Umbilic foliations and curvature (Q1358720)

The author shows that there are no \(k\)-dimensional umbilic foliations of an \(n\)-dimensional manifold of positive sectional curvature with \(k> {n-1 \over 2}\). In the paper, there are studied umbilic foliations of codimension one on manifolds of nonpositive curvature. The author also proves that an umbilic \(k\)-dimensional foliation \({\mathcal F} (k>1)\) of the complete simply connected space \(M\) of constant curvature \(c\) is flat if \(c\leq 0\), and homogeneous if \(c \geq 0\).