Minimizability of developable Riemannian foliations

TL;DR

This paper proves that developable Riemannian foliations on closed manifolds with polynomial growth fundamental groups are minimizable, by showing the vanishing of certain characteristic classes and applying existing theorems.

Contribution

It establishes the minimizability of developable Riemannian foliations under polynomial growth conditions, extending previous results to new classes of foliations.

Findings

Secondary characteristic classes vanish for developable foliations with polynomial growth fundamental group.

Developable Riemannian foliations are minimizable under the same conditions.

Foliations of codimension 2 with polynomial growth fundamental group are minimizable.

Abstract

Let (M,F) be a closed manifold with a Riemannian foliation. We show that the secondary characteristic classes of the Molino's commuting sheaf of (M,F) vanish if (M,F) is developable and the fundamental group of M is of polynomial growth. By theorems of \'{A}lvarez L\'{o}pez, our result implies that (M,F) is minimizable under the same conditions. As a corollary, we show that (M,F) is minimizable if F is of codimension 2 and the fundamental group of M is of polynomial growth.