dg.differential geometry - 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 $f:UMto mathbb{R}$ 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.