Global results on Bonnet Surfaces

Abstract

The local theory of the Bonnet Surfaces in the three dimensional Euclidean Space of the type of non-constant mean curvature that admit infinitely many non-trivial and geometrically distinct isometries preserving the mean curvature has been developed, in the literature, under the assumption that the surfaces contain no umbilic points and no critical points of the mean curvature function. Here, we prove that these restrictions do not create any difference from the possible global results, except in one case in which we prove that the set of the umbilic points, known to consist of exactly one point, is equal to the set of the critical points of the mean curvature function. Furthermore, we show that the index of this umbilic point, as isolated singularity of the foliation of the principal curves is one. In our proofs we use: (a) An intrinsic characterization of these surfaces, which we derive in a manner easier and including more details than those already found in the literature. From this characterization we conclude that all surfaces of this type are analytic. (b) The harmonic functions of the angles by which the respected isometries rotate the principal frames, which we compute.