Circles and Quadratic Maps Between Spheres

TL;DR

This paper studies special analytic maps called roundings that transform lines into circles and shows they are equivalent to fractional quadratic maps, leading to insights on quadratic sphere maps.

Contribution

It introduces an equivalence relation on roundings and proves their classification as fractional quadratic maps when the differential rank is at least 2.

Findings

Roundings are equivalent to fractional quadratic maps.

Any rounding induces a quadratic map between spheres.

Implications for known results on quadratic sphere maps.

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.