An equivalent description is obtained for homeomorphisms $ \varphi $ of a domain $ \Omega $ in a Riemannian space $ 𝕄 $ onto a metric space $ 𝕐 $ , which guarantees the boundedness of the composition operator from the space of Lipschitz functions $ \operatorname{Lip}(𝕐) $ into the homogeneous Sobolev space on $ 𝕄 $ with first generalized derivatives integrable to the power $ 1\leq q\leq\infty $ , along with other new properties of such homeomorphisms.The new approach makes it possible to effectively prove a theorem on homeomorphisms of domains in an arbitrary Riemannian space $ 𝕄 $ that induce a bounded composition operator between Sobolev spaces with first generalized derivatives.The new proof, which is considerably shorter compared to the original one, relies on a minimal set of tools and allows us to establish new properties of the homeomorphisms under study.
