Suppose $M$ is a parallelizible Riemannian manifold with metric tensor $g\_x(\cdot,\cdot)$ Let $F\_x(\cdot,\cdot)$ denote the flat metric on $M$ that we get from parallelization. Is it true that there exist c, C such that for any $x$ and $v$ in $T\_xM$, $cg\_x(v,v)\leq F\_x(v,v)\leq Cg\_x(v,v)$?



Lipschitz equivalence of Riemannian metrics
As Dmitri says any two Riemannian metrics on a **compact** manifold are Lipschitz equivalent. The proof is quite simple. 


Consider $g$ and $h$ two metrics on $M$, Let $UM$ be the unit tangent bundle, since $M$ is compact, $UM$ is compact. Then you see that $f:UM\to \mathbb{R}$ defined by $f(x)=\frac{g(x,x)}{h(x,x)}$ is continuous and strictly positive. By compactness, it is bounded above and below by positive constants.


