Bonnet Pairs in the 3 - Sphere

Two non-congruent surfaces that are isometric and have the same mean curvature at corresponding points are called a Bonnet pair of surfaces or simply a Bonnet pair. The interest in such pairs arises from the following considerations: One of the fundamental problems of surface theory is to find invariants which characterize surfaces geometrically. The Bonnet theorem states that a surface is determined up to congruence by its first and second fundamental forms. However, this description has some redundancy since the first and second fundamental forms must satisfy the Gauss and Codazzi equations. So it may be that some pieces of information can be eliminated from this description, but which? It seems wise to retain the metric. But given the metric, we know the Gaussian curvature, i.e. the product of the two principal curvatures. The second fundamental form contains the following extra pieces of information: the mean curvature, i.e. half the sum of the two principal curvatures, and the principal curvature directions. It seems worthwhile to investigate what happens if we do not prescribe the curvature directions and ask the question: Is a surface already determined by its metric and mean curvature?