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

Abstract

Let $(M^{n},g)$ be a closed, connected, oriented, $C^{\infty}$, Riemannian, n-manifold with a transversely oriented foliation $\boldkey F$. We show that if $\lbrace X,Y \rbrace$ are basic vector fields, the leaf component of $[X,Y]$, $\Cal{V}[X,Y]$, has vanishing leaf divergence whenever $κ\wedge χ_{\boldkey F}$ is a closed (possibly zero) de Rham cohomology (p+1)-form. Here $κ$ is the mean curvature one-form of the foliation ${\boldkey F}$ and ${χ_{\boldkey F}}$ is its characteristic form. In the codimension-2 case, $κ\wedge χ_{\boldkey F}$ is closed if and only if $κ$ is horizontally closed. In certain restricted cases, we give necessary and sufficient conditions for $κ\wedge{χ_{\boldkey F}}$ 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.