New properties of composition operators in Sobolev spaces on Riemannian manifolds

An equivalent description is obtained for homeomorphisms $\varphi$ of a domain $\Omega$ in a Riemannian space $\Bbb{M}$ onto a metric space $\Bbb{Y}$, which guarantees the boundedness of the composition operator from the space of Lipschitz functions $\operatorname{Lip}(\Bbb{Y})$ into the homogeneous Sobolev space on $\Bbb{M}$ 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 $\Bbb{M}$ 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.