Sergey Vodopyanov (Sobolev Institute of Mathematics)
Sobolev functions and mappings on Riemannian manifolds
Abstract.
It is well known a classical result: composition of smooth functions is also a smooth function. It is supposed to show a solution of the same problem for Sobolev functions with the first generalized derivatives. More exactly, we consider an arbitrary mapping from one spatial domain to another one and give necessary and sufficient conditions for the composition operator with Sobolev functions to be bounded. If the composition operator is an isomorphism we have a rigidity of the geometry of the given mapping in the following sense: the mapping coincides almost everywhere with a quasi-isometry if the summability does not equal the topological dimension of the space. Some applications of this approach to the global characterization of Riemannian manifolds will be considered.

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