On totally geodesic foliations with bundle-like metric (Q860220)

The authors state the following theorem: ``Let \({\mathcal F}\) be a totally geodesic \(n\)-foliation of an \((n+p)\)-dimensional Riemannian manifold \((M,g)\) satisfying the following conditions: (i) the metric \(g\) is bundle-like for \({\mathcal F}\); (ii) there exists a point \(x\in M\) such that none of the mixed sectional curvatures of \(M\) at \(x\) vanishes. Then the inequality \(n\leq p-1\) must be satisfied''. Here ``mixed sectional curvature'' means the sectional curvature of a plane span \(\{u,v\}\) with \(u\) tangent to \({\mathcal F}\) and \(v\) orthogonal to \({\mathcal F}\) at the point \(x\in M\). Moreover, it is shown that the inequality \(n\leq p-1\) is optimal and that none of the conditions (i) and (ii) in the theorem can be removed. Also, some interesting corollaries of the theorem are derived.