Affine transverse foliations in sphere bundles

Abstract

Let~$S^{n-1}\rightarrow E \rightarrow M^n$ be an oriented sphere bundle supporting an affine transverse foliation. We give an upper bound for the Euler number of the bundle. We also give a new and elementary proof of the following fact: if the fundamental group of $M^n$ is amenable, then the Euler number of the bundle vanishes.