Harmonic foliations on the sphere (Q1192572)

Let \(F\) be a harmonic foliation of codimension \(q\) of an \(n\)-dimensional unit sphere \(S^ n(1)\) for which the standard metric is bundle-like. If \(\| h\|^ 2\leq (n-q)/(2-1/q)\) on \(S^ n(1)\) (\(h\) being the second fundamental form of leaves), then \(F\) is totally geodesic. If the sectional curvature \(K\) of leaves satisfy \(K\geq(q-1)/(2q-1)\) on \(S^ n(1)\), then \(F\) is totally geodesic.