Contact Schwarzian Derivatives

Abstract

H. Sato introduced a Schwarzian derivative of a contactomorphism of three-dimensional Euclidean space and with T. Ozawa described its basic properties. In this note their construction is extended to all odd dimensions and to non-flat contact projective structures. The contact projective Schwarzian derivative of a contact projective structure is defined to be a cocycle of the contactomorphism group measuring the extent to which a contactomorphism fails to be an automorphism of the contact projective structure. For the flat model contact projective structure this gives a contact Schwarzian derivative associating to a contactomorphism of Euclidean space a tensor which vanishes if and only if the given contactomorphism is an element of the linear symplectic group acting by linear fractional transformations.