Lobachevskii Journal of Mathematics Vol. 4, 1999, 89–98

\begin{matrix}c \\ ◯ \end{matrix}H. Sato

H. Sato

SCHWARZIAN DERIVATIVES OF CONTACT DIFFEOMORPHISMS

(submitted by B.N.Shapukov)

1. Introduction

In this note, we give the definition of Schwarzian derivative of contact diffeomorphism ϕ : K^{3} → K^{3} where K is ℝ or 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.

2. Schwarzian derivative of nondegenerate maps

Suppose that a nondegenerate map(diffeomorphism) ϕ : K^{n} → K^{n} is given:

ϕ : x = (x_{1}, …, x_{n}) → (X_{1}, … X_{n})

for n ≥ 2. According to Yoshida[13] (see also [14]), by using Einstein convention, Schwarzian derivative S_{ij}^{k}(ϕ, x) is given by

S_{ij}^{k}(ϕ, x) = \frac{∂^{2}X^{p}}{∂x^{i}∂x^{j}} \frac{∂x^{k}}{∂X^{p}} − \frac{1}{n + 1}(δ_{i}^{k} \frac{∂^{2}X^{p}}{∂x^{q}∂x^{j}} \frac{∂x^{q}}{∂X^{p}} + δ_{j}^{k} \frac{∂^{2}X^{p}}{∂x^{q}∂x^{i}} \frac{∂x^{q}}{∂X^{p}}),

(cf. Gunning[5], Kobayashi-Ochiai[7]). Clearly S_{ij}^{k}(ϕ, x) = S_{ji}^{k}(ϕ, x) and further they satisfy the canonical-form relation

∑_{m=1}^{n} S_{im}^{m}(ϕ, x) = 0 for i = 1, 2, …, n.

So the number of Schwarzian derivatives is (n^{3} + n^{2} − 2n)/2. The Schwarzian derivatives are P GL(n + 1, K)-invariant. That is, for A ∈ P GL(n + 1, K),

S_{ij}^{k}(ϕ, x) = S_{ij}^{k}(Aϕ, x) for 1 ≤ i, j, k ≤ n.

They vanish for any projective linear transformation ϕ.

Conversely, if all S_{ij}^{k}(ϕ, x) vanish for 1 ≤ i, j, k ≤ n, then ϕ is a projective linear transformation.

For n = 2, as a Schwarzian derivative, we have following quadruple of four functions where we write ϕ(x, y) = (X(x, y), Y (x, y)) and X_{x}, X_{xx} … mean the partial differentials ∂X/∂x, ∂^{2}X/∂x^{2} … etc., and Δ is the Jacobian ∂(X, Y)/∂(x, y).

\{ \begin{matrix}S_{11}^{2}(ϕ, x) & = Δ^{−1}(Y_{xx}X_{x} − X_{xx}Y_{x}), \\ S_{11}^{1}(ϕ, x) & = −S_{12}^{2}(ϕ, x) \\ & = \frac{1}{3}Δ^{−1}(X_{xx}Y_{y} − Y_{xx}X_{y} + 2(X_{xy}Y_{x} − Y_{xy}X_{x})), \\ S_{22}^{2}(ϕ, x) & = −S_{12}^{1}(ϕ, x), \\ & = \frac{1}{3}Δ^{−1}(Y_{yy}X_{x} − X_{yy}Y_{x} + 2(Y_{xy}X_{y} − X_{xy}Y_{y})), \\ S_{22}^{1}(ϕ, x) & = Δ^{−1}(X_{yy}Y_{y} − Y_{yy}X_{y}). \end{matrix}  (1)

These four functions appear when we consider the ordinary differential equation y′′ = 0 as follows. In the space K^{2} = {(x, y)}, regard the second variable y as a function with variable x. We study the problem of deciding the ordinary differential equations y′′ = f(x, y, y′) which are transformed to the equation y′′ = 0 by a diffeomorphism of the (x, y) space (point transformation).

Taking y′ = dy/dx, we get K^{3} = {(x, y, z = y′)}. A diffeomorphism ϕ : K^{2} → K^{2} with ϕ(x, y) = (X, Y) induces the prolongation ϕ̃ : K^{3} → K^{3} given by

ϕ̃(x, y, z) = (X, Y, Z = \frac{Y_{x} + zY_{y}}{X_{x} + zX_{y}}),

where X_{x} = ∂X/∂x, … etc. By using this prolongation, we get the following.

Proposition 2.1. The inverse ϕ^{−1} of a diffeomorphism ϕ : K^{2} → K^{2} maps the differential equation Y′′ = 0 to

y′′ = −S_{11}^{2}(ϕ, x) + 3S_{11}^{1}(ϕ, x)y′ − 3S_{22}^{2}(ϕ, x)(y′)^{2} + S_{22}^{1}(ϕ, x)(y′)^{3}

Proof. Since Y′ = Z, we calculate the differential form dZ on K^{3}. Using the equation Z = (Y_{x} + zY_{y})/(X_{x} + zX_{y}) and putting the equality dy = zdx, we get

dZ = \frac{X_{x}Y_{y} − Y_{x}X_{y}}{(X_{x} + zX_{y})^{2}} (dz − Udx)

where

U = −S_{11}^{2}(ϕ, x) + S_{11}^{1}(ϕ, x)z − S_{22}^{2}(ϕ, x)z^{2} + S_{22}^{1}(ϕ, x)z^{3}.

Since Y′′ = 0 is equivalent to dZ = 0, we get dz − Udx = 0. Using z = y′, we obtain the result.

Conversely, the condition that four functions are the Schwarzian derivatives of some diffeomorphism is given as follows.

Theorem 2.1. Let P_{11}^{2}, P_{11}^{1}, P_{22}^{2}, P_{11}^{1} be four functions on K^{2}. Then the condition that there exists a diffeomorphism ϕ : K^{2} → K^{2} such that

P_{ij}^{k} = S_{ij}^{k}(ϕ, x)

(i.e. four functions are Schwarzian derivatives of ϕ) is that P_{ij}^{k} satisfies the following two partial differential equations, where we put P_{11}^{2} = −P, P_{11}^{1} = Q, P_{22}^{2} = −R, P_{11}^{1} = S.

\begin{cases}R_{xx} − 2Q_{xy} + P_{yy} + SP_{x} + 2P S_{x} & − 3P R_{y} \\ & − 3RP_{y} − 3QR_{x} + 6QQ_{y} = 0 \\ Q_{yy} − 2R_{xy} + S_{xx} − P S_{y} − 2SP_{y} & + 3SQ_{x} \\ & + 3QS_{x} + 3RQ_{y} − 6RR_{x} = 0 \end{cases} (2)

Proof. By Proposition 2.1, the condition is equivalent to that the second order ordinary differential equation y′′ = P + 3Qy′ + 3R(y′)^{2} + S(y′)^{3} is mapped by a point transformation to y′′ = 0. There are old results due to Tresse[12], E.Cartan[1]. E.Cartan regarded a differential equation y′′ = f(x, y, y′) as a projective structure and calculated the curvature. If the curvature vanishes then the structure is flat and y′′ = f(x, y, y′) is equivalent to y′′ = 0 by a point transformation. The curvature vanishes when f(x, y, y′) is polynomial of y′ with order ≤ 3 and the coefficients of y′ satisfy the relation of the theorem. A modern treatment is given in Grissom-Thompson-Wilkens[3].

For n ≥ 3, we must consider the system of ordinary equations y′′ = f(x, y, y′) where y = ^{t}(y_{1}, y_{2}, …, y_{n−1}). For n = 3, the author has analogous result to Proposition 2.1, but, for n ≥ 4, concrete expressions by Schwarzian derivatives are still unknown.

3. Construction of diffeomorphism

When given four functions P, Q, R, S on K^{2} which satisfy the conditions of Theorem 2.1, we consider the problem of constructing by partial differential equations the diffeomprphism whose Schwarzian derivatives are the functions P, Q, R, S. This is solved by Gronwall[4] when he considered the problem of making nomogram(cf. Hénaut[6]). We follow his arguments extending to general case. For a diffeomorphism ϕ(x, y) = (X, Y), as in (1), the Schwarzian derivatives are expressed by partial derivatives

X_{xx}, X_{xy}, X_{yy}, Y_{xx}, Y_{xy}, Y_{yy}, X_{x}, X_{y}, Y_{x}, Y_{y}

and Δ = X_{x}Y_{y} − X_{y}Y_{x}. Thus we have four equations. Put θ = log Δ (Δ = e^{θ}). By the partial differentiation, we get

X_{xx}Y_{y} − Y_{xx}X_{y} − (X_{xy}Y_{x} − Y_{xy}X_{x}) = e^{θ}θ_{x}
Y_{yy}X_{x} − X_{yy}Y_{x} − (Y_{xy}X_{y} − X_{xy}Y_{y}) = e^{θ}θ_{y}

Together with the expressions of the Schwarzian derivative, we get six equations. We solve these equations linearly relative to second derivatives X_{xx}, X_{xy}, X_{yy}, Y_{xx}, Y_{xy}, Y_{yy}. We get six equations as follows;

\{ \begin{matrix}π_{xx} & = (Q + \frac{2}{3}θ_{x})π_{x} − P π_{y} \\ π_{xy} & = (R + \frac{1}{3}θ_{y})π_{x} + (−Q + \frac{1}{3}θ_{x})π_{y} \\ π_{yy} & = Sπ_{x} + (−R + \frac{2}{3}θ_{y})π_{y} \end{matrix}  (3)

where π is either X or Y. A constant function is a solution. Since a projective linear transformation preserves Schwarzian derivatives, the solution vector space should be of dimension 3. This is equivalent to that the system is integrable. The integrable condition of the equations (3) is given by

\{ \begin{matrix}θ_{xx} & = \frac{1}{3}(θ_{x})^{2} + Qθ_{x} − P θ_{y} + 3(Q_{x} − P_{y} + 2P R − 2Q^{2}) \\ θ_{xy} & = \frac{1}{3}θ_{x}θ_{y} + Rθ_{x} − Qθ_{y} + 3(R_{x} − Q_{y} + P S − QR) \\ θ_{yy} & = \frac{1}{3}(θ_{y})^{2} + Sθ_{x} − Rθ_{y} + 3(S_{x} − R_{y} + 2QS − 2R^{2}) \end{matrix}  (4)

A projective linear transformation by an element (\begin{matrix}a & b & c \\ d & e & f \\ g & h & i \end{matrix}) ∈ SL(3, K)

changes the Jacobian Δ to (ax + by + c)^{−3}Δ. The dimension of the space of θ’s by all projective linear transformations is equal to 3. The solution of (4) must depend on θ(x_{0}), θ_{x}(x_{0}), θ_{y}(x_{0}) and the integrable condition must be satisfied. By a simple calculation, we know that the integrable condition of the equations (4) is just equal to the condition (2) of Theorem 2.1. Put

ϑ = e^{−1/3θ}.

Then the system (4) is equivalent to the following linear system.

\{ \begin{matrix}ϑ_{xx} & = Qϑ_{x} − P ϑ_{y} − (Q_{x} − P_{y} + 2P R − 2Q^{2})ϑ \\ ϑ_{xy} & = Rϑ_{x} − Qϑ_{y} − (R_{x} − Q_{y} + P S − QR)ϑ \\ ϑ_{yy} & = Sϑ_{x} − Rϑ_{y} − (S_{x} − R_{y} + 2QS − 2R^{2})ϑ \end{matrix}  (5)

The solution vector space of (5) is of dimension 3 if and only if the condition (2) is satisfied.

On the other hand, put

Π = e^{−1/3θ}π.

The key point is that, under the condition (2), by a simple calculation, we know that the system (3) is just equivalent to (5), with ϑ = Π. The system (5) has ϑ_{0} ≡ e^{−1/3θ} as a solution. Under the condition (2), we have two other independent solutions ϑ_{1} and ϑ_{2} which we can write

ϑ_{1} = e^{−1/3θ}π_{1}, ϑ_{2} = e^{−1/3θ}π_{2}.

Then

π_{1} = ϑ_{1}/ϑ_{0}, π_{2} = ϑ_{2}/ϑ_{0}.

Both π_{1} and π_{2} are solutions of (5) which are independent of θ.

We have proved the following.

Theorem 3.1. Let ϕ(x_{1}, x_{2}) = (X_{1}, X_{2}) be a diffeomorphism of K^{2} whose Schwarzian derivatives are given by P_{11}^{2} = −P, P_{11}^{1} = Q, P_{22}^{2} = −R, P_{11}^{1} = S. Then the equations (2) hold. The solution vector space of (5) is of dimension 3. Each X_{1} or X_{2} is equal to the ratio of three independent solutions of (5).

Conversely, we have the following.

Theorem 3.2. Suppose that four functions P, Q, R, S satisfy the condition (2). Then the solution vector space of (5) is of dimension 3. Let ϑ_{0}, ϑ_{1}, ϑ_{2} be a basis. Then the mapping ϕ : K^{2} → K^{2} defined by ϕ(x_{1}, x_{2}) = (ϑ_{1}/ϑ_{0}, ϑ_{2}/ϑ_{0}) for x = (x_{1}, x_{2}) ∈ K^{2} is a diffeomorphism such that S_{11}^{2}(ϕ, x) = −P, S_{11}^{1}(ϕ, x) = Q, S_{22}^{2}(ϕ, x) = −R, S_{22}^{1}(ϕ, x) = S.

Proof. Put

δ = |\begin{matrix}ϑ_{0} & ϑ_{1} & ϑ_{2} \\ (ϑ_{0})_{x} & (ϑ_{1})_{x} & (ϑ_{2})_{x} \\ (ϑ_{0})_{y} & (ϑ_{1})_{y} & (ϑ_{2})_{y} \end{matrix}| = ϑ_{0}^{3} \frac{∂(ϑ_{1}/ϑ_{0}, ϑ_{2}/ϑ_{0})}{∂(x, y)}.

Since ϑ_{i} are bases of the solutions of (5), δ never vanishes. Further, since ϑ_{i} are solution of (5), we obtain by calculation, that δ_{x} = δ_{y} = 0. Thus δ is a nonzero constant k. Put θ = −3 log k − 3 log ϑ_{0}. Then the Jacobian ∂(ϑ_{1}/ϑ_{0}, ϑ_{2}/ϑ_{0})/∂(x, y) is equal to e^{θ}. Since ϑ_{0} satisfies (5), θ satisfies (4) and ϑ_{1}/ϑ_{0} and ϑ_{2}/ϑ_{0} satisfy (3). Thus the functions ϑ_{1}/ϑ_{0}, ϑ_{2}/ϑ_{0} and θ satisfy the original six relations defining the Schwarzian derivatives.

4. Schwarzian derivative of contact diffeomorphism

On the 3-dimensional space K^{3} = {(x, y, z)}, we give the contact form ω = dy − zdx. A contact diffeomorphism ϕ : K^{3} → K^{3} is a diffeomorphism such that ϕ^{∗}ω = fω for some non-vanishing function f.

The 3-dimensional Heisenberg group H^{3} is defined by

H^{3} = \{(\begin{matrix}1 & r & q \\ 0 & 1 & p \\ 0 & 0 & 1 \end{matrix}) | p, q, r ∈ K\}

and its Lie algebra 𝔥^{3} is given by 𝔥^{3} = \{(\begin{matrix}0 & r & q \\ 0 & 0 & p \\ 0 & 0 & 0 \end{matrix}) | p, q, r ∈ K\}. We may

identify the contact space K^{3} with H^{3} by a diffeomorphism mapping x, y, z to p, q, r respectively. Then H^{3} acts on K^{3} as a left multiplication which keeps the contact form ω invariant. The invariant vector fields on K^{3} are given by

v_{1} = \frac{∂}{∂x} + z \frac{∂}{∂y} , v_{2} = \frac{∂}{∂z} , v_{3} = \frac{∂}{∂y}

with

[v_{2}, v_{1}] = v_{3}, [v_{1}, v_{3}] = 0, [v_{2}, v_{3}] = 0.

Put v_{4} = v_{1}v_{2} + v_{2}v_{1}. We regard the contact space K^{3} = {(x, y, z)} as {(x, y, y′)} and we have the space K^{4} = {(x, y, y′, y′′)} with 1-forms dy − y′dx, dy′ − y′′dx. In [11], we have studied the problem to decide the ordinary differential equations y′′′ = f(x, y, y′, y′′) which are transformed to the equation y′′′ = 0 by a contact diffeomorphism of the (x, y, y′) space.

For a contact diffeomorphism ϕ : K^{3} → K^{3} with

ϕ(x, y, z) = (X(x, y, z), Y (x, y, z), Z(x, y, z)),

we put

Δ = |\begin{matrix}v_{1}(X) & v_{1}(Z) \\ v_{2}(X) & v_{2}(Z) \end{matrix}| ,

and define the contact Schwarzian derivatives as follows.

\{ \begin{matrix}C_{11}^{2}(ϕ, x) & = Δ^{−1}(v_{1}^{2}(Z)v_{1}(X) − v_{1}^{2}(X)v_{1}(Z)), \\ C_{11}^{1}(ϕ, x) & = −C_{12}^{2}(ϕ, x) \\ = \frac{1}{3}Δ^{−1} & (v_{1}^{2}(X)v_{2}(Z) − v_{1}^{2}(Z)v_{2}(X) + v_{4}(X)v_{1}(Z) − v_{4}(Z)v_{1}(X)), \\ C_{22}^{2}(ϕ, x) & = −C_{12}^{1}(ϕ, x), \\ = \frac{1}{3}Δ^{−1} & (v_{2}^{2}(Z)v_{1}(X) − v_{2}^{2}(X)v_{1}(Z) + v_{4}(Z)v_{2}(X) − v_{4}(X)v_{2}(Z)), \\ C_{22}^{1}(ϕ, x) & = Δ^{−1}(v_{2}^{2}(X)v_{2}(Z) − v_{2}^{2}(Z)v_{2}(X)). \end{matrix} (6)

A diffeomorphism ϕ : K^{3} → K^{3} with ϕ(x, y, z) = (X, Y, Z) induces the prolongation ϕ̃: K^{4} → K^{4} given by

ϕ̃(x, y, z, w) = (X, Y, Z, W = \frac{Z_{x} + zZ_{y} + wZ_{z}}{X_{x} + zX_{y} + wX_{x}}).

By using this prolongation, we get the following.

Proposition 4.1. The inverse ϕ^{−1} of a contact diffeomorphism ϕ : K^{3} → K^{3} maps the differential equation Y′′′ = 0 to

y′′′ = −C_{11}^{2}(ϕ, x) + 3C_{11}^{1}(ϕ, x)y′′ − 3C_{22}^{2}(ϕ, x)(y′′)^{2} + C_{22}^{1}(ϕ, x)(y′′)^{3}

Proof. Since Y′′ = W, we calculate the differential form dW on K^{4}. Using the equation W = (Z_{x} + zZ_{y} + wZ_{z})/(X_{x} + zX_{y} + wX_{x}) and putting the equality dy = zdx, dz = w, we get

dW = \frac{Δ}{(X_{x} + zX_{y} + wX_{z})^{2}} (dw − V dx)

where

V = −C_{11}^{2}(ϕ, x) + C_{11}^{1}(ϕ, x)z − C_{22}^{2}(ϕ, x)z^{2} + C_{22}^{1}(ϕ, x)z^{3}.

Since Y′′′ = 0 is equivalent to dW = 0, we get dw − V dx = 0. Using w = y′′, we obtain the result.

As a corollary, we obtain the following.

Corollary 4.1. Let ϕ : K^{3} → K^{3} be a contact diffeomorphism. If all contact Schwarzian derivatives C_{ij}^{k}(ϕ, x) vanish, then ϕ keeps the differential equation y′′′ = 0 invariant.

Conversely, the condition that four functions are the contact Schwarzian derivatives of some contact diffeomorphism is given as follows.

Theorem 4.1. Let P_{11}^{2}, P_{11}^{1}, P_{22}^{2}, P_{11}^{1} be four functions on the contact space K^{3}. Then the condition that there exists a contact diffeomorphism ϕ : K^{3} → K^{3} such that

P_{ij}^{k} = C_{ij}^{k}(ϕ, x)

(i.e. four functions are the contact Schwarzian derivatives of ϕ) is that P_{ij}^{k} satisfies the following four partial differential equations, where we put P_{11}^{2} = −P, P_{11}^{1} = Q, P_{22}^{2} = −R, P_{22}^{1} = S.

\begin{cases}\frac{3}{2}v_{1}v_{2}(P)− & v_{2}v_{1}(P) − \frac{1}{2}v_{1}^{2}(Q) + P v_{2}(Q) − Qv_{2}(P) − 2P v_{1}(R) \\ − Rv_{1}(P) + 3Qv_{1}(Q) − P^{2}S − 2Q^{3} + 3P QR = 0 \\ \frac{1}{2}v_{2}^{2}(P) + 4v & _{1}v_{2}(Q) − \frac{7}{2}v_{2}v_{1}(Q) − v_{1}^{2}(R) − 2P v_{1}(S) + 3Qv_{2}(Q) \\ −3Rv_{2} & (P) + 3Rv_{1}(Q) − Sv_{1}(P) + 6P R^{2} − 3Q^{2}R − 3P QS = 0 \\ −\frac{1}{2}v_{1}^{2}(S) − 4 & v_{2}v_{1}(R) + \frac{7}{2}v_{1}v_{2}(R) + v_{2}^{2}(Q) − 2Sv_{2}(P) + 3Rv_{1}(R) \\ −3Qv_{1} & (S) + 3Qv_{2}(R) − P v_{2}(S) − 6Q^{2}S + 3QR^{2} + 3P RS = 0 \\ −\frac{3}{2}v_{2}v_{1}(S)+ & v_{1}v_{2}(S) + \frac{1}{2}v_{2}^{2}(R) + Sv_{1}(R) − Rv_{1}(S) − 2Sv_{2}(Q) \\ − Qv_{2}(S) + 3Rv_{2}(R) + P S^{2} + 2R^{3} − 3QRS = 0 \end{cases} (7)

Proof. By Proposition 4.1, the condition is equivalent to that the third order ordinary differential equation y′′′ = P + 3Qy′′ + 3R(y′′)^{2} + S(y′′)^{3} is mapped by a contact transformation to y′′′ = 0. By [11], the condition that y′′′ = f(x, y, y′, y′′) is mapped to y′′′ = 0 by a contact transformation is the vanishing of two curvatures A and b. The curvature b is equal to a^{2}∂^{4}f/6c^{5}∂q^{4} on a principal bundle (see. [11, p.1002]). We obtain that b = 0 is equivalent to ∂^{4}f/∂x^{4} = 0. This is satisfied in our case. The curvature A is found by Chern [2] and the condition A = 0 is equal to

\frac{∂f}{∂y} + \frac{1}{3} \frac{∂f}{∂q} \frac{∂f}{∂p} + \frac{2}{27} (\frac{∂f}{∂q} )^{3} − \frac{1}{2} \frac{d}{dx} \frac{∂f}{∂q} − \frac{1}{3} \frac{∂f}{∂q} \frac{d}{dx} \frac{∂f}{∂q} + \frac{1}{6} \frac{d^{2}}{dx^{2}} \frac{∂f}{∂q} = 0,

where we write p = y′, q = y′′, d/dx = ∂/∂x + p∂/∂y + q∂/∂p + f∂/∂q. Arranging the left-hand side for f = P + 3Qq + 3Rq^{2} + Sq^{3} as a polynomial of q, we know that degree is three. Vanishing of the coefficients of each degree gives the conditions of our theorem.

5. Construction of contact diffeomorphism

When given four functions P, Q, R, S on K^{3} which satisfy the conditions of Theorem 4.1, we consider the problem of constructing the contact diffeomprphism whose contact Schwarzian derivatives are the functions P, Q, R, S. For a contact diffeomorphism ϕ(x, y, z) = (X, Y, Z), as in (6), the contact Schwarzian derivatives are expressed by partial derivatives

v_{i}v_{j}(X), v_{i}v_{j}(Z), v_{i}(X), v_{i}(Z), v_{4}(X), v_{4}(Z), 1 ≤ i, j ≤ 2

and Δ = v_{1}(X)v_{2}(Z) − v_{1}(Z)v_{2}(X). Thus we have four equations. Put θ = log Δ (Δ = e^{θ}). By differentiating Δ by v_{i}, we get

v_{1}^{2}(X)v_{2}(Z) − v_{1}^{2}(Z)v_{2}(X) − v_{1}v_{2}(X)v_{1}(Z) + v_{1}v_{2}(Z)v_{1}(X) = e^{θ}v_{1}(θ)
v_{2}^{2}(Z)v_{1}(X) − v_{2}^{2}(X)v_{1}(Z) − v_{2}v_{1}(Z)v_{2}(X) + v_{2}v_{1}(X)v_{2}(Z) = e^{θ}v_{2}(θ).

Thus we have two equations.

We have another condition that there should exist Y such that (x, y, z) → (X, Y, Z) is a contact diffeomorphism.

Proposition 5.1. On the contact space K^{3} = {(x, y, z)} with the contact form ω = dy − zdx, suppose that two function X = X(x, y, z), Z = Z(x, y, z) are given. The condition that there exists a function Y = Y (x, y, z) such that

dY − ZdX = g(dy − zdx)  (8)

for some function g = g(x, y, z) is given by the following two equations;

v_{1}^{2}(X)v_{2}(Z) − v_{1}^{2}(Z)v_{2}(X)+v_{2}v_{1}(X)v_{1}(Z) − v_{2}v_{1}(Z)v_{1}(X)
+ 2(v_{1}v_{2}(Z)v_{1}(X) − v_{1}v_{2}(X)v_{1}(Z)) = 0
v_{2}^{2}(X)v_{1}(Z) − v_{2}^{2}(Z)v_{1}(X)+v_{1}v_{2}(X)v_{2}(Z) − v_{1}v_{2}(Z)v_{2}(X)
+ 2(v_{2}v_{1}(Z)v_{2}(X) − v_{2}v_{2}(X)v_{2}(Z)) = 0

Proof. From (8), we have

Y_{x} − Z_{x} = −gz, Y_{y} − Z_{y} = g, Y_{z} − ZX_{z} = 0.

Thus we get the condition

v_{1}(Y) = Zv_{1}(X), v_{2}(Y) = Zv_{2}(X).

Consider the following problem. Given two functions T_{i} = T_{i}(x, y, z), (i = 1, 2) on K^{3} and consider the following system of partial differential equations with unknown function Y = Y (x, y, z);

v_{i}(Y) = T_{i} for i = 1, 2.  (9)

By usingthe structure of the universal enveloping algebra of the Heisenberg Lie algebra 𝔥, we know that the condition of the existence of Y is the following two equations

\begin{aligned}(2v_{1}v_{2} − v_{2}v_{1})(T_{1}) − v_{1}^{2}(T_{2}) & = 0 \\ (2v_{2}v_{1} − v_{1}v_{2})(T_{2}) − v_{2}^{2}(T_{1}) & = 0. \end{aligned}  (10)

Applying these equation, we get the condition.

Together with the six equations expressing the contact Schwarzian derivatives and the differentiations of Δ, we get eight equations. We solve these equations linearly relative to eight second derivatives v_{i}v_{j}(X), v_{i}v_{j}(Z), 1 ≤ i, j ≤ 2. Then we get eight equations as follows;

\{ \begin{matrix}v_{1}^{2}(π) & = (Q + v_{1}(θ))v_{1}(π) − P v_{2}(π) \\ v_{1}v_{2}(π) & = (R + v_{2}(θ))v_{1}(π) − Qv_{2}(π) \\ v_{2}v_{1}(π) & = Rv_{1}(π) + (−Q + v_{1}(θ))v_{2}(π) \\ v_{2}^{2}(π) & = Sv_{1}(π) + (−R + v_{2}(θ))v_{2}(π) \end{matrix}  (11)

where π is either X or Z.

Any constant function is a solution. When the system is integrable, the solution vector space is of dimension three. Necessary integrability conditions of the equations (11) are given by the following equations.

\{ \begin{matrix}v_{1}^{2}(θ) & = \frac{1}{2}v_{1}(θ)^{2} + Qv_{1}(θ) − P v_{2}(θ) + \frac{1}{2}v_{1}(Q) − \frac{1}{2}v_{2}(P) + P R − Q^{2} \\ v_{4}(θ) & = v_{1}(θ)v_{2}(θ) + 2(Rv_{1}(θ) − Qv_{2}(θ)) + v_{1}(R) − v_{2}(Q) + P S − QR \\ v_{2}^{2}(θ) & = \frac{1}{2}v_{2}(θ)^{2} + Sv_{1}(θ) − Rv_{2}(θ) + \frac{1}{2}v_{1}(S) − \frac{1}{2}v_{2}(R) + QS − R^{2} \end{matrix} (12)

This is the first order condition for the integrability of (12) and we have second order integrability condition, both of which give the complete integrability condition of (12).

Put

ϑ = e^{− \frac{1}{2} θ}.

Then the system (12) is equivalent to the following linear system.

\{ \begin{matrix}v_{1}^{2}(ϑ) & = Qv_{1}(ϑ) − P v_{2}(ϑ) − \frac{1}{2}(\frac{1}{2}v_{1}(Q) − \frac{1}{2}v_{2}(P) + P R − Q^{2})ϑ \\ v_{4}(ϑ) & = 2(Rv_{1}(ϑ) − Qv_{2}(ϑ)) − \frac{1}{2}(v_{1}(R) − v_{2}(Q) + P S − QR)ϑ \\ v_{2}^{2}(ϑ) & = Sv_{1}(ϑ) − Rv_{2}(ϑ) − \frac{1}{2}(\frac{1}{2}v_{1}(S) − \frac{1}{2}v_{2}(R) + QS − R^{2})ϑ \end{matrix} (13)

In the following paper [9], we prove that the condition (7) leads the integrability condition of (13). Further we show that (7) leads both the first and the second order integrability conditions of (12).

Theorem 5.1 ([9]). Under the condition (7), the equations (13) (hence (12)) and the equations (11) are integrable.

Consequently, given four functions P, Q, R, S on the contact space K^{3} which satisfy the integrable conditions (7), we can construct the contact diffeomorphism ϕ : K^{3} → K^{3} whose contact Schwarzian derivatives are P, Q, R, S by the solutions of linear partial differential equations.

H. Sato 98

References

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[3] Grissom, C., Thompson, G. and Wilkens, G., Linearization of Second Order Ordinary Differential Equations via Cartan’s Equivalence Method, J. Diff. Equ. 77, pp. 1-15(1989).

[4] Gronwall, T. H., Sur les équations entre trois variables représentables par des nomogrammes à points alignés, J. Math. Pures Appl. 8, pp 59-102(1912)

[5] Gunning, R., On uniformization of complex manifolds: the role of connections Math. Notes No.22, Princeton, Princeton University Press, 1978

[6] Hénaut, A., Sur la linéarisation des tissue de C^{2}, Topology, 32, pp. 531-542(1993).

[7] Kobayashi, S. and Ochiai, T., Holomorphic Projective Structures on Compact Complex Surfaces, Math. Ann. 249, pp. 75-94(1980).

[8] Ozawa, T. and Sato, H., Linearizations of ordinary differential equations by area preserving maps, to appear in Nagoya Math. J.

[9] Ozawa, T and Sato, H., in preparation.

[10] Sasaki, T, Projective Differential Geometry and Linear Homogeneous Differential Equations, Rokko Lectures in Math. 5, Kobe, Kobe University, 1999

[11] Sato, H. and Yoshikawa, A. Y., Third order ordinary differential equations and Legendre connections J. Math. Soc. Japan, 50, pp. 993-1013 (1998).

[12] Tressse, A., Détermination des invariants ponctuels de l’éuation différentielle ordinaire du second order y′′ = ω(x, y, y′), Leipzig, Hirzel, 1896.

[13] Yoshida, M., Fuchsian Differential Equations, Aspects of Mathematics, Vieweg, Braunschweig, 1987.

[14] Yoshida, M., Schwarzian program (in Japanese), Sûgaku, 40, pp. 36-46 (1988).

Graduate School of Mathematics, Nagoya University, Chikusa-ku, Nagoya, 464-8602, Japan E-mail address: hsato@math.nagoya-u.ac.jp

Received July 27, 1999