Foliations by minimal submanifolds and Ricci curvature. (English) Zbl 1230.57024
The main result assumes a transversely oriented codimension 2 foliation \(\mathcal F\) on a closed oriented Riemannian manifold \((M,g)\) with a basic transverse volume form \(\mu\) such that the Ricci curvature of each leaf of \(\mathcal F\) is quasi-positive. Suppose that the restriction to each leaf of \(\mathcal F\) of the 1-form \(\beta\) on \(M\) is closed, where \(\beta(E)=g(\mathcal V[X,Y],E)\) for basic vector fields \(X\), \(Y\) with \(\mu(X,Y)=1\), and \(\kappa\wedge\chi_\mathcal F\) is closed on \(M\), where \(\kappa\) is the mean curvature 1-form and \(\chi_\mathcal F\) the characteristic form for \(\mathcal F\). Then the distribution orthogonal to that of the leaves of \(\mathcal F\) is integrable with its leaves being minimal surfaces of \(M\).
Reviewer: David B. Gauld (Auckland)
MSC:
57R30 | Foliations in differential topology; geometric theory |
53C25 | Special Riemannian manifolds (Einstein, Sasakian, etc.) |