Rigidity of the \'Alvarez classes of Riemannian foliations with nilpotent structure Lie algebras

TL;DR

This paper proves that for Riemannian foliations with nilpotent structure Lie algebras on closed manifolds, the Alvarez class exhibits rigidity properties, remaining invariant under deformations when the fundamental group has polynomial growth.

Contribution

It establishes the rigidity of the Alvarez class for Riemannian foliations with nilpotent structure Lie algebras, extending invariance results under deformations.

Findings

Integral of the Alvarez class along closed paths is exponential of an algebraic number.

Alvarez class and tautness are invariant under deformation for manifolds with polynomial growth fundamental group.

Rigidity results apply specifically to foliations with nilpotent structure Lie algebras.

Abstract

We show that if the structure algebra of a Riemannian foliation F on a closed manifold M is nilpotent, then the integral of the \'Alvarez class of (M,F) along every closed path is the exponential of an algebraic number. By this result and the continuity of the \'Alvarez class under deformations shown in arXiv:1009.1098v2, we prove that the \'Alvarez class and the geometrically tautness of Riemannian foliations on a closed manifold M are invariant under deformation, if the fundamental group of M has polynomial growth.