Rigidity of the Alvarez class

TL;DR

This paper investigates the properties of the e1lvarez class in Riemannian foliations on closed manifolds, revealing that its integrals along closed paths are logarithms of algebraic integers under certain conditions.

Contribution

It establishes a new rigidity result showing the integrals of the e1lvarez class are logarithms of algebraic integers when the fundamental group is polycyclic or the foliation has polynomial growth.

Findings

Integral of e1lvarez class is the log of an algebraic integer.

Rigidity of the e1lvarez class under specific geometric conditions.

Connection between fundamental group properties and foliation invariants.

Abstract

Let $(M,\mathcal{F})$ be a closed manifold with a Riemannian foliation. The \'{A}lvarez class is the cohomology class of degree 1 of $M$ whose triviality characterizes the minimizability of $(M,\mathcal{F})$. We show that the integral of the \'{A}lvarez class along every closed path in $M$ is the logarism of an algebraic integer if $\pi_{1}M$ is polycyclic or $\mathcal{F}$ is of polynomial growth.