Orbit decomposition of space of differential equations

Hajime SATO Graduate School of Mathematics, Nagoya Universty

0. Picture.

The contents of this paper can be caught by the following picture [Figure 1]. Let 𝒢 be a continuous group which in general is of infinite dimension. For

example, 𝒢 is the diffeomorphism group Diff(M) of a finite dimensional manifold M or the contact diffeomorphism group ContDiff(M) of a contact manifold M.

Let ℳ be the (infinite dimensional) set of all differential equations of a fixed type or the set of all structures of a fixed type on a manifold M. For example, let ℳ be the set of all second order ordinary differential equations {y′′ = f(x, y, y′)} where y = y(x) is a function on ℝ, or the set of all projective structures ProjStr(N) on a finite dimensional manifold N.

The equivalence problem of differential equations or geometric structures is to decide whether one element in ℳ is transformed to another element in ℳ by an element of 𝒢.

The quadrature of differential equations is the method to give a solution of an differential equation μ ∈ ℳ by finding an element μ_{0} ∈ ℳ and an element g ∈ 𝒢 such that μ_{0} is an easily integrable equation and g(μ) = μ_{0}. If we have an explicit expression of the inverse element g^{−1} ∈ 𝒢, we can integrate μ_{0} to obtain solutions.

Specifically these problems are formulated as follows.

Given a natural action of 𝒢 on ℳ, determine the orbit 𝒢μ = {gμ | g ∈ 𝒢} for μ ∈ ℳ. If g(μ) = μ_{0} (μ, μ_{0} ∈ ℳ), find g ∈ 𝒢.

In other words, the problem is to know the orbit space 𝒪 and to determine the orbit decomposition of ℳ by 𝒢.

Define the isotropy group 𝒢_{μ} at μ ∈ ℳ by

𝒢_{μ} = {g ∈ 𝒢 | g(μ) = μ}.

The interesting cases for our study are when 𝒢_{μ} are finite dimensional Lie groups for all μ ∈ ℳ. Then an important problem is to know the orbit 𝒢_{μ_{0}} for μ_{0} ∈ ℳ such that dim 𝒢_{μ_{0}} is of maximum. The orbit 𝒢_{μ_{0}} is of minimal dimension and very singular, which we call a minimal orbit.

In studying the orbit decomposition of ℳ, we can find interesting phenomena by comparing two different sets ℳ for one 𝒢. In some cases, we have ℳ_{1} and ℳ_{2} consisting of different types of differential equations such that the actions of 𝒢 are isomorphic. Then we say that 𝒢 spaces ℳ_{1} and ℳ_{2} are dual to each other.

μ μ_{0} 𝒢 ↷ ℳ 𝒪  ^{orbit}

Figure 1: image picture

A discovery of such phenomena will be helpful in understanding of each type of differential equations.

Moreover, the following explanation may be possible. In the classical mechanics, a movement is an element of Diff(M) which acts on the space M. In the quantum mechanics, we consider corresponding differential equations. Our setting is to consider the action of Diff(M) on function spaces of M and the movement is determined by differential equations. Thus our purpose may be said to quantize the classical mechanics.

The purpose of this paper is to collect and survey some results. The author sincerely thanks to the coauthors and coworkers Atsuko Yoshikawa, Tetsuya Ozawa, Keizo Yamaguchi and Hiroshi Suzuki.

1. 2nd order ordinary differential equations.

In this section, we consider the set of all 2nd order ordinary differential equations with one independent variable x and one dependent variable y. Thus we put

ℳ = {y′′ = f(x, y, y′)}.

If we write z = y′, then {(x, y, y′)} is equal to the space ℝ^{3} = {(x, y, z)} with the contact form dy − zdx. The space ℳ is isomorphic to the set of all functions with three independent variables {f(x, y, z)}.

Since we use the method of exterior differential forms, arguments may be global. But in the following, we are mainly concerned with local situations near a neighborhood of a point (x_{0}, y_{0}, z_{0}) in ℝ^{3}.

As a continuous group, we put

𝒢 = ContDiff(x, y, z).

This contact diffeomorphism group 𝒢 acts naturally on ℳ.

The classical Lie’s theorem asserts that any 2nd order ordinary differential equation is locally equivalent to y′′ = 0 under contact transformations. Thus the action of 𝒢 on ℳ is locally transitive.

Then what happens when we choose 𝒢 smaller than ContDiff(x, y, z)? Let us put 𝒢 = Diff(x, y).

For an element ϕ : (x, y) → (X(x, y), Y (x, y)) ∈ 𝒢, put z = \frac{dy}{dx}. Since

Y′ = \frac{dY}{dX} = \frac{Y_{x}dx + Y_{y}dy}{X_{x}dx + X_{y}dy} = \frac{Y_{x} + Y_{y} \frac{dy}{dx}}{X_{x} + X_{y} \frac{dy}{dx}} = \frac{Y_{x} + Y_{y}z}{X_{x} + X_{y}z} ,

we get an element ϕ̃ ∈ ContDiff(x, y, z) by

ϕ̃: (x, y, z) ↦ (X, Y, Z = \frac{Y_{x} + Y_{y}z}{X_{x} + X_{y}z} ) .

This contact diffeomorphism ϕ̃ is called the prolongation of ϕ ∈ Diff(x, y). The prolongation ϕ̃ induces a transformation of ℳ by f(x, y, z) ↦ f(ϕ̃(x, y, z)). Thus we get an action of 𝒢 = Diff(x, y) on ℳ = {y′′ = f(x, y, y′)}. The induced transformations are classically called point transformations. A work of Tresse([Tr]) in the 19th century is to classify 2nd order ordinary differential equations by point transformations.

Further we can choose following subgroups each of which gives an interesting object to study;

control system transformation (x, y) ↦ (x, Y (x, y)), Painlevé type transformation (x, y) ↦ (X(x), \frac{ψ_{1}(x)y+ψ_{2}(x)}{ψ_{3}(x)y+ψ_{4}(x)}), web type transformation (x, y) ↦ (X(x), Y (y)).

Let us return to the case when 𝒢 = Diff(x, y). ¿From the works of Tresse, we know that a point in the orbit space of the action of 𝒢 = Diff(x, y) on ℳ = {y′′ = f(x, y, y′)} is locally expressed by two functions

A = A(f) = \frac{d^{2}f_{zz}}{dx^{2}} − 4\frac{df_{zy}}{dx} − 3f_{y}f_{zz} + 6f_{yy} + f_{z}(4f_{zy} − \frac{df_{zz}}{dx}),
B = B(f) = f_{zzzz}.  (\frac{d}{dx} = \frac{∂}{∂x} + z \frac{∂}{∂y} + f \frac{∂}{∂z})

This means that two differential equations y′′ = f(x, y, y′) and y′′ = g(x, y, y′) are transformable to one another by point transformations if and only if A(f) = A(g) and B(f) = B(g).

The set of solutions of a differential equation y′′ = f(x, y, y′) defines a structure of path geometry on the contact manifold (x, y, y′). Cartan defined a connection on the structure and showed that two functions A(f) and B(f) are just equal to the curvatures of that connection ([Ca1], cf. [IL, Chap. 8]). He extended this result to that of projective differential geometry and clarified the geometric feature.

The minimal orbit of this 𝒢 = Diff(x, y)−action on ℳ = {y′′ = f(x, y, y′)} is the orbit 𝒢_{μ_{0}} of the flat point μ_{0} = {y′′ = 0}. We can write

𝒢_{μ_{0}} = {y′′ = P (x, y) + 3Q(x, y)y′ + 3R(x, y)(y′)^{2} + S(x, y)(y′)^{3}},

where ϕ(x, y) = (X(x, y), Y (x, y)) ∈ 𝒢. This satisfies the condition A = 0. The functions P, Q, R, S are defined by

\{ \begin{matrix}P (x, y) & = Δ^{−1}(Y_{xx}X_{x} − X_{xx}Y_{x}), \\ Q(x, y) & = \frac{1}{3}Δ^{−1}(X_{xx}Y_{y} − Y_{xx}X_{y} + 2(X_{xy}Y_{x} − Y_{xy}X_{x})), \\ R(x, y) & = \frac{1}{3}Δ^{−1}(Y_{yy}X_{x} − X_{yy}Y_{x} + 2(Y_{xy}X_{y} − X_{xy}Y_{y})), \\ S(x, y) & = Δ^{−1}(X_{yy}Y_{y} − Y_{yy}X_{y}),  Δ = \frac{∂(X, Y)}{∂(x, y)}. \end{matrix}

This differential equation y′′ = P +3Qy′+3R(y′)^{2}+S(y′)^{3} satisfies the condition A = B = 0.

Conversely, the condition that an element in ℳ

y′′ = P (x, y) + 3Q(x, y)y′ + 3R(x, y)(y′)^{2} + S(x, y)(y′)^{3}

(satisfying B = 0) is contained in the orbit 𝒢_{μ_{0}} of the flat point μ_{0} = {y′′ = 0} is equal to that four functions P, Q, R, S satisfy the following condition (IC) derived from the condition A = 0 ;

\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}

Cartan regarded (IC) as the vanishing of the curvature of the path geometry defined by the differential equation μ = {y′′ = P (x, y) + 3Q(x, y)y′ + 3R(x, y)(y′)^{2} + S(x, y)(y′)^{3}}.

The isotropy group 𝒢_{μ_{0}} of 𝒢 = Diff(x, y) at μ_{0} = {y′′ = 0} is isomorphic to the fractional linear group in P GL(3, ℝ). This 𝒢_{μ_{0}} is equal to the affine transformation group Aff(ℝ^{2}) consisting of elements which maps the infinite point to itself.

This argument works globally. On the 2-shere S^{2}, we have

Diff(S^{2})/P GL(3, ℝ) ≅ {(P, Q, R, S) with (IC) }.

Now let

μ = {y′′ = P (x, y) + 3Q(x, y)y′ + 3R(x, y)(y′)^{2} + S(x, y)(y′)^{3} with (IC)}

be an element in the minimal orbit 𝒢_{μ_{0}} of the flat point μ_{0} = {y′′ = 0}. Then the element g ∈ Diff(x, y) such that g(μ) = μ_{0} is captured by solving the following system of linear partial differential equations (SD) called Schwarzian differential equation:

(SD) \{ \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}

If we choose the functions P, Q, R, S to be some special ones with singularity, the Schwarzian equation is equal to the Appel’s hypergeometric function.

The condition (IC) is just equal to the integrability condition of the system (SD).

If (IC) is satisfied, the dimension of the solution space of (SD) is equal to 3. Let ϑ_{0}, ϑ_{1}, ϑ_{2} be a basis of the solution. Define a diffeomorphism g : (x, y) ↦ (X, Y) by

X = \frac{ϑ_{1}}{ϑ_{0}} , Y = \frac{ϑ_{2}}{ϑ_{0}} .

Then we obtain g(μ) = μ_{0}. The ambiguity of the choice of the basis which is isomorphic to P SL(3, ℝ) comes from the isotropy group 𝒢_{μ_{0}}.

Dual equations

In this situation of the action of 𝒢 on ℳ, we can find the dual objects in the same ℳ.

An differential equation y′′ = f(x, y, y′) defines the Legendre vector field V_{f} = \frac{∂}{∂x} + z \frac{∂}{∂y} + f(x, y, z)\frac{∂}{∂z} on (x, y, z). The integral curves are the solutions of the differential equation.

To consider point transformations is to specify the Legendre vector field \frac{∂}{∂z} . Thus we have a Legendre 2-web structure.

In this situation, we can show that there exists a contact transformation

(x, y, y′ = z) ↦ (X, Y, Z)

such that

V_{f} = \frac{∂}{∂Z} .

Then we can write

V_{F} ≡ \frac{∂}{∂X} + Z \frac{∂}{∂Y} + F (X, Y, Z)\frac{∂}{∂Z} = \frac{∂}{∂z}

for some function F = F (X, Y, Z). Thus Y′′ = F (X, Y, Y′) is a dual equation of y′′ = f(x, y, y′).

By taking the dual equation, we have an involution of ℳ , and so, an involution of 𝒪 which is expressed as

𝒪 ∋ (A, B) ↦ (B, A).

The reason that we can choose the dual in the same space ℳ is explained by the following fact.

The double fibration of the twistor picture defined by the model Legendrian 2-web Legendre 2-web structure

SL(3, ℝ)/H = SO(3)
S^{1} ↙ ↘^{S^{1}} SL(3, ℝ)/H_{1} = P^{2}(ℝ)  SL(3, ℝ)/H_{2} = P^{2}(ℝ)

is symmetric.

Expression of dual equations

If the solutions of y′′ = f(x, y, y′) is given by

y = h(x, X, Y), z ≡ y′ = h_{x}(x, X, Y)

with two parameters X, Y, then a dual equation Y′′ = F (X, Y, Z) is given concretely by the following formula.

We take the coordinate of the space (x, y, z) by (x, X, Y). Further we take its coordinate by (Z, X, Y) such that dy − zdx = g (dY − ZdX), g ≠ 0. Then (x, y, z) ↦ (X, Y, Z) is a contact transformation. By a calculation, we get

g = h_{Y} , Z = −\frac{h_{x}}{h_{y}} .

Thus

\frac{dZ}{dX} = \frac{h_{X}h_{XY} − h_{Y} h_{XX} + Z(h_{X}h_{Y Y} − h_{Y} h_{XY})}{h_{Y} ^{2}}.

Putting F to be the function of the right-hand-side, we obtain a dual equation Y′′ = F (X, Y, Y′) .

2. 3rd order ordinary differential equations.

In this section, we consider the set of all 3rd order ordinary differential equations with one independent variable x and one dependent variable y. Thus we put

ℳ = {y′′′ = f(x, y, y′, y′′)}.

If we write z = y′, w = y′′, then {(x, y, y′, y′′)} is equal to the Engel manifold ℝ^{4} = {(x, y, z, w)} with two Pfaffian forms dy − zdx, dz − wdx.

Here, a 4-dimensional manifold M is an Engel manifold if there are two Pfaffian forms on M such that the annihilating space D^{2} is a 2-dimensional distribution such that

dim[D^{2}, D^{2}] = 3, dim[ [D^{2}, D^{2}], [D^{2}, D^{2}] ] = 4.

A transformation of an Engel manifold which keeps the distribution D^{2} invariant is called an Engel transformation. By a classical theorem of Bäcklund, any Engel transformation is locally a prolongation of a contact transformation of a 3-dimensional contact manifold. A prolongation of a contact transformation ϕ : (x, y, z) ↦ (X(x, y, z), Y (x, y, z), Z(x, y, z)) of (x, y, z) is given by ϕ̃ : (x, y, z, w) → (X, Y, Z, W = \frac{Y_{x}+Y_{y}z+Y_{z}w}{X_{x}+X_{y}z+X_{z}w} ) , which is an Engel transformation.

Thus even if we choose 𝒢 to be the set of all Engel transformations, the action is not locally transitive and the Lie’s theorem does not hold in this situation.

Now let 𝒢 be the set of all contact transformations of (x, y, y′); 𝒢 = ContDiff (x, y, y′). We study the action of this 𝒢 on the space ℳ of all 3rd order ordinary differential equations.

We take the following three vector fields on the contact space {x, y, z}:

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

They are the basis of the Lie algebra of the 3-dimensional heisenberg group. We have the 2nd order differential operator v_{4} = v_{2}v_{1} + v_{1}v_{2} and the contact Jacobian

Δ = v_{1}(X)v_{2}(Z) − v_{2}(X)v_{1}(Z).

The geometric structure corresponding to the space of 3rd order differential equations under contact transformation can be formulated as an isotropic path geometry on an Engel manifold. We can use Cartan’s moving frame method. A point in the orbit space 𝒪 of the action of 𝒢 = ContDiff (x, y, y′) on ℳ = {y′′′ = f(x, y, y′, y′′)} is represented by the following two functions ([SY])．

A = A(f) = −f_{y} − \frac{1}{3}f_{zw} − \frac{2}{27}(f_{w})^{3} + \frac{1}{2} \frac{df_{z}}{dx} + \frac{1}{3}f_{w} \frac{df_{w}}{dx} − \frac{1}{6} \frac{d^{2}f_{w}}{dx^{2}} ,
B = B(f) = f_{wwww}. (\frac{d}{dx} = \frac{∂}{∂x} + z \frac{∂}{∂y} + w \frac{∂}{∂z} + f \frac{∂}{∂w})

The minimal orbit of this 𝒢−action on ℳ is the orbit 𝒢_{μ_{0}} of the flat point μ_{0} = {y′′′ = 0}. We can write

𝒢_{μ_{0}} = {y′′′ = P + 3Qy′′ + 3R(y′′)^{2} + S(y′′)^{3}},

where P, Q, R, S are functions with variables (x, y, z = y′), and are given from a contact transformation (x, y, z) ↦ (X, Y, Z) by the following;

\{ \begin{matrix}P = & Δ^{−1}(v_{1}^{2}(Z)v_{1}(X) − v_{1}^{2}(X)v_{1}(Z)), \\ Q = & \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)), \\ R = & \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)), \\ S = & Δ^{−1}(v_{2}^{2}(X)v_{2}(Z) − v_{2}^{2}(Z)v_{2}(X)). \end{matrix}

Conversely, the condition that an element in ℳ

y′′′ = P + 3Qy′′ + 3R(y′′)^{2} + S(y′′)^{3}

(satisfying B = 0) is contained in the orbit 𝒢_{μ_{0}} of the flat point μ_{0} = {y′′′ = 0} is equal to that four functions P, Q, R, S satisfy the following condition (CIC) derived from the condition A = 0 ;

\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) − 4v_{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}

Now let

μ = {y′′′ = P + 3Qy′′ + 3R(y′′)^{2} + S(y′′)^{3} with (IC)}

be an element in the minimal orbit 𝒢_{μ_{0}} of the flat point μ_{0} = {y′′′ = 0}. Then the element g ∈ Diff(x, y) such that g(μ) = μ_{0} is captured by solving the following system of linear partial differential equations (CSD) called contact Schwarzian differential equation ([OS]);

\{ \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}

The condition (CIC) is just equal to the integrability condition of the system (CSD). If (CIC) is satisfied, the dimension of the solution space of (CSD) is equal to 4.

Let ϑ_{0}, ϑ_{1}, ϑ_{2} be a basis of the solution. Define a diffeomorphism g : (x, y) ↦ (X, Y) by

X = \frac{ϑ_{1}}{ϑ_{0}} , Y = \frac{ϑ_{2}}{ϑ_{0}} .

Then we obtain g(μ) = μ_{0}. The ambiguity of the choice of the basis which is isomorphic to P SL(3, ℝ) comes from the isotropy group 𝒢_{μ_{0}}.

For solutions ϑ, η of (CSD), a bilinear form

I(ϑ, η) = \frac{1}{2}ϑv_{3}(η) − \frac{1}{2}v_{3}(ϑ)η + v_{1}(ϑ)v_{2}(η) − v_{2}(ϑ)v_{1}(η)

is a symplectic form. Let {ϑ, ξ, ζ, η} be a symplectic basis of the solution of (CSD). Then the map

g : (x, y, z) ↦ (\frac{ξ}{ϑ}, \frac{1}{2} (\frac{η}{ϑ} + \frac{ξζ}{ϑ^{2}} ) , \frac{ζ}{ϑ})

is a contact transformation such that g(μ) = μ_{0}.

Considering globally, on S^{3}, we have

ContDiff(S^{3})/PSp(4, ℝ) ≅ {(P, Q, R, S) with (CIC)}.

Dual Equations

The differential equation y′′′ = f(x, y, y′, y′′) defines an isotropic vector field V_{f} = \frac{∂}{∂x} + z \frac{∂}{∂y} + w \frac{∂}{∂z} + f \frac{∂}{∂w} on the Engel manifold (x, y, z, w). Since we consider the equivalence problem under contact transformations of (x, y, z), we distinguish the vector field \frac{∂}{∂w} . The distributions V_{f} and \frac{∂}{∂w} span the Engel distribution D^{2}. But the two vector fields are not equivalent. For the Cauchy characteristic Char([D^{2}, D^{2}]) of the commutator distribution [D^{2}, D^{2}], we have the following equation;

Char([D^{2}, D^{2}]) = ⟨ \frac{∂}{∂w} ⟩.

Thus an en Engel transformation necessarily maps ⟨\frac{∂}{∂w} ⟩ onto itself. Thus a dual equation of a 3rd order differential equation must lie in other ℳ.

We have the following double fibration;

S^{3} × S^{1} = Sp(2, ℝ)/H̃ ≈ SO(3, 2)/H = P^{3}(ℝ) × S^{1}
S^{1} ↙  ↘^{S^{1}} S^{3} = Sp(2, ℝ)/H_{1}  SO(3, 2)/H_{2} = S^{2} × S^{1}.

The fibration in the left side can be regarded as an S^{1}-fibration of Legendre subspaces on a contact 3-manifold, while the right side is the S^{1}-fibration of null subspaces of the tangent bundle of a conformal compactification of a flat (2, 1)-metric.

To abstract the geometry of the null geodesics of (2, 1)-metrics, we proceed as follows.

Let M be a 3-dimensional manifold and let ℳ̂ be the set of all 1-parameter family of 1-dimensional foliations of general type of M. Here a 1-parameter family of 1-dimensional foliations is of general quadric type means the following. A 1-dimensional foliation is determined by a nonsingular vector field V. Let s be a parameter. Then a 1-parameter family of 1-dimensional foliations is determined by a 1-parameter family V_{s} of nonsingular vector fields. Then for x ∈ M, ⋃_{s} V_{s}(x) defines a curve v_{x}(s) in the projective space P (T_{x}M) which is homeomorphic to P^{2}(ℝ).

Definition. A 1-parameter family of 1-dimensional foliations V_{s} is of general

type if the velocity vector \frac{dv_{x}(s)}{ds} never vanishes for all s and for all x ∈ M. If V_{s} is of general type, the union

W = {⋃_{s} v_{x}(s) | x ∈ M^{3}}

is a 4-dimensional submanifold of the projective tangent bundle P (T M) which is a P^{2}(ℝ) bundle over M.

A point p in the projective tangent bundle P (T_{x}M) represents a 1-dimensional line L(p) in T_{x}(M). Let π : P (T M) → M be the projection. On P (T M), we have the tautological 2-dimensional distribution E defined by

E = {X ∈ T_{p}(P (M)) | {π_{∗}X} = L_{p}, p ∈ P (M)}.

Definition. A 1-parameter family of 1-dimensional foliations V_{s} is of general quadric type if the restriction D = E |_{W} is an Engel distribution of the 4dimensional manifold W.

In such case, the vector fields Char([D, D]) and the projection direction of π : W → M span the Engel distribution D. Define 𝒢̂ to be the group of diffeomorphisms of M whose prolongation on P (T M) keep W invariant. The pair (𝒢̂, ℳ̂) is the dual of our (𝒢, ℳ).

This is related to the works of Frittelli, Kozameh, Newman, Kamran etc. concerning the null surface formalism of GR (see e.g. [FKN]).

3. Goursat equations.

Any partial differential equation with two independent and one dependent variables z = x(x, y) is written as

F (x, y, z, p, q, r, s, t) = 0,

using the variables p = \frac{∂z}{∂x}, q = \frac{∂z}{∂y} , r = \frac{∂^{2}z}{∂x^{2}} , s = \frac{∂^{2}z}{∂x∂y} , t = \frac{∂^{2}z}{∂y^{2}} .

The following system

\begin{cases}r + λs + 2ψ − λψ′ = 0, \\ s + λt + ψ′ = 0 \end{cases}

expressed by a parameter λ and a function ψ = ψ(x, y, z, p, q, λ) with ψ′ = \frac{∂ψ}{∂λ}

defines a hypersurface of the space (x, y, z, p, q, r, s, t) by the elimination of λ. Thus this system is a partial differential equation with two independent and one dependent variables. If the condition (GC):

\begin{aligned}& 2ψ_{y} + 2qψ_{z} − 2(ψ′ − λψ′′)ψ_{p} − 2ψ′′ψ_{q} \\ = & ψ′_{x} + λψ′_{y} + (p + λq)ψ′_{z} − (2ψ − λψ′)ψ′_{p} − ψ′ψ′_{q} \end{aligned}

is satisfied, we call the differential equation Goursat equation ( [Ca2], cf [Ya], [Sa]).

A condition that a partial differential equation with two independent and one dependent variables is a Goursat equation is that it is parabolic and the Monge characteristic system is integrable. Such equations are studied first by Goursat and consequently deeply by Cartan.

Let ℳ be the set of all Goursat equations such that the degree on λ of the function ψ is greater than 2. The 5-dimensional space {(x, y, z, p, q)} is a contact manifold. Put 𝒢 = ContDiff (x, y, z, p, q). The group 𝒢 acts naturally on ℳ. Then, for any μ ∈ ℳ, the isotropy group 𝒢_{μ} is a finite dimensional Lie group. The minimal orbit of this 𝒢 action on ℳ is the orbit of the differential equation

μ_{0} : {9 r^{2} + 12t^{2}(rt − s^{2}) + 32s^{3} − 36rst = 0}.

This equation is a Goursat equation defined by ψ(x, y, z, p, q, λ) = \frac{1}{6}λ^{3}. The isotropy group 𝒢_{μ_{0}} is the non-compact exceptional simple Lie group G_{2}. The differential equation μ_{0} can be solved by quadratures. The space of the

solutions is parameterized by two functions of one variable.

The orbit space of this 𝒢 action on ℳ is parameterized by one function of one variable.

4. Overdetermined involutive systems.

A system consisting of two partial differential equations with two independent variables and one dependent variable z = z(x, y) is an overdetermined system and expressed as F_{i}(x, y, z, p, q, r, s, t) = 0 (i = 1.2) where p, q, r, s, t are as before.

The following system

\{ \begin{matrix}r + 2ψ − 2λψ′ + λ^{2}ψ′′ = 0, \\ s + ψ′ − λψ′′ = 0, \\ t + ψ′′ = 0 \end{matrix}

expressed by a parameter λ and a function ψ = ψ(x, y, z, p, q, λ) defines a codimension two submanifold of the space (x, y, z, p, q, r, s, t) by the elimination of λ. Thus this system is an overdetermined system as above. The condition (GC) defined in the previous section is equal to the condition that the overdetermined system is involutive in the classical sense. We can show that by the Cartan test.

Let ℳ be the set of all involutive overdetermined system such that the degree on λ of the function ψ is greater than 2. Put 𝒢 = ContDiff (x, y, z, p, q) as previous. Then, for any μ ∈ ℳ, the isotropy group 𝒢_{μ} is a finite dimensional Lie group. The minimal orbit of this 𝒢 action on ℳ is the orbit of the system

μ_{0} = \{\frac{∂^{2}z}{∂x^{2}} = \frac{1}{3} (\frac{∂^{2}z}{∂y^{2}} )^{3} , \frac{∂^{2}z}{∂x∂y} = \frac{1}{2} (\frac{∂^{2}z}{∂y^{2}} )^{2}\} .

This system is given by ψ(x, y, z, p, q, λ) = \frac{1}{6}λ^{3}.

The isotropy group 𝒢_{μ_{0}} is the non-compact exceptional simple Lie group G_{2} as before.

The differential equation μ_{0} can be solved by quadratures. But the space of the solutions is parameterized by only one function of one variable.

The orbit space of this 𝒢 action on ℳ is parameterized by one function of one variable as before.

Thus the pairs of group and the space (𝒢, ℳ) in sections 3 and 4 are dual to each other, Geometrically, this duality is equal to the correspondence between space curves and their tangent surfaces, or conversely ruled surfaces and their lines of regression .

References

[Ca1] E. Cartan, Sur les variétés à connection projective , Bull. Soc. math. Fr., vol 52 (1924), 205–241.

[Ca2] E. Cartan, Les systèmes de Pfaff à cinq variables et les équations aux dérivées partielles du second ordre, Ann. Sci. Ecole Norm., vol 27 (1910), 109–192.

[FKN] S. Frittelli, N. Kamran, E. T. Newman Differnetial equations and conformal geometry, J. Geom. Phy. , vol 43 (2002), 133–145.

[IL] T. A. Ivey, J. M. Landsberg Cartan for Beginners, Graduate. Studies in Math. , vol 61 (2003), AMS.

[OS] T. Ozawa, H. Sato, Contact transformations and their Schwarzian derivatives, Advanced Studies in Pure math. , 37 (2002), 337–366.

[Sa] H. Sato, Contact geometry of second order partial differential equations (in Japanese) , Sugaku, 55 (2003), 155–165, English translation is to appear in Sugaku Expositions.

[SY] H. Sato, A. Y. Yoshikawa, Third order ordinary differential equations and Legendre manifolds, J. Math. Soc. Japan, 50 (1998), 993–1013.

[Tr] A. Tresse, Détermination des invaiants ponctuels de l’Éuation différentielle ordinaire du second ordre y′′ = ω(x, y, y′), Hirzel, Leipzig (1896).

[Ya] K. Yamaguchi, G_{2}-geometry of overdetermined systems of second order, Analysis and geometry in several complex variables (Katata, 1997) , 289– 314, Trends Math., Birkhäuser, Boston.