On the integrability of Bertrand curves and Razzaboni surfaces

Abstract

Based on classical but apparently little known results due to Razzaboni, the integrable nature of Bertrand curves and their geodesic embedding in surfaces is discussed in the context of modern soliton theory. The existence of parallel Razzaboni surfaces which constitute the surface analogues of the classical offset Bertrand mates is recorded. It is shown that the natural geodesic coordinate systems on Razzaboni surfaces and their mates are related by a reciprocal transformation. The geodesic coordinate system on the Razzaboni transform generated by a Bäcklund transformation is given explicitly in terms of Razzaboni’s pseudopotential obeying a compatible Frobenius system. The Razzaboni transformation and the duality transformation which links a Razzaboni surface and its mate are proven to commute. A canonical quantity introduced by Razzaboni is recognized as an invariant of the Razzaboni and duality transformations. Finally, Razzaboni surfaces are shown to be amenable to the Sym–Tafel formula.