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

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