  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 ${\kappa}\wedge
\chi_{\boldkey F}$ is a closed (possibly zero) de Rham cohomology (p+1)-form.
Here ${\kappa}$ is the mean curvature one-form of the foliation ${\boldkey F}$
and ${\chi_{\boldkey F}}$ is its characteristic form. In the codimension-2
case, ${\kappa}\wedge \chi_{\boldkey F}$ is closed if and only if ${\kappa}$ is
horizontally closed. In certain restricted cases, we give necessary and
sufficient conditions for ${\kappa}\wedge{\chi_{\boldkey F}}$ 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.
