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 $T_xM$, $cg_x(v,v)\leq F_x(v,v)\leq Cg_x(v,v)$?
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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. 


$X_1,..., X_n$
are vector fields on$M$
that parallelize$M$
. Can't we define$F_p(v,u)$ = \left\langle (a_1,...,a_n), (b_1,...,b_n)\right\rangle$
where$v = a_1X_1(p) + \cdots + a_nX_n(p)$
and$u = b_1X_1(p) + \cdots + b_nX_n(p)$
? – William Sep 2 '10 at 8:56