Contact Transformations and Their Schwarzian Derivatives

Abstract

The second author introduced Schwarzian derivatives for 3-dimensional contact transformations in [Sat]. Our purposes of this paper are firstly to investigate the fundamental properties of the contact Schwarzian derivatives of 3-dimensional contact transformations that are satisfied by the usual Schwarzian derivatives, secondly to consider systems of linear PDE's with contact Schwarzian derivatives as coefficients and their integrability conditions, and finally to reconstruct the contact transformation from the solutions of the systems of linear PDE's. We obtain the necessary and sufficient condition for functions to be contact Schwarzian derivatives of a 3-dimensional contact transformation.