Let $M$ be a smooth manifold. If $\nabla$ is a linear connection on $M$, I would like to induce a unique linear connection on an open subset $U\subseteq M$. I know that for all $p\in U$ there is a natural isomorphism $T_pU\cong T_pM$, so I can restrict global vector fields to local vector fields on $U$. Unfortunately there are some local vector fields on $U$ that don't came from a restriction of global vector fields.
For this reason I can't find a reasonable linear connection $\nabla^U$ over $U$ induced by $\nabla$. I need help.