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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