It is well-known that there are two equivalent definitions of Sobolev space on open subset $\Omega\subset\mathbb{R}^n$:
D1. The completion of $C^\infty(\Omega)$ under $H^p_k$ norm.
D2. All functions with the $i$-times weak derivative $L^p$-integrable, $1\leq i\leq k$.
I have already come across some representative references on Sobolev spaces on Riemannian manifolds, such as E. Hebey(1999) and T.Aubin(1982). Surprisingly, these materials only cover the first definition on Riemannian manifold $M$. Soon I realized that (D2) required the duals of those operators $\nabla^k$, thus the author had to append a huge amount of additional conceptions on differential geometry to validate it. Maybe that's why they chose not to mention the second definition.
However, even burdened with extra work, the second definition should be working somehow. This post https://mathoverflow.net/questions/126419/density-of-smooth-functions-in-sobolev-spaces-on-manifolds has provided some references, but I have no access to them.
Is there any other reference studying their relationship?