I've seen this mentioned in a few physics books (Hawking-Ellis, Stewart), but never with any details. Let $(M,g)$ be an analytic Riemannian manifold, and $U\subset M$ an open set with compact closure. Define the Sobolev space $H^k(U)$ as usual. Is $C^\omega(\bar U)$ dense in $H^k(U)$? I think a proof could make use of Whitney's $C^k$-norm analytic approximation theorem (Hirsch, p. 66, or maybe the stronger Grauert-Remmert approximation theorem), but I'm not sure how to proceed without partitions of unity. Namely, if $\bar U$ is contained in a chart with compact closure, one should dominate the Sobolev norm by the $C^k$ norm: $$||f||_{H^k}\le C\sup_{0\le p\le k}|D^p f|,$$ where $C$ depends on $\dim M$, and bounds of Christoffel symbols, curvature tensors, and derivatives of curvatures (hence it is important that $U$ is precompact). Whitney's theorem then gives $C^\omega\subset C^k$ being dense in $H^k$-norm, which gives density in $H^k$.
I don't understand Hirsch's proof of Whitney's analytic atlas theorem, which might be useful?
I'm looking an idea of how to proceed. Also, what is the situation when $\bar U$ is noncompact?