Let $(M,g)$ be a smooth Riemannian manifold of dimension $n\geq1$ . Let $p\geq1$ . Then the Sobolev space $H_1^p(M)$ on the manifold is defined by the completion of $C_1^p(M)$ in $L^p(M)$ , where $C_1^p(M)$ is defined as $$C_1^p(M)=\bigg\{u\in C^\infty(M) \ \bigg| \ \int_M|\nabla^iu|^p_g\,dv_g<+\infty \ \ \text{for} \ \ i=0,1\bigg\}$$ where $dv_g=\sqrt{|g|}\,dx$ is the volume element , and $\displaystyle|\nabla u|_g=\sqrt{g^{kl}\frac{\partial u}{\partial x^k}\frac{\partial u}{\partial x^l}}$ in local coordinates .
I want to verify why this definition is consistent with the standard Sobolev space $W_1^p(\Omega)$ we define in terms of distributional derivatives in open subsets $\Omega$ of $\mathbb{R}^n$ . Moreover , is the space $H_1^p(M)$ endowed with the norm $$||u||_{H_1^p}=||u||_{L^p}+||\nabla u||_{L^p}$$ a Banach space ? If so , how to prove it ? Any help is appreciated .