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 $\mathbb{F}$. We show that if $\lbrace X,Y \rbrace$ are basic vector fields, the leaf component of $[X,Y]$, $\mathcal{V}[X,Y]$, has vanishing leaf divergence whenever ${\kappa}\wedge \chi_{\mathbb{F}}$ is a closed (possibly zero) de Rham cohomology $(p+1)$-form. Here ${\kappa}$ is the mean curvature one-form of the foliation $\mathbb{F}$ and $\chi_{\mathbb{F}}$ is its characteristic form. In the codimension-$2$ case, ${\kappa}\wedge \chi_{\mathbb{F}}$ is closed if and only if ${\kappa}$ is horizontally closed. In certain restricted cases, we give necessary and sufficient conditions for ${\kappa}\wedge{\chi_{\mathbb{F}}}$ 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