Deformation of Surfaces Preserving Principal Curvatures

The isometric deformation of surfaces preserving the principal curvatures was first studied by O. Bonnet in 1867. Bonnet restricted himself to the complex case, so that his surfaces are analytic, and the results are different from the real case. After the works of a number of mathematicians, W. C. Graustein took up the real case in 1924-, without completely settling the problem. An authoritative study of this problem was carried out by Elie Cartan in [2], using moving frames. Based on this work, we wish to prove the following: Theorem: The non-trivial families of isometric surfaces having the same principal curvatures are the following: 1)a family of surfaces of constant mean curvature;2)a family of surfaces of non-constant mean curvature. Such surfaces depend on six arbitrary constants, and have the properties: a) they are W-surfaces; b) the metric $$d{s^2} = {\left( {gradH} \right)^2}d{s^2}/\left( {{H^2} - K} \right)$$, where d s2 is the metric of the surface and H and K are its mean curvature and Gaussian curvature respectively, has Gaussian curvature equal to — 1.