Bonnet ruled surfaces (Q2581188)

Bonnet surfaces are those immersions into Euclidean 3-space whose congruence class is not uniquely determined by their metric and mean curvature. As already proven by Bonnet, all Bonnet surfaces belong to one of the following classes: constant mean curvature surfaces, Bonnet families and Bonnet pairs. The author of the present paper studies ruled surfaces that are Bonnet pairs. This is done using a result by \textit{Z. Soyucok} [see: J. Geom. 52, No. 1--2, 173--188 (1995; Zbl 0828.53004)] according to which Bonnet surfaces are characterized by the existence of a so called A-net, i.e., a conformal chart with respect to which the off-diagonal coefficient of the second fundamental form is constant.