Riemannian foliations of spheres (Q309012)

The main result of this paper is the following: Suppose \(\mathcal F\) is a Riemannian foliation by \(k\)-dimensional leaves of a compact manifold \((M, g)\) which is homeomorphic to \(\mathbb S^{n}\). If \(0< k< n\), then one of the following holds:NEWLINENEWLINE (a) \(k = 1\) and the foliation is given by an isometric flow with respect to some Riemannian metric.NEWLINENEWLINE(b) \(k = 3\) , \(n \equiv3 \) mod \(4\) and the generic leaves are diffeomorphic to \(\mathbb R \mathbb P ^{3}\) or \( \mathbb S ^{3}\) .NEWLINENEWLINE(c) \(k = 7\), \(n = 15\) and \(\mathcal F\) is given by the fibers of a Riemannian submersion \((M, g) \rightarrow (B,{g})\), where \((B,{g})\), is homeomorphic to \(\mathbb S ^{8}\) and the fiber is homeomorphic to \(\mathbb S ^{7}\) .NEWLINENEWLINEThe authors also show that all the above cases may occur. Moreover, their topological result allows to classify Riemannian foliations of the round sphere up to metric congruence.