discussion in Section \[final\] below, we give a proof of it with some details.

\[riem-mm\] Suppose $(X, g)$ is a compact Riemannian $n$-manifold. Then for every $p$ such that $1 < p  < \infty$ the spaces $L^p_1(X, V_g)$ and $M^p_1(X, V_g)$ coincide with equivalent norms.

By Proposition 10.1 from [@HajlaszKoskela] $$M^p_1(X,V_g) \hookrightarrow L^p_1(X,V_g),$$ hence we need the opposite inclusion. Since $X$ is compact, there exists a finite number of charts $$\{(U_{\alpha},\phi_{\alpha})\ :\ \alpha \in A \},$$ such that for every $\alpha$ the components $g_{i j}^\alpha$ of $g$ in $(U_{\alpha},\phi_{\alpha})$ satisfy $$\begin{aligned}
 \frac{1}{2} \delta_{ij} \leq g^{\alpha}_{ij} \leq 2 \delta_{ij}\end{aligned}$$ as bilinear forms. Let $\{ \eta_{\alpha} \}$ be smooth partition of unity subordinate to covering $\{ U_{\alpha} \}$. We proceed in two steps.\
[**Step 1.**]{} Let $\mathcal{L}^n$ be the $n$-dimensional Lebesgue measure, and fix $u$ in $C^{\infty}(X)$. Since $M^p_1(\mathbb{R}^n, \mathcal{L}^n)$ and $L^p_1(\mathbb{R}^n, \mathcal{L}^n)$ are equivalent, see [@Hajlasz] for example, there exists a constant $C > 1$ such that for every $\alpha$ in $A$ $$\begin{aligned}
\label{ru}
\frac{1}{C} \| (\eta_{\alpha} u ) \circ \phi_{\alpha}^{-1} \|_{L^p_1(\mathbb{R}^n, \mathcal{L}^n)} \leq 
\| (\eta_{\alpha} u ) \circ \phi_{\alpha}^{-1} \|_{M^p_1(\mathbb{R}^n, \mathcal{L}^n)} \leq C \| (\eta_{\alpha} u ) \circ \phi_{\alpha}^{-1}\|_{L^p_1(\mathbb{R}^n,\mathcal{L}^n)}.\end{aligned}$$ Furthermore $$\int_X |\eta_{\alpha} u|^p d V_g =  \int_{U_{\alpha}} |\eta_{\alpha} u|^p d 