheeger deformations for associated bundles
  5. Some new rigidity results for fat bundles
  6. Fat Riemannian submersions
  7. Fat submersions and "twisted foliations"
  8. A rigidity result for non-negatively curved fat Riemannian submersions
  9. On the fibers of non-negatively curved fat submersions
  10. Riemannian Foliations on Lie Groups with bi-invariant metrics
  11. On the fibers of non-negatively curved fat Riemannian submersions
  12. A sketch of the proof of Theorem

Key Result

Proposition 2.1

Let $\pi: F\hookrightarrow (M,\mathop{\mathrm{\textsl{g}}}\nolimits) \rightarrow (B,\mathop{\mathrm{\textsl{g}}}\nolimits_B)$ be a Riemannian submersion with totally geodesic fibers $F$. Then, $\pi$ is a fiber bundle, and $\mathop{\mathrm{\textsl{g}}}\nolimits$ is a connection metric.

Theorems & Definitions (44)

  • Definition 1
  • Definition 2
  • Remark : The geometric flavor of the fatness assumption
  • Proposition 2.1: Theorem 2.7.2, p.98 in gw
  • Proposition 2.2: Proposition 2.6, p.9 in Ziller_fatnessrevisited
  • Definition 3: The $A^*_X$-dual to the O'Neill tensor
  • Proposition 2.3: Proposition 2.4, p.8 in Ziller_fatnessrevisited
  • Proposition 2.4: Proposition 2.5 in Ziller_fatnessrevisited
  • Remark : What do we mean by a fat Riemannian submersion
  • Definition 4
  • ...and 34 more