Logo
ScienceStack
Table of Contents
Fetching ...
Sign In

A note on the vertizontal curvature of fat bundles

Leonardo F. Cavenaghi

TL;DR

The note establishes a rigidity result for fat Riemannian foliations with bounded holonomy: if a curvature constraint $R_{\mathsf g}(X,V)A^*_X V=0$ holds for all good triples, then vertizontal curvatures are determined at a point and, if they coincide there, everywhere on a compact connected manifold. This implies the existence of a global scaling making vertizontal curvatures identically $1$ under suitable conditions, providing a partial answer to Ziller's question for fat principal bundles with compact structure groups, particularly when the total space is locally symmetric. The results apply to known fat $S^3$ and $SO(3)$ bundles, showing the total spaces can be $3$-Sasakian, and specialize to homogeneous constructions where $K/H \to G/H \to G/K$ yields explicit curvature formulas, such as $K_g(X,V)=|[X,V]|^2_g$ and quaternionic Hopf-type fibrations attaining vertizontal curvature $1$.

Abstract

In his unpublished notes on fat bundles, W. Ziller poses a compelling question: given a fat principal $G$-bundle $(P, g) \rightarrow (B, h)$ with $\dim G = 3$, and $g$ representing a Riemannian submersion metric ensuring that the $G$-orbits are totally geodesic, can one modify $h$ to render all vertical curvatures equal to $1$? In this note, we establish a rigidity result for fat Riemannian foliations with bounded holonomy and a specific curvature constraint. Our result addresses Ziller's question for fat fiber bundles with compact structure groups, considering connected compact total spaces under a curvature constraint that holds on various examples, such as locally symmetric spaces. Additionally, we assume that all vertizontal curvatures coincide at a point.

A note on the vertizontal curvature of fat bundles

TL;DR

The note establishes a rigidity result for fat Riemannian foliations with bounded holonomy: if a curvature constraint holds for all good triples, then vertizontal curvatures are determined at a point and, if they coincide there, everywhere on a compact connected manifold. This implies the existence of a global scaling making vertizontal curvatures identically under suitable conditions, providing a partial answer to Ziller's question for fat principal bundles with compact structure groups, particularly when the total space is locally symmetric. The results apply to known fat and bundles, showing the total spaces can be -Sasakian, and specialize to homogeneous constructions where yields explicit curvature formulas, such as and quaternionic Hopf-type fibrations attaining vertizontal curvature .

Abstract

In his unpublished notes on fat bundles, W. Ziller poses a compelling question: given a fat principal -bundle with , and representing a Riemannian submersion metric ensuring that the -orbits are totally geodesic, can one modify to render all vertical curvatures equal to ? In this note, we establish a rigidity result for fat Riemannian foliations with bounded holonomy and a specific curvature constraint. Our result addresses Ziller's question for fat fiber bundles with compact structure groups, considering connected compact total spaces under a curvature constraint that holds on various examples, such as locally symmetric spaces. Additionally, we assume that all vertizontal curvatures coincide at a point.
Paper Structure (3 sections, 4 theorems, 11 equations)

This paper contains 3 sections, 4 theorems, 11 equations.

Table of Contents

  1. Introduction and Preliminaries
  2. Proof of Theorem
  3. Proof of Corollary

Key Result

Theorem 1.5

On any locally symmetric space $(M,\mathop{\mathrm{\mathsf{g}}}\nolimits)$ it holds that $R_{\mathop{\mathrm{\mathsf{g}}}\nolimits}(X,V)(A^*_XV) = 0$ where $R_{\mathop{\mathrm{\mathsf{g}}}\nolimits}$ is the Riemannian curvature tensor of $\mathop{\mathrm{\mathsf{g}}}\nolimits$.

Theorems & Definitions (10)

  • Definition 1.1
  • Definition 1.2: Holonomy Fields and Dual Holonomy Fields
  • Definition 1.3: Bounded Holonomy
  • Example 1.4: Proposition 3.4 in speranca2017on
  • Theorem 1.5: Munteanu--Tapp, tappmunteanu2
  • Theorem 1.6
  • Corollary 1.7
  • Corollary 1.8
  • proof
  • proof