Two Riemannian Metrics $g$ and $h$ on a Manifold $M$ are Quasi-isometric if there exist positive constants $A$ and $B$ such that $$A\|u\|_g\leq\|u\|_h\leq B\|u\|_g$$ for any $u \in T(N)$. where $\|\cdot\|_g$ and $\|\cdot\|_h$ are the norms determined by $g$ and $h$, respectively.
The questiones are
- Is it possible that a complete riemannian metric is quasi-isometric to incomplete one?
- Is it true that if the manifold is compact then any two Riemannian metrics on it are quasi-isometric?