Abstract
Consider an analytic map of 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. Such a map is called rounding. We introduce a natural equivalence relation on roundings and prove that any rounding, whose differential at 0 has rank at least 2, is equivalent to a fractional quadratic rounding. A fractional quadratic map is just the ratio of a quadratic map and a quadratic polynomial. We also show that any rounding gives rise to a quadratic map between spheres. The known results on quadratic maps between spheres have some interesting implications concerning roundings.
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Timorin, V. Circles and Quadratic Maps Between Spheres. Geom Dedicata 115, 19–32 (2005). https://doi.org/10.1007/s10711-005-8673-4
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DOI: https://doi.org/10.1007/s10711-005-8673-4