I have a reference request concerning equivalent norms on Sobolev Spaces on manifolds of bounded geometry. This may be obvious to the experts but I am not working in the field and only want to use this result. Let $(M,g)$ be a smooth Riemannian manifold. The Sobolev Space $W_p^m(M)$ with $m\in\mathbb{N}$ and $1\leq p< \infty$ is defined via completion of the space of smooth functions on $M$ in $L^p(M)$ with respect to the norm

$\left\lVert f\right\rVert_{W_p^m(M)}:=\sum_{l=0}^m \left(\int_M \vert\nabla^lf\vert^p\,\mathrm{d}V_g\right)^{\frac{1}{p}}\,.$

Now it is very frequently used that on manifolds with bounded geometry and positive injectivity radius one can use geodesic normal coordinates and appropriate partitions of unity to define an equivalent norm on $W_p^m(M)$ via looking at the norms of the "pulled backed" function in euclidean space. (see for example: Triebel, Theory of function spaces II 7.4.5., identity (8) on page 320).

I am looking for a reference where the equivalence of the norms is shown explicitely.

Thank you!