Analysis on Riemannian foliations of bounded geometry

TL;DR

This paper revisits and extends the theory of Riemannian foliations with bounded geometry, focusing on leafwise Hodge decomposition, smoothing operators, and the Novikov complex, providing new insights and tools for geometric analysis.

Contribution

It offers a new proof of leafwise Hodge decomposition and extends the framework to leafwise Novikov complexes, enhancing understanding of Riemannian foliations with bounded geometry.

Findings

Reproved leafwise Hodge decomposition in bounded geometry setting

Constructed smoothing operators with explicit Schwartz kernels

Extended analysis to leafwise Novikov differential complex

Abstract

A leafwise Hodge decomposition was proved by Sanguiao for Riemannian foliations of bounded geometry. Its proof is explained again in terms of our study of bounded geometry for Riemannian foliations. It is used to associate smoothing operators to foliated flows, and describe their Schwartz kernels. All of this is extended to a leafwise version of the Novikov differential complex.