Circles and quadratic maps between spheres
Geometriae Dedicata. 2005. Vol. 115. P. 19-32.
Consider an analytic map from a neighborhood of 0 in a vector space to a Euclidean space. Suppose that this map takes all germs of lines passing through 0 to germs of circles. We call such map a rounding. Two roundings are equivalent if they take the same lines to the same circles. We prove that any rounding whose differential at 0 has rank at least 2 gives rise to a quadratic map between spheres. Results of P. Yiu on quadratic maps between spheres have some interesting implications concerning roundings.