Totally umbilic Riemannian foliations (Q910016)

The author proves the following theorem: Let F be an umbilicalisable Riemannian foliation of a compact connected manifold M. Then F is dense, and the following conditions are equivalent: (i) the basic cohomology of F verifies Poincaré duality; (ii) F is taut; (iii) F is geodesible. Moreover, if F does not satisfy these conditions then there is a Riemannian metric on M for which F is totally umbilic and isoparametric, and for which the leaves of F have zero sectional curvature. Furthermore, there is an open dense subset of M that is saturated by simply connected leaves. The above theorem implies some sufficient conditions for the geodesibility of F. A classification of umbilical and non-geodesible Riemannian foliations of codimension 2 of a compact connected manifold M, is given.