  Let $M=\Sigma_1\times \Sigma_2$ be the product of two compact Riemannian
manifolds of dimension $n\geq 2 $ and two, respectively. Let $\Sigma$ be the
graph of a smooth map $f:\Sigma_1\mapsto \Sigma_2$, then $\Sigma$ is an
$n$-dimensional submanifold of $M$. Let ${\frak G}$ be the Grassmannian bundle
over $M$ whose fiber at each point is the set of all $n$-dimensional subspaces
of the tangent space of $M$. The Gauss map $\gamma:\Sigma\mapsto \frak{G} $
assigns to each point $x\in \Sigma$ the tangent space of $\Sigma$ at $x$. This
article considers the mean curvature flow of $\Sigma$ in $M$. When
  $\Sigma_1$ and $\Sigma_2$ are of the same non-negative curvature, we show a
sub-bundle $\frak{S}$ of the Grassmannian bundle is preserved along the flow,
i.e. if the Gauss map of the initial submanifold $\Sigma$ lies in $\frak{S}$,
then the Gauss map of $\Sigma_t$ at any later time $t$ remains in $\frak{S}$.
We also show that under this initial condition, the mean curvature flow remains
a graph, exists for all time and converges to the graph of a constant map at
infinity . As an application, we show that if $f$ is any map from $S^n$ to
$S^2$ and if at each point, the restriction of $df$ to any two dimensional
subspace is area decreasing, then $f$ is homotopic to a constant map.
