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