Metric foliations of homogeneous three-spheres

A smooth foliation of a Riemannian manifold is metric when its leaves are locally equidistant and is homogeneous when its leaves are locally orbits of a Lie group acting by isometries. Homogeneous foliations are metric foliations, but metric foliations need not be homogeneous foliations. We prove that a homogeneous three-sphere is naturally reductive if and only if all of its metric foliations are homogeneous.