  We study global regularity properties of Sobolev homeomorphisms on
$n$-dimensional Riemannian manifolds under the assumption of $p$-integrability
of its first weak derivatives in degree $p\geq n-1$. We prove that inverse
homeomorphisms have integrable first weak derivatives. For the case $p>n$ we
obtain necessary conditions for existence of Sobolev homeomorphisms between
manifolds. These necessary conditions based on Poincar\'e type inequality: $$
\inf_{c\in \mathbb R} \|u-c\mid L_{\infty}(M)\|\leq K \|u\mid
L^1_{\infty}(M)\|. $$ As a corollary we obtain the following geometrical
necessary condition:
  {\em If there exists a Sobolev homeomorphisms $\phi: M \to M'$, $\phi\in
W^1_p(M, M')$, $p>n$, $J(x,\phi)\ne 0$ a. e. in $M$, of compact smooth
Riemannian manifold $M$ onto Riemannian manifold $M'$ then the manifold $M'$
has finite geodesic diameter.}}
