Deformation of surfaces preserving principal curvatures. (English) Zbl 0566.53002

Differential geometry and complex analysis, Vol. dedic. H. E. Rauch, 155-163 (1985).
[For the entire collection see Zbl 0561.00010.]
Using moving frames the author continues the investigations of Bonnet (1867), Graustein (1924) and Elie Cartan (1942) on isometric deformations of surfaces \(\subset {\mathbb{E}}^ 3\) preserving the principal curvatures. He proves that beside the surfaces of constant mean curvature (cf. Bonnet) there exists a six parameter family of surfaces admitting such deformations; they are W-surfaces, and if their metric \(ds^ 2\) is replaced by (grad H)\({}^ 2ds^ 2/(H^ 2-K)\) then they have Gaussian curvature -1.
Reviewer: H.Reckziegel


53A05 Surfaces in Euclidean and related spaces


Zbl 0561.00010