The low-dimensional metric foliations of Euclidean spheres
A metric foliation of a Riemannian manifold is a foliation where leaves are locally equidistant. The main result of the paper is a classification theorem for metric foliations of the Euclidean spheres. It states: Let \({\mathcal F}^ k\) be a k-dimensional metric foliation of the Euclidean sphere \(S^{n+k}\). Then, up to congruence: (i) \({\mathcal F}^ i\) is given uniquely by a direct sum of rotations \[ \ell^{2\pi i\theta_ 1\ell}\oplus...\oplus \ell^{2\pi i\theta_ s\ell} \] where \(n=2s-1\) and \(0\leq \theta_ 1\leq...\leq \theta_ s=1\); (ii) no \({\mathcal F}^ 2\) exists; (iii) any \({\mathcal F}^ 3\) is given uniquely by a direct sum of irreducible unitary representations \(\rho_{m_ 1}\oplus...\oplus \rho_{m_ s}\) of SU(2) where all \(m_{\sigma}\) are odd, \(n=4\ell\) and \(\ell \leq m_ 1\leq...\leq m_ s;\quad m_ 1+...+m_ s=2(\ell +1)-s.\)
scientific article; zbMATH DE number 4119133