Two remarks about foliations and minimal foliations of codimension greater than two (Q2771403)

Let \({\mathcal F}\) be a foliation on a closed Riemannian manifold \(W\). \({\mathcal F}\) is said to be minimal if the leaves are minimal submanifold of \textit{G. W. Oshikiri} [Tohoku Math. J. 33, 133-137 (1981; Zbl 0437.57013)] proved that for a foliation \({\mathcal F}_1\) of codimension 1 and \(W\) with non-negative Ricci curvature \({\mathcal F}_1\) minimal implies that \({\mathcal F}_1\) and \({\mathcal F}_2\) are totally geodesic, where \({\mathcal F}_2\) denotes the normal bundle to \({\mathcal F}_1\). \textit{F. Brito} [Tohoku Math. J. 36, 341-350, (1984; Zbl 0554.53024)] extend these results to the case of codimension two under the additional hypothesis that \({\mathcal F}_2\) is integrable. The purpose of this paper is to extend these results to the arbitrary codimension. Let \(\tau_{{\mathcal F}_2}\) be the scalar curvature of the foliation \({\mathcal F}_2\). The authors prove the following theorem: Let \(W^{n+N}\) be an oriented closed Riemannian manifold with two complementary and orientable foliations, \({\mathcal F}_1\) and \({\mathcal F}_2\), where \({\mathcal F}_1\) is minimal. Then: 1) If \(\text{Ricc}(W)>0\), then \(\tau_{{\mathcal F}_2}\neq 0\); 2) If \(\text{Ricc} (W)\geq 0\), then either \({\mathcal F}_1\) is totally geodesic or \(\tau_{{\mathcal F}_2} \neq 0\) (both can occur simultaneously); 3) If \(W\) has non-negative sectional curvature, then either \(\tau_{{\mathcal F}_2}\neq 0\) or \({\mathcal F}_1\) is totally geodesic (both can occur simultaneously). Also, the authors obtain an integral formula for the scalar curvature of the leaves of two complementary foliations.NEWLINENEWLINEFor the entire collection see [Zbl 0953.00036].