Let $(M,g)$ be a $d$-dimensional smooth, riemannian manifold without boundary, which is geodesically complete, connected, has positive injectivity radius and a bounded geometry (i.e. all derivatives of the metric are bounded and $\det(g)\geq c>0$ uniformly on every local chart repsectively). Additionally we furnish this manifold with a geodesic atlas, which is at most countable and uniformly locally finite if we choose the radius of the geodesic balls small enough (and we assume this in the following), and consider the $L_2$-based Sobolev space $H^1_2(M)$ defined via the covariant derivative. For this space we still have the usual continuous Sobolev embeddings $H^1_2(M)\hookrightarrow L_q(M)$ for $\frac{1}{2}\geq \frac{1}{q}\geq \frac{1}{2}-\frac{1}{d}>0$. Of course in this generality, they cannot be compact, but I wonder if there is some "local compactness" comparable to the situation on $R^d$, where the sobolev embeddings are compact for bounded domains with Lipschitz boundary. More precisely I'm interested in the following: Let $(O,\kappa)$ be a local geodesic chart. Can $\overline{O}$ be defined as a compact smooth riemannian manifold with boundary? (For such manifolds the above embeddings $H^1_2(\overline{O})\hookrightarrow L_q(\overline{O})$ are compact)
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