  Our results concern geometry of a manifold endowed with a pair of
complementary orthogonal distributions (plane fields) and a time-dependent
Riemannian metric. The work begins with formulae concerning deformations of
geometric quantities as the Riemannian metric varies conformally along one of
the distributions. Then we introduce the Extrinsic Geometric Flow depending on
the mean curvature vector field of the distribution, and show
existence/uniquenes and convergence of a solution as $t\to\infty$, when the
complementary distribution is integrable with compact leaves. We apply the
method to the problem of prescribing mean curvature vector field of a
foliation, and give examples for harmonic and umbilical foliations and for the
double-twisted product metrics, including the codimension-one case.
