I want to define Sobolev spaces for sections on a vector bundle, basically I want that a section will belong to the Sobolev space $W^{k,p}$ if its coordinates in any aceptable patch belong to the corresponding Sobolev space of functions. So here is my try:
Let $\mathcal{A}$ be the collection of atlases of the differential structure of $M$ on which the coordinate transformations belong to $\widetilde{C}^{\infty,p}(U,\mathbb{R}^n):=\{f \in C^\infty(U) : \|f\|_{W^{k,p}(U)} < \infty \}$ where $U$ is some open set of $\mathbb{R}^n$. We see that by making the coordinate charts small enough $\mathcal{A}$ is not empty.
A section (not necessarily smooth) $s$ of the vector bundle $\pi: E \rightarrow M$ is in the Sobolev space $W^{k,2}(E)$, if for any $(V,\psi)\in \mathcal{A}$, and for any $U$ compactly contained in $V$, the map $$ pr_2 \circ \phi_V \circ s|_U \circ \psi^{-1}:\psi(U) \rightarrow \mathbb{R}^m $$ lies in $W^{k,2}(\psi(U),\mathbb{R}^m)$. Where $\phi_V$ is a trivialization of $E$ over $V$, and $pr_2$ is the projection onto the second factor.
I would like to know if this definition is consistent in the sense that it does not depend on the choices made. And if is not what changes should I put. Thanks
