Consider $M$ a manifold and $g_1, g_2$ two different Riemannian metrics. I want to know how the condition $|\nabla^{g_1,k}(g_1-g_2)|_{g_1}\leq C$ implies that the norms of $|\nabla^{g_1,i}u|_{T^{\otimes i}M, g_1}$ and $|\nabla^{g_2,i}u|_{T^{\otimes i}M, g_2}$ are equivalent for $i=1,...,k+1$, where $\nabla^{g_1,i}=\nabla^{g_1}\dots\nabla^{g_1}$ $i$-times, and therefore the Sobolev spaces up to order $k+1$ defined by $g_1$ and $g_2$ are equivalent. For $k=0$, this is quasi-isometry and for first order Sobolev spaces I know how to show it. But for arbitrary order I haven't found a way. So I post it here if someone can give me a reference. Because every time I saw this statement there was not an indication of a proof.