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Showing papers by "David E. Blair published in 1982"


Journal ArticleDOI
TL;DR: In this paper, it was shown that a conformal map of Euclidean space is the product of a similarity and an inversion and similarities preserve flatness for a single inversion.
Abstract: In [1] B.-Y. Chen proved that if ~o:E\" ~ E\" is a conformal transformation of Euclidean space and M is a compact flat surface in E\" such that M = ~o(M) is also a compact flat surface, then there exists a similarity of E\" which also maps M to ~t. Of course, since a conformal map of E\"(n >/3) is the product of a similarity and an inversion and similarities preserve flatness, it is enough to prove this for a single inversion. Chen then shows that M must lie in a sphere concentric with the sphere of inversion. As a simple counterexample in the non-compact case, consider M to be an irregular patch on a cone with vertex at the center of inversion. The inversion then maps M to another such patch )~t, but there is no similarity mapping M to h4. In this paper we prove that the theorem of Chen and this example describe the general behavior of a conformal map mapping a flat submanifold of dimension ~> 3 to another such submanifold. Moreover the phenomena are essentially local in nature.