A Cohomology (p+1) Form Canonically Associated with Certain Codimension-q Foliations on a Riemannian Manifold

Original · EN

Let (Mⁿ,g) be a closed, connected, oriented, C∞, Riemannian, n-manifold with a transversely oriented foliation F. We show that if X,Y are basic vector fields, the leaf component of [X,Y], V[X,Y], has vanishing leaf divergence whenever κ χ F is a closed (possibly zero) de Rham cohomology (p+1)-form. Here κ is the mean curvature one-form of the foliation F and χ F is its characteristic form. In the codimension-2 case, κ χ F is closed if and only if κ is horizontally closed. In certain restricted cases, we give necessary and sufficient conditions for κχ F to be harmonic. As an application, we give a characterization of when certain closed 3-manifolds are locally Riemannian products. We show that bundle-like foliations with totally umbilical leaves with leaf dimension greater than or equal to two on a constant curvature manifold, with non-integrable transversal distribution, and with Einstein-like transversal geometry are totally geodesic.