# Is the definition of the Sobolev space H^1(M) on a compact manifold that simple?

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?