Some properties of mean curvature vectors for codimension-one foliations

Abstract

Given a codimension-one foliation $\mathcal F$ of a closed manifold $M$ and a vector field $X$ on $M$, we show that if $X$ is transverse to $\mathcal F$, then there are many functions $f$ on $M$ so that $fX$ is the mean curvature vector of $\mathcal F$ with respect to some Riemannian metric on $M$. Further we give a necessary and sufficient condition for $X$ to become the mean curvature vector of $\mathcal F$ with respect to some Riemannian metric on $M$.