Riemannian foliations of bounded geometry

TL;DR

This paper introduces a new, chart-free definition of bounded geometry for Riemannian foliations, establishing equivalences with normal foliation charts and exploring their properties, with applications to trace formulas for foliated flows.

Contribution

It provides a novel, chart-free characterization of bounded geometry for Riemannian foliations and analyzes properties of normal foliation charts.

Findings

Equivalence between chart-free bounded geometry and conditions on normal foliation charts

Uniform boundedness of derivatives of coordinate changes in these charts

Existence of partitions of unity compatible with bounded geometry

Abstract

Continuing the study of bounded geometry for Riemannian foliations, begun by Sanguiao, we introduce a chart-free definition of this concept. Our main theorem states that it is equivalent to a condition involving certain normal foliation charts. For this type of charts, it is also shown that the derivatives of the changes of coordinates are uniformly bounded, and there are nice partitions of unity. Applications to a trace formula for foliated flows will given in a forthcoming paper.