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It is well-known that if a complete Riemannian manifold has bounded curvature and injectivity radius bounded away from zero, then the space $C^\infty_c(M)$ is dense in the Sobolev spaces $W^{k, p}(M)$ for $k=0, 1, 2$.

My question: Is this an if and only if? That is, if $C^\infty_c(M)$ is dense in these Sobolev spaces, does $M$ necessarily have bounded curvature and injectivity radius bounded away from zero?

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The general answer to this question is no. Global bound on the Ricci curvature is not necessary for the density of smooth functions with compact supports.

Indeed, when $(M,g)$ is a smooth complete Riemannian manifold with positive injectivity radius and lower bound for the Ricci curvature, then the smooth functions with compact support are density in the Sobolev spaces for $p$ equals 2. This can be found for instance on Emmanuel Hebey's book: Nonlinear Analysis on Manifolds: Sobolev Spaces and Inequalities.

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  • $\begingroup$ What is an example for such a manifold that does not have bounded curvature? Do such manifolds even exist? $\endgroup$ – Matthias Ludewig Dec 29 '14 at 13:40
  • $\begingroup$ @MatthiasLudewig The horn $\endgroup$ – Fan Zheng Dec 12 '16 at 23:19
  • $\begingroup$ What is the horn? $\endgroup$ – Matthias Ludewig Dec 13 '16 at 9:49
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I'd say it is always dense.

Certainly, for any point $x\in M$ there is a nbd $U$ such that any $f\in W^{p,k}(M)$ supported in $U$ can be approximated by smooth functions supported in $U$ (e.g. for the same result you are quoting: we could modify $M$ in the complement of $U$ to make it compact, and nobody that lives inside $U$ would realize it).

Smooth partitions of unity do exist in any Riemannian manifold $M$. Therefore, any $f\in W^{p,k}(M)$ can be written as a locally finite sum $f=\sum_n f_n$ with $f_n\in W_c^{p,k}(U_n)$, for some locally finite open cover $\{U_n\}_n$ of $M$, by small nbd's $U_n$ were the density is true. Given $\epsilon>0$, we can choose $g_n\in C^\infty_c(U_n)$ such that $\|f_n-g_n\|_{p,k}\le 2^{-n}\epsilon$, so that the corresponding $g=\sum_n g_n$ is a smooth function with $\|f-g\|_{p,k}\le\epsilon$.

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  • $\begingroup$ But $g$ is not necessarily compactly supported it your index set is infinite! $\endgroup$ – Matthias Ludewig Dec 3 '14 at 17:19
  • $\begingroup$ ops, I hadn't noticed the subscript $c$ $\endgroup$ – Pietro Majer Dec 3 '14 at 17:48

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