Transverse sphere theorems for Riemannian foliations

TL;DR

The paper develops transverse analogues of classical sphere rigidity theorems for Riemannian foliations, focusing on Killing foliations and their leaf-closure spaces. The authors deploy a deformation technique to approximate Killing foliations by closed foliations, preserving transverse curvature bounds and enabling the use of orbifold and Alexandrov geometry tools. They prove a transverse diameter sphere theorem, showing that under $\sec_\mathcal{F}>1$ and $\mathrm{diam}_\perp(\mathcal{F})\ge\pi/2$, the leaf-closure space $M/\overline{\mathcal{F}}$ is a sphere-quotient $\mathbb{S}^q/G$ with $G$ an extension of a finite group by $\mathbb{T}^d$. Additionally, a transverse quarter-pinched theorem shows that for $\tfrac{1}{4}<\sec_\mathcal{F}<1$ and codimension $q\ge3$, the universal cover develops to a simple foliation over $\mathbb{S}^q$, i.e., a fibration over the sphere. These results connect transverse foliation geometry to classical sphere theorems via GH convergence and orbifold theory, yielding explicit quotient descriptions and fibration structures.

Abstract

We prove two transverse analogs of classical theorems involving the rigidity of the sphere: the Grove--Shiohama diameter sphere theorem and the Berger--Klingenberg quarter-pinched sphere theorem. We obtain, more precisely, that if a Killing foliation of a compact, connected manifold has transverse sectional curvature greater than 1 and transverse diameter greater than $π/2$, then the space of its leaf closures is homeomorphic to the orbit space of a continuous action on a sphere. We also prove that the space of leaf closures of a Killing foliation of a compact manifold is the Gromov--Hausdorff limit of a sequence of orbifolds. This, in turn, enables us to prove that a transversely quarter-pinched, complete Riemannian foliation of a connected manifold, with codimension greater than or equal to 3, develops on the universal cover to a fibration over a sphere.

Theorems & Definitions (16)

• Theorem A: Theorem \ref{['theorem transverse diameter sphere']}
• Theorem B: Theorem \ref{['theorem regular convergence implies GH convergence']}
• Theorem C: Theorem \ref{['teoprincipal']}
• Theorem 2.1: caramello
• Proposition 3.1
• proof
• Corollary 3.2
• Proposition 3.3
• Corollary 3.4
• Theorem 3.5: Transverse diameter sphere theorem