Suppose I have a map $u:\mathbb{R}\times [0,1]\rightarrow M$ , where $M$ is a smooth manifold with a metric of bounded geometry ,such that $\lim_{s\rightarrow \pm \infty} \partial_s u=0, \lim_{s\rightarrow \pm \infty}D_t\partial_s u=0$ and $\lim_{s\rightarrow \pm \infty}D_s\partial_s u=0$. Now I can look at the pull-back bundle $u^*(TM)$, and since $\mathbb{R}\times [0,1]$ is contractible I find a unitary trivialization $\Psi$ of this bundle.
Now I would like to undesrtand what happens with $D_s\Psi$ and $D_t\Psi$. It is claimed that these will be bounded and that $\lim_{s\rightarrow \pm \infty}D_s\Psi =0$. However I am not sure why this will be the case. I tried writting everything in local coordinates , and when we do this there will be term of the derivatives of $\Psi$ appearing and I am not sure how we can control this to be bounded ? Is it because it is a unitary trivialization? And even if we can control these derivatives, I am not sure how we control the term coming from $\partial_t u$.
Any help is appreciated, thanks in advance.