*(English)*Zbl 1003.53007

Two surfaces in Euclidean 3-space (identified with the imaginary quaternions) ${\mathbb{R}}^{3}\cong \text{Im}H$ are said to form a Bonnet pair if they induce the same metric and have the same mean curvature. The authors construct such Bonnet pairs $({f}_{+},{f}_{-})$ from isothermic surfaces $f$ and their Christoffel transforms ${f}^{*}$ by means of spin transforms, $d{f}_{\pm}=(\pm \epsilon +{f}^{*})df(\pm \epsilon -{f}^{*})$. Here the multiplication is quaternionic multiplication, and the special form of the spin transforms ensure that the differentials $d{f}_{\pm}$ are integrable. Moreover, it is shown that any Bonnet pair defined on a simply connected domain can be obtained in this way.

Note that a relation between isothermic surfaces in space forms and Bonnet pairs was known to *M. Servant* [Comptes Rendus 134, 1291-1293 (1902; JFM 33.0651.01)] who established a relation on the level of fundamental forms, and that *L. Bianchi* [Rom. Acc. L. Rend. 12, 511-520 (1903; JFM 34.0654.01)] derived formulas for the differentials of the surfaces of a Bonnet pair from the frame of an isothermic surface in the 3-sphere.