Define the fractional Sobolev spaces (for $0<s<1$) in some Riemannian $N$-manifold $M$ as follows : $$W^{s,p}(M):=\left\{u\in L^p(M) \ \bigg| \ \iint_{M\times M}\frac{|u(x)-u(y)|^p}{(d_g(x,y))^{N+ps}}\,dv_g(x)dv_g(y)<\infty\right\} \\ W_0^{s,p}(M):=\{u\in W^{s,p}(M) \ \mid \ \text{supp}(u) \ \text{is a compact subset of} \ M\}$$ Then are $W^{s,p}(M)$ and $W_0^{s,p}(M)$ equivalent when $M$ is a compact Riemannian manifold with or without boundary $\partial M$? Also what if $M$ is a complete Riemannian manifold?
We know that for compact manifolds $M$ one has $W^{k,p}(M)=W_0^{k,p}(M)$ for $k\in\mathbb{N}$ and $p\geq1$ (cf. Hebey) which are defined as completion of the spaces $C^\infty(M)$ and $C_0^\infty(M)$ respectively. I am curious to know whether this will extend for the fractional case as well.