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Abstract:
A classical theorem of Liouville states that any
conformal, or angle-preserving, map between open subsets of R^n, for n
at least 3, is a composition of an inversion and an affine similarity.
I will present a proof, due to Charles Frances, of the main step in
Liouville's theorem, showing that any angle-preserving map sends
spheres to spheres. The idea is to extend the map to C^n, where it
will still be conformal in some sense, and to use the fact that these
conformal maps send certain complex lines, called null geodesics, to
other null geodesics.
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