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Cyclides. (English) Zbl 0970.53004

A cyclide is a surface in Euclidean three-space defined by a quartic equation of the form \[ (x_1^2+x_2^2+x_3^2)^2+2(x_1^2+x_2^2+x_3^2)\sum_{i=1}^3b_ix_i+ \sum_{i,j=1}^3a_{ij}x_ix_j+2\sum_{i=1}^3a_ix_i+a=0. \] It is proved in the paper that a nonsingular cyclide is conformally equivalent to cyclide of the form \((x_1^2+x_2^2+x_3^2)^2-2a_1x_1^2-2a_2x_2^2-2a_3^2x_3^2+a=0\) where \(a\not=0\), and that such a surface is topologically a torus, a sphere or a union of two spheres. Such a cyclide contains \(n\) circles through each point which is not an umbilic and \(n-1\) circles through the isolated umbilics where \(n=1,2,3,4,5,\) or \(6\). An example of R. Blum of a genus one surface which contains six circles through each point is a cyclide; see [Lect. Notes Math. 792, 213-221 (1980; Zbl 0428.53001)]. The author conjectures that a surface is a cyclide if it contains two circles through almost every point. The author has proved that a simply connected surface is a plane or a sphere if contains three cicles through a point [J. Geom. 24, 123-130 (1985; Zbl 0571.53001)] and that a closed genus one surface cannot contain seven circles through every point [Proc. Am. Math. Soc. 100, 145-147 (1987; Zbl 0617.53053)].

MSC:

53A05 Surfaces in Euclidean and related spaces
53A04 Curves in Euclidean and related spaces
53A30 Conformal differential geometry (MSC2010)
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