Flows of metrics on a fiber bundle

Let $(M^{n+p},g)$ be a closed Riemannian manifold, and $\pi: M\to B$
a smooth fiber bundle with compact and orientable $p$-dimensional
fiber $F$. Denote by $D_F$ ($D$) the distribution tangent
(orthogonal, resp.) to fibers.

We discuss conformal flows of the metric restricted to $D$ with the
speed proportional to

(i) the divergence of the mean curvature vector $H$ of $D$,

(ii) the mixed scalar curvature $Sc_{mix}$ of the distributions.
     (If $M$ is a surface, then $Sc_{mix}$ is the gaussian curvature $K$).

For (i), we show that the flow is equivalent to the heat flow of the
1-form dual to $H$, provided the initial 1-form is $D_F$-closed. We
use known long-time existence results for the heat flow to show that
our flow has a global solution $g_t$. It converges to a limiting
metric, for which $D$ is harmonic (i.e., $H=0$); actually under some
topological assumptions we can prescribe $H$.

For (ii) on a twisted product, we observe that $H$ satisfies the
Burgers type PDE, while the warping function satisfies the heat
equation; in this case the metrics $g_t$ converge to the product.

We consider illustrative examples of flows similar to (i) and (ii)
on a surface (of revolution), they yield convection-diffusion PDEs
for curvature of $D$-curves (parallels) and solutions -- non-linear
waves.

For $M$ with general $D$, we modify the flow (ii) with the help of a
measure of ``non-umbilicity" of $D_F$, and the integrability tensor
of $D$, while the fibers are totally geodesic. Let $\lambda_0$ be
the smallest eigenvalue of certain Schrödinger operator on the
fibers. We assume $H$ to be $D_F$-potential and show that

-- $H$ satisfies the forced Burgers type PDE;

-- the flow has a unique solution converging to a metric, for which
$Sc_{mix}\ge-n\lambda_0$,
   and $H$ depends only on the $D$-conformal class of the initial metric.

-- if $D$ had constant rate of ``non-umbilicity" on fibers, then the
limiting metric
   has the properties: $Sc_{mix}$ is quasi-positive, and $D$ is harmonic.