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Bonnet pairs and isothermic surfaces. (English) Zbl 1003.53007

Two surfaces in Euclidean 3-space (identified with the imaginary quaternions) 3 ImH 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, df ± =(±ε+f * )df(±ε-f * ). Here the multiplication is quaternionic multiplication, and the special form of the spin transforms ensure that the differentials df ± 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.

53A10Minimal surfaces, surfaces with prescribed mean curvature
53A05Surfaces in Euclidean space