ar X iv : m at h / 04 05 36 9 v 1 [ m at h . D G ] 1 9 M ay 2 00 4 CONTACT SCHWARZIAN DERIVATIVES DANIEL

H. Sato introduced a Schwarzian derivative of a contactomorphism of R 3 and 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 taking values in the space of sections of a certain vector bundle associated to the contact structure, and 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 R 2n−1 a tensor which vanishes if and only if the given contactomorphism is an element of the linear symplectic group acting by linear fractional transformation.