In https://hebey.u-cergy.fr/NotesSharpSP.pdf right at the beginning Hebey says

Given $(M,g)$ a smooth compact $n$-dimensional Riemannian manifold, one easily defines the Sobolev spaces $H^p_k(M)$, following what is done in the more traditionnal Euclidean context. For instance, when $k = 1$, and $p = 2$, one may define the Sobolev space $H^1_2(M)$ as follows: for $u \in C^{\infty}(M)$, we let $$\Vert u \Vert^2_{H^1_2(M)}=\Vert u \Vert^2_2+\Vert \nabla u \Vert_2^2$$ where $\Vert \cdot \Vert_p$ is the $L^p$-norm with respect to the Riemannian measure $dv(g)$. We then define $H^1_2(M)$ as the completion of $C^{\infty}(M)$ with respect to $\Vert \cdot \Vert_{H^1_2(M)}$ .

How does he define it so "easily" while I see other sources where they use things such as partition of the unity to define Sobolev spaces on manifolds etc? Also, what might go wrong if we don't require $M$ to be compact?


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