  We investigate the convergence of the mean curvature flow of arbitrary
codimension in Riemannian manifolds with bounded geometry. We prove that if the
initial submanifold satisfies a pinching condition, then along the mean
curvature flow the submanifold contracts smoothly to a round point in finite
time. As a consequence we obtain a differentiable sphere theorem for
submanifolds in a Riemannian manifold.
