Schwarzian derivatives of contact diffeomorphisms

In this note, we give the definition of Schwarzian derivative of contact diffeomorphism $\phi\colon K^3\to K^3$ where $K$ is $\mathbb R$ or $\mathbb C$. The Schwarzian derivative is a quadruple of functions and plays an analogous role to the alreadydefined Schwarzian derivatives of nondegenerate maps of multi-variables. See the books of M. Yoshida [13] and T. Sasaki [10]. We give a survey of known results in sections 2 and 3. In sections 4 and 5, we define the Schwarzian derivative and consider analogous results in the contact case. The contact Schwarzian derivative vanishes if the contact diffeomorphism keep the third order ordinary differential equation $y'''=0$ invariant. We also give the condition for a quadruple of functions to be the contact Schwarzian derivative of a contact diffeomorphism. These results are consequences of our paper Sato–Yoshikawa [11]. In a forthcoming paper [9] with Ozawa, we give a system of linear partial differential equations whose coefficients are given by contact Schwarzian derivatives. If a quadruple of functions satisfies the condition, the system of partial differential equations is integrable and the solution gives the contact diffeomorphism.