Circles on surfaces

TL;DR: Celestials in 3-space that are obtained from translating a circle along a circle, in either Euclidean or elliptic space are considered, and a classically avored theorem in elliptic geometry is obtained: if the authors translate a line along acircle but not along a line then exactly 2 translated lines will coincide.

Abstract: The sphere in 3-space has an innite number of circles through any closed point. The torus has 4 circles through any closed point. Two of these circles are known as Villarceau circles ([0]). We dene a \celestial" to be a real surface with at least 2 real circles through a generic closed point. Equivalently, a celestial is a surface with at least 2 families of real circles. In 1980 Blum [1] conjectured that a real surface has either at most 6 families of circles or an innite number. For compact surfaces this conjecture has been proven by Takeuchi [2] in 1987 using topological methods. In 2001 Schicho [3] classied complex surfaces with at least 2 families of conics. This result together with Moebius geometry led to a classication of celestials in 3-space [4]. In 2012 Pottmann et al. [5] conjectured that a surface in 3-space with exactly 3 circles through a closed point is a Darboux Cyclide. We conrm this conjecture as a corollary from our classication in [4]. We recall that a translation is an isometry where every point moves with the same distance. In this talk we consider celestials in 3-space that are obtained from translating a circle along a circle, in either Euclidean or elliptic space. This is a natural extension of classical work by William Kingdon Cliord and Felix Klein on the Cliord torus. Krasauskas, Pottmann and Skopenkov conjectured, that celestials in 3-space of Moebius degree 8 are Moebius equivalent to an Euclidean or Elliptic translational celestial. This conjecture is true if its Moebius model has a family of great circles ([6]). Moreover, its real singular locus consist of a great circle. As a corollary we obtain a classically avored theorem in elliptic geometry: if we translate a line along a circle but not along a line then exactly 2 translated lines will coincide ([6]).