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In one or
two dimensions, the rank preservation requirement is satisfied locally
by any well-behaved mapping. Specifically, a mapping realized
by an analytic function with a non-vanishing Jacobian in a given
region is conformal there [Cohn, 1967]. Such a function preserves
similitude of small triangles; in particular, a scalene triangle
formed by a triplet of points will be mapped into a triangle with the
same ranking of side lengths (see Figure 6). In
higher dimensions, conformality is much more restrictive. As proved
by Liouville in 1850, already for n=3 there are no conformal
mappings from [IMAGE ] to itself besides those which are composed of
finitely many inversions with respect to spheres. Such mappings,
called Möbius transformations, constitute a finite-dimensional Lie
group which includes the group of motions in [IMAGE ] and is only
slightly broader than that group [Reshetnyak, 1989].
Edelman Shimon
Tue Nov 28 13:24:55 IST 1995