Minimizability of developable Riemannian foliations

Abstract

Let ${(M,\mathcal{F})}$ be a closed manifold with a Riemannian foliation. We show that the secondary characteristic classes of the Molino’s commuting sheaf of ${(M,\mathcal{F})}$ vanish if ${(M,\mathcal{F})}$ is developable and π ₁ M is of polynomial growth. By theorems of Álvarez López in (Álvarez López, Ann. Global Anal. Geom., 10:179–194, 1992) and (Álvarez López, Ann. Pol. Math., 64:253–265, 1996), our result implies that ${(M,\mathcal{F})}$ is minimizable under the same conditions. As a corollary, we show that ${(M,\mathcal{F})}$ is minimizable if ${\mathcal{F}}$ is of codimension 2 and π ₁ M is of polynomial growth.