Hi, I have the following question: take a Riemannian manifold M, with a family of smooth metrics $g(t)$ in $[0,T)$, call $D_0$ the Levi-Civita connection of $g(0)$ and assume that for every $m\geq 0$  
$\int_0^T \sup_M|D_0^m \frac{\partial}{\partial t} g(t)|_{g(0)} dt< \infty$, then why $g(t)$ converges in $C^{\infty}$ to a smooth tensor?