There is extensive literature on Sobolev spaces on complete Riemannian manifolds but are there any standard references regarding the definition and properties of Sobolev spaces on domains (possessing certain regularity, at least) in complete Riemannian manifolds?

In particular, I'm interested in a definition of such spaces that has the following two properties:

- The Sobolev embedding theorems hold (assuming the usual geometric restrictions on the ambient manifold).
- Passing to a smaller domain induces a restriction map on the corresponding Sobolev spaces.

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