Moscow-Petersburg seminar on Low-Dimensional Mathematics
(Joint session with A.M.Vershik's seminar)
May 28, 2003, 17:00

Vladlen Timorin (Moscow--Toronto)
Circles, Quadratic Maps Between Spheres and Representations of 
Clifford Algebras.

Abstract:
Consider an analytic map from a neighborhood of 0 in a vector space to
a Euclidean space. Suppose that this map takes germs
of all lines passing through 0 to germs of circles. Such map is called
rounding.
Two roundings are equivalent if they take the same lines to the same
circles. It turns out that any rounding whose differential at 0 has rank
at least 2 is equivalent to a fractional quadratic rounding. The latter
gives rise to a quadratic map between spheres. We give some
interesting applications of this result including
- a description of rectifiable bundles of circles in dimension 4 which
is surprisingly different from the corresponding results in dimensions
2 (Khovanskii) and 3 (Izadi) due to quaternions,
- a classification of all Kahler metrics in complex dimension 2 whose real
geodesics are circles.
- a geometric description of Clifford algebras representations as
nonlinear projections taking lines to circles.
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