2026-07-072026-07-07http://salesiana.dossiersoluciones.com/handle/123456789/162206We 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 Poincaré 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 $ϕ: M \to M'$, $ϕ\in W^1_p(M, M')$, $p>n$, $J(x,ϕ)\ne 0$ a. e. in $M$, of compact smooth Riemannian manifold $M$ onto Riemannian manifold $M'$ then the manifold $M'$ has finite geodesic diameter.}}In the first version, there was an inaccuracy in Theorem 4. In the revised version added additional assumptionsFunctional AnalysisComplex Variables46E35; 30C65text