Hebbey defines the Sobolev space of functions on a Riemannian manifold (M,g) as the completion of smooth functions under the Sobolev norm. However, I have seen (elsewhere) that Sobolev spaces have been defined as the collection of locally integrable functions whose weak derivatives (w.r.t to the LeviCivita connection) are in L^p. For compact manifolds, or for manifolds with bounded geometry and positive injectivity radius (like $\mathbb{R}^n$), I can see why these two definitions are equivalent. Are they equivalent for general noncompact manifolds?
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According to pages 14 and 15 of:
The poof is given in:
The results are: The closure of smooth functions with compact support, the closure of smooth function in the Sobolev space, and the Sobolev space are all different in general on open Riemannian manifolds. If the manifold is of bounded geometry (of order $k$) then all these spaces coincide up to Sobolev order $k+2$. This holds even for sections of vector bundles. Thus on compact manifolds all these spaces coincide also. 

