Two surfaces in Euclidean 3-space (identified with the imaginary quaternions) 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 from isothermic surfaces and their Christoffel transforms by means of spin transforms, . Here the multiplication is quaternionic multiplication, and the special form of the spin transforms ensure that the differentials 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.