##
*Hajime Sato*

Schwarzian derivatives of contact diffeomorphisms

*(Lobachevskii Journal of Mathematics,
Vol.4, pp.89-98)*

In this note, we give the definition of Schwarzian derivative of contact
diffeomorphism $\phi : K^3 \to K^3$ where $K$ is $\R$ or $\C$. The \Sch
\ is a quadruple of functions and plays an analogous role to the already-defined
Schwarzian derivatives of nondegenerate maps of multi-variables. See the
books of M.Yoshida{\cite{Yo1} and T.Sasaki{\cite{Sa}}. 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 \cite{SY}. In a forthcoming paper \cite{SO2}
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.

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