Differential equations and moving frames

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Differential equations and moving frames

Odinette Renée Abib

To cite this version:

Odinette Renée Abib. Differential equations and moving frames. Matemática Contemporânea, So-

ciedade Brasileira de Matemática, 2006, 30, pp.01-27. �hal-00096436�

ccsd-00096436, version 1 - 19 Sep 2006

Differential equations and moving frames

Odinete Ren´ee Abib

Laboratoire de Math´ ematiques Rapha¨ el Salem UMR 6085 CNRS

Universit´ e de Rouen Avenue de l’Universit´ e, BP.12 76801 Saint Etienne du Rouvray, FRANCE

[email protected]

September 19, 2006

Abstract

We shall study the foundations of the differential geometric consideration for differential equations. We show a local structure theorem. The main idea lies in the structure equations. The Lie algebra aspects of local differential equations is studied too.

1 Introduction

The purpose of the present paper is to study the relationship between differential equa- tions, Pfaffian systems and geometric structures, via the method of moving frames of E. Cartan [5, 9]

Following Cartan, we deal with every differential equation as a Pfaffian system on a suitable manifold (Section 3). This is the fundamental idea of Cartan. Further, we shall consider the structure equations which are satisfied by Pfaffian systems determined by differential equations. The intergration of a given differential equation is deeply related to the structure equation associated with the differential equation. We shall show it by means of some examples.

In Section 4, we shall establish a local structure theorem (Theorem 4.1). By virtue of this theorem, differential equations can be regarded as a differential geometric structure on a manifold. In Section 6, we shall consider the Lie algebraic aspect of local differ- ential equations; each differential Lie algebra (definition 6.1) determines locally a local differential equation (Theorem 6.1); if g is a semi-simple graded Lie algebra, then g has a structure of fundamental Lie algebra (Theorem 6.2). Moreover sl(2, R ) has a structure of differential Lie algebra which is not fundamental.

In Section 7 we study one system, which is one of the typical examples in Cartan’s

paper [6], related with G-structures and the local automorphism group of the given system.

The Section 2 conscerns remarks on Pfaffian systems, Cauchy characteristics and solvable systems.

In this paper, by the language differentiable we mean differentiable of class C ^∞ . I thank Marco Antonio Teixeira, Luiz San Martin, Paulo R´egis Ruffino for encour- agement and IMECC-UNICAMP, BRASIL for their hospitality during the preparation of this work.

2 Cauchy characteristic system

We begin with the preliminary remarks on Pfaffian systems. Let M be a differentiable manifold. F (M ) denotes the ring of real-valued differentiable functions on M and Λ ¹ (M ) the F (M )-module of all 1-forms (Pfaffian forms) on M. A F (M )-submodule Σ of Λ ¹ (M ) is said a Pfaffian system of rank n on M if Σ is generated by n linearly independent Pfaffian forms θ ¹ , . . . , θ ⁿ . A submanifold N of M is said an integral manifold of Σ if i ^∗ θ = 0 for all θ ∈ Σ, where i denotes the inclusion N ֒ → M . A differentiable function f on M is said a first integral of Σ if the exterior derivative df belongs to Σ. By the symbol Σ = hθ ¹ , . . . , θ ⁿ i we mean that the Pfaffian system Σ is generated by the linearly independent Pfaffian forms θ ¹ , . . . , θ ⁿ defined on M .

For each Pfaffian system Σ on M , we can construct the dual system, that is, the differentiable subbundle D(Σ) of the tangent bundle T (M ) on M such that the fiber dimension of D(Σ) is equal to dim M − n. Let D(Σ) be the sheaf of germs of local vector fields which belong to D(Σ) and D(Σ)

x (x ∈ M ) the stalk of D(Σ) at x. We set Ch(D(Σ))

x = n

A ∈ D(Σ)

x

h A , D(Σ)

x

i ⊂ D(Σ)

x

o

where [ , ] denotes the natural bracket operation. Further, for each x ∈ M , we define the subspaces Ch(D(Σ)) _x of T x (M ) by

Ch(D(Σ)) _x = n

X x ∈ D(Σ) x

X _x ∈ Ch(D(Σ)) _x o ,

where X denotes a vector field and X _x the germ at x determined by X. We suppose that dim Ch(D(Σ)) _x is constant on M. Thus, we obtain the subbundle Ch(D(Σ)) of T (M ).

Ch(D(Σ)) is called the Cauchy charateristic of D(Σ). The dual system of Ch(D(Σ)) is called the Cauchy characteristic system of Σ. The following theorem is due to Cartan [3, 4].

Theorem 2.1 Let Σ = hθ ¹ , . . . , θ ⁿ i be a Pfaffian system.

1. If Σ is completely integrable, i.e. dθ ⁱ = 0 (mod. θ ¹ , . . . , θ ⁿ ) i = 1, 2, . . . , n, then Ch(Σ) = Σ.

2. If Σ is not completely integrable, then there exist linearly independent Pfaffian forms ω ¹ , . . . , ω ^m satisfying the following conditions:

(i) θ ¹ , . . . , θ ⁿ , ω ¹ , . . . , ω ^m are also linearly independent;

(ii) (θ ¹ , . . . , θ ⁿ , ω ¹ , . . . , ω ^m ) forms a (local) generator of Ch(Σ);

(iii) dθ ⁱ = P m

j,k=1 C _jk ⁱ ω ^j ∧ ω ^k (mod. θ ¹ , . . . , θ ⁿ ), where C _jk ⁱ denotes a differentiable function (i = 1, 2, . . . , n; j, k = 1, 2, . . . , m).

3. Ch(Σ) is completely integrable.

4. Let x ¹ , . . . , x ^n+m be independent first integrals of Ch(Σ). Then there exist linearly independent Pfaffian forms θ ⁱ = P n+m

j=1 A ⁱ _j (x ¹ , . . . , x ^n+m ) dx ^j , i = 1, 2, . . . , n, such that (θ ¹ , . . . , θ ⁿ ) forms a (local) generator of Σ.

By making use of property 2.(ii), we can construct the Cauchy characteristic system Ch(Σ).

Example 2.1 Consider the Pfaffian system Σ = hθi, θ = dz + pdx + p ² dy, on R ⁴ = {(x, y, z, p)}. We have dtheta = dp ∧ (dx + 2p dy) and

ω ¹ = dp , ω ² = dx + 2p dy , ω ³ = p ;

determine the Cauchy characteristic system of Σ. We can find by quadrature three inde- pendent first integrals as follows:

u 1 = z + xp + yp ² , u 2 = x + 2yp , u 3 = p ; and θ itself is expressed as θ = du 1 − u 2 du 3 .

Definition 2.1 A system (ω ¹ , . . . , ω ^m ) of linearly independent Pfaffian forms on M will be said a solvable system of Σ = hθ ¹ , . . . , θ ⁿ i if it satisfies the following conditions:

(i) (ω ¹ , . . . , ω ^m ) forms a generator of Ch(Σ);

(ii) dω ¹ = 0 and dω ^p ≡ 0 (mod. ω ¹ , . . . , ω ^{p − 1} ) for all p = 2, 3, . . . , m.

If we can find a solvable system of Σ, then m independent first integrals of Ch(Σ) are given by quadrature. In the above example, the system (ω ¹ , ω ² , ω ³ ) is a solvable system of Σ = hθi.

3 Differential equations and structure equations

In this section we shall consider, by means of simple examples, the relation between the

differential equations and Pfaffian systems.

a) Take the first order equation on R ² = {(x, y)}

∂z

∂x + 1 2

∂z

∂y 2

= 0 . (3.1)

Setting on R ⁴ = {(x, y, z, q)}, ω ¹ = dx, ω ² = dy − q dx, ω ³ = dz + ¹ ₂ q ² dy − q dy, ω ⁴ = dq,

we have 

 



 



dω ¹ = 0 , dω ² = ω ¹ ∧ ω ⁴ , dω ³ = ω ² ∧ ω ⁴ , dω ⁴ = 0 .

(3.2)

Each integral of (3.1) defines a 2-dimensional integral manifold of hω ³ i on which ω ¹ and ω ² are linearly independent. The equation (3.1) is left invariant by the automorphism group of the absolute parallelism ω ¹ , . . . , ω ⁴ on R ⁴ . The structure of this group is determined by the equation (3.2). The integration of the equation (3.1) depends deeply on the structure equation (3.2) of this group. In this case (ω ¹ , ω ² , ω ³ ) forms a solvable system of hω ³ i.

Therefore three independent first integrals of Ch(hω ³ i) are given by quadrature as u 1 = q , u 2 = z + 1

2 xq ² − yq , u 3 = y − xq ; and we have ω ³ = du ₂ + u ₃ du ₁ . The formula

z + ¹ ₂ xq ² − yq = f(q) , y − xq + f ^′ (q) = 0

gives an integral surface of the equation (3.1), where f is a differentiable function and f ^′ denotes its derivative.

Conversely, we consider an absolute parallelism ω ¹ , ω ² , ω ³ , ω ⁴ on R ⁴ satisfying the

equations

dω ¹ ≡ 0 , dω ² ≡ 0 (mod. ω ¹ , ω ² ) ,

dω ³ ≡ ω ² ∧ ω ⁴ (mod. ω ³ ) . (3.3)

Let x and y be two independent first integals of the completely integrable Pfaffian system ω ¹ = ω ² = 0. If we reduce ω ³ to the submanifold defined by the equations x =const., y =const., then from the equation dω ³ ≡ ω ² ∧ ω ⁴ (mod. ω ³ ) we have dω ³ ≡ 0 (mod. ω ³ ) on this submanifold. Therefore ω ³ must be of the form

ω ³ = a(dz − p dx − q dy) ,

where a is a non-zero function. Since ω ¹ , ω ² , ω ³ are linearly independent, the functions x, y and z are also independent.

By this procedure we can determine the functions p and q of the variables x, y, z and another t:

p = p(x, y, z, t) , q = q(x, y, z, t) ; (3.4) and the same equation dω ³ ≡ ω ² ∧ ω ⁴ (mod. ω ³ ) implies

rank ∂p

∂t , ∂q

∂t

= 1 .

On a 2-dimensional integral manifold of ω ³ = 0 on which x and y are still independent, p and q can be considered as the first partial derivatives of z = z(x, y). Therefore the equation (3.4) can be regarded as a first-order differential equation. For example, the differential equation

∂z

∂x + 1 2

∂z

∂y − f (x, y ) 2

= g(x, y)

belongs to the family determined by the structure equation (3.2), where f(x, y) and g(x, y) are differentiable functions satisfying the equation

∂f

∂x = ∂g

∂y .

b) Next, we consider an absolute parallelism ω ¹ , ω ² , ω ³ , ω ⁴ , ω ⁵ , ω ⁶ on R ⁶ satisfying

 

 

 



dω ¹ ≡ 0 , dω ² ≡ 0 (mod. ω ¹ , ω ² ) , dω ³ ≡ ω ¹ ∧ ω ⁴ + ω ² ∧ ω ⁵ (mod. ω ³ ) , dω ⁴ = 0 (mod. ω ³ , ω ⁴ , ω ⁵ ) ,

dω ⁵ = ω ² ∧ ω ⁶ (mod. ω ³ , ω ⁴ , ω ⁵ ) .

(3.5)

Let x and y be two independent first integrals of the completely Pfaffian integrable system ω ¹ = ω ² = 0; ω ³ is expressed as

ω ³ = a(dz − p dx − qd y) (a 6= 0) .

The functions x, y, z, p and q are independent first integrals of the completely integrable Pfaffian system ω ¹ = ω ² = ω ³ = ω ⁴ = ω ⁵ = 0. Therefore ω ⁴ and ω ⁵ can be written by means of the exterior derivatives dx, dy, dz, dp, dq and the formulas

dp − r dx − s dy = a ₁ ω ⁴ + a ₂ ω ⁵ + a ₃ ω ³ dq − s ^′ dx − t dy = a ₄ ω ⁴ + a ₅ ω ⁵ + a ₆ ω ³

determine the functions r, s, s ^′ , t and a i ’s of the variables x, y, z, p, q and another u.

From the equation dω ³ ≡ ω ¹ ∧ ω ⁴ + ω ² ∧ ω ⁵ (mod. ω ³ ), one can verify that the function s coincides with s ^′ . Moreover, the equations dω ⁴ = 0 , dω ⁵ = ω ² ∧ ω ⁶ (mod. ω ³ , ω ⁴ , ω ⁵ ) imply

rank ∂r

∂u , ∂s

∂u , ∂t

∂u

= 1 . Therefore the functions

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

determine a system of second-order partial differential equations. This family of systems of differential equations determined by an absolute parallelism satisfying (3.5) is the main subject of Cartn’s researches in his paper [6].

For example, take the system of differential equations (c.f. [6, §§ 13, 14])

∂ ² z

∂x ² = 0 , ∂ ² z

∂x ∂y = z − x ∂z

∂x . (3.6)

Putting on R ⁶ = {(x, y, z, p, q, t)} ω ¹ = dx, ω ² = dy, ω ³ = dz − p dx − q dy, ω ⁴ = dp − (z − xp) dy, ω ⁵ = dq − (z − xp) dx − t dy and ω ⁶ = dt − (q − x(z − xp)) dx we have the structure equations

 

 

 

 

 



dω ¹ ≡ 0 , dω ² ≡ 0 , dω ³ ≡ ω ¹ ∧ ω ⁴ + ω ² ∧ ω ⁵ , dω ⁴ ≡ ω ² ∧ ω ³ − x ω ² ∧ ω ⁴ ,

dω ⁵ = ω ² ∧ ω ⁶ + ω ¹ ∧ ω ³ − x ω ¹ ∧ ω ⁴ ,

dω ⁶ = ω ¹ ∧ ω ⁵ − x ω ¹ ∧ ω ³ − x ² ω ¹ ∧ ω ⁴ + K ω ¹ ∧ ω ² ,

where K = t − xq + x ² (z − xp). The absolute parallelism satisfies the equations (3.5). It is easy to see that the system (ω ² , ω ³ , ω ⁴ , ω ⁵ , ω ⁶ ) forms a solvable system of Σ = hω ³ , ω ⁴ , ω ⁵ i.

Five independent first integrals of the solvable system are given by quadrature as follows:

u 1 = y , u 2 = z − xp , u 3 = p , u 4 = q − x(z − xp) , u 5 = K , and we have (c.f. [6, §10, IV])

 



ω ³ − x ω ⁴ = du ₂ − u ₄ du ₁ ω ⁴ = du ₃ − u ₂ du ₁ , ω ⁵ − x ω ³ = du 4 − u 5 du 1 .

By this expression, the general integral surface of (3.6) is given by the formulas:

p = f(y) , z − xp = f ^′ (y) , q − x(z − xp) = f ^′′ (y) , t − x(q − x(z − xp)) = f ^′′′ (y) where f is a differentiable function and f ^′ , f ^′′ and f ^′′′ denote its derivatives.

4 Differential geometric structures

In the previous section we have seen that the integration of differential equation is deeply related to the structure equations of differential equations. In this section we shall consider the differential geometric structures for differential equations.

Let V ⁻ 1 and V 0 be finite dimensional real vector spaces. We define by induction the real vector spaces V k , k = 1, 2, . . . as follows. Let V ₁ = Hom(V ⁻ 1 , V ₀ ); V k − 1 (k ≥ 2) being determined, we set

V k = n

X ∈ Hom(V ⁻ 1 , V k − 1 )

X(u)(v) = X(v)(u) , u, v, ∈ V ⁻ 1

o .

We have V _k ∼ = V ₀ ⊗ S ^k (V − ^∗ 1 ) as a vector space (k = 0, 1, 2, . . .), where S ^k (V − ^∗ 1 ) denotes the symmetric tensor space of the dual space V − ^∗ 1 . For an integer k ≥ 1 we set

W k (V ⁻ 1 , V 0 ) = V ⁻ 1 ⊕ V 0 ⊕ · · · ⊕ V k (direct sum),

and we define the bracket operation [ , ] on W k (V ⁻ 1 , V ₀ ) as follows:

(i) For all X ⁻ 1 ∈ V ⁻ 1 , X p ∈ V p (p ≥ 1),

[X p , X ⁻ ₁ ] = −[X ⁻ 1 , X p ] = X p (X ⁻ 1 ) ; (ii) [X, Y ] = 0 for any other combination

By this bracket operation, W k (V ⁻ 1 , V 0 ) becomes a nilpotent Lie algebra. It is easy to prove the following.

Proposition 4.1

(i) For a non-zero element X ⁻ ₁ ∈ V ⁻ ₁ , [X ⁻ 1 , V p ] = V p − 1 (p ≥ 1).

(ii) If [X ⁻ 1 , V p ] = (0) for X p ∈ V p (p ≥ 1), then X p = 0.

(iii) For an arbitrary subspace V _k ⁰ of V k ,

W _k ⁰ (V ⁻ ₁ , V ₀ ) = V ⁻ ₁ ⊕ V ₀ ⊕ · · · ⊕ V _k ⁰ (direct sum) is a Lie subalgebra of W k (V ⁻ 1 , V 0 ).

Example 4.1

(1) dim V ⁻ 1 = 1, dim V 0 = 1. We have dim V k = 1 for any k ≥ 1. There exists a basis X ⁻ 1 , X 0 , X 1 , . . . , X k of W k (V ⁻ 1 , V 0 ) such that X p ∈ V p (−1 ≤ p ≤ k) and [X ⁻ 1 , X p ] = −X p − 1 (1 ≤ p ≤ k).

(2) dim V ⁻ 1 = 2, dim V 0 = 1. We have dim V k = k + 1 for any k ≥ 1.

(i) k = 1. There exists a basis X ₁ , X ₂ , X ₃ , X ₄ , X ₅ of W _k (V ⁻ 1 , V ₀ ) such that X ₁ , X ₂ ∈ V ⁻ ₁ ; X ₃ ∈ V ₀ ; X ₄ , X ₅ ∈ V ₁ and

[X 1 , X 4 ] = −X 3 , [X 2 , X 5 ] = −X 3

and otherwise [X, Y ] = 0.

(ii) k = 2. There exists a basis X ₁ , X ₂ , X ₃ , X ₄ , X ₅ , X ₆ , X ₇ , X ₈ of W ₂ (V ⁻ 1 , V ₀ ) such that X ₁ , X ₂ ∈ V ⁻ ₁ ; X ₃ ∈ V ₀ ; X ₄ , X ₅ ∈ V ₁ ; X ₆ , X ₇ , X ₈ ∈ V ₂ and

[X 1 , X 4 ] = −X 3 , [X 2 , X 5 ] = −X 3 , [X 1 , X 6 ] = −X 4

[X 1 , X 7 ] = −X 5 , [X 2 , X 7 ] = −X 4 , [X 2 , X 8 ] = −X 5

and otherwise [X, Y ] = 0.

Let π : M → N be a fibered manifold on a differentiable manifold N and J ^k (M, π)

the space of k-jets of local sections of π. If dim N = dim V − 1 and dim M = dim(V ⁻ 1 ⊕ V ₀ ),

W k (V ⁻ 1 , V 0 ) is regarded as the local structure of J ^k (M, π), i.e. J ^k (M, π) ∼ = W k (V ⁻ 1 , V 0 )

(locally diffeomorphic).

Let u ^p : W k (V ⁻ 1 , V 0 ) → V p (−1 ≤ p ≤ k) be the natural projection. We regard u ^p as a vector-space valued function on W k (V ⁻ 1 , V ₀ ), so that the system (u ^{− 1} , u ⁰ , . . . , u ^k ) can be considered as a linear coordinate system on W k (V ⁻ ₁ , V ₀ ). We set θ ^{− 1} = du ^{− 1} , θ ^p = du ^p − [u ^{p − 1} , du ^{− 1} ] (0 ≤ p ≤ k − 1), θ ^k = du ^k and θ = θ ^{− 1} + θ ⁰ + · · · + θ ^k . θ is a W k (V ⁻ 1 , V ₀ )-valued 1-form on W k (V ⁻ 1 , V ₀ ). We have

dθ ^{− 1} = 0

dθ ^p + [θ ^{− 1} ∧ θ ^p+1 ] = 0 (0 ≤ p ≤ k − 1).

For example, making use of the notations in Example 4.1, 2.(ii), we set u ^{− 1} = xX 1 + yX 2 , u ⁰ = zX ₃ , u ¹ = pX ₄ + qX ₅ , u ₂ = rX ₆ + sX ₇ + tX ₈ and θ = P 8

i=1 ω ⁱ X i . Then we have ω ¹ = dx, ω ² = dy, ω ³ = dz − p dx − q dy, ω ⁴ = dp − r dx − s dy, ω ⁵ = dq − s dx − t dy ω ⁶ = dr, ω ⁷ = ds, ω ⁸ = dt and

 

 

 



dω ¹ = 0 , dω ² = 0 , dω ³ = ω ¹ ∧ ω ⁴ + ω ² ∧ ω ⁵ , dω ⁴ = ω ¹ ∧ ω ⁶ + ω ² ∧ ω ⁷ , dω ⁵ = ω ¹ ∧ ω ⁷ + ω ² ∧ ω ⁸ .

Let ρ ^k : W k (V ⁻ ₁ , V ₀ ) → W k − 1 (V ⁻ ₁ , V ₀ ) (k ≥ 1) be the natural projection, where we put W 0 (V ⁻ 1 , V 0 ) = V ⁻ 1 ⊕ V 0 ; W k (V ⁻ 1 , V 0 ) can be considered as a fibered manifold on W _k − 1 (V ⁻ 1 , V ₀ ) with the fibering ρ ^k .

Definition 4.1 We shall say that a submanifold R k of W k (V ⁻ 1 , V ₀ ) is a local differen- tial equation of order k if R k admits an absolute parallelism and if there exist an open submanifold U of W k − 1 (V ⁻ 1 , V 0 ) such that ρ ^k (R k ) = U and ρ ^k | _R

: R k → U is a fibered submanifold of ρ ^k : (ρ ^k ) ^{− 1} (U ) → U .

Let i : R k ֒ → W k (V ⁻ ₁ , V ₀ ) be the inclusion and ω = i ^∗ θ the induced W k (V ⁻ ₁ , V ₀ )-valued 1-form on R k . According to the direct sum decomposition of W k (V ⁻ 1 , V 0 ), we decompose ω as ω = ω ^{− 1} +ω ⁰ +· · · ω e ^k , where ω ^p (resp. ω e ^k ) is a V _p -valued 1-form (resp. V _k -valued 1-form) on R k (−1 ≤ p ≤ k − 1). Let n be the fiber dimension of ρ : R k → U . Then there exist n linearly independent Pfaffian forms ω ₁ ^k , . . . , ω _n ^k which are also linearly independent of the Pfaffian forms obtained from ω ^{− 1} , ω ⁰ , . . . , ω ^{k − 1} . We fix a n-dimensional vector subspace V _k ⁰ of V k and its basis X ₁ , . . . , X n and we set ω ^k = P n

j=1 ω ^k _j X j . ω ^k is a V _k ⁰ -valued 1-form on R k . We define the differentiable mappings

F _p ^k : R k −→ Hom(V p , V k ) (−1 ≤ p ≤ k − 1) , F _k ^k : R k −→ Hom(V _k ⁰ , V k )

by the formula

d(u ^k ◦ i) = F − ^k 1 (ω ^{− 1} ) + · · · + F _k ^k (ω ^k ) ; and we define the differentiable mapping

T : R k −→ Hom(V ⁻ 1 × V _k ⁰ , V k − 1 )

by the formula

T x (X ⁻ 1 , X k ) = [X ⁻ 1 , F _k ^k (x)(X k )] (x ∈ R k , X ⁻ 1 ∈ V ⁻ 1 , X k ∈ V _k ⁰ ) .

Since the rank of the inclusion i is maximal on R k , the linear mapping F _k ^k (x) : V _k ⁰ → V k

(x ∈ R k ) is injective. Therefore T has the following property:

(C 1 ) For each x ∈ R k , T x (V ⁻ 1 , X k ) = 0 (X k ∈ V _k ⁰ ) implies X k = 0.

It is also easy to prove the following properties:

(C 2 ) (i) dω ^{− 1} ≡ 0 (mod. ω ^{− 1} );

(ii) (k ≥ 2) For p = 0, 1, . . . , k − 2,

dω ^p + [ω ^{− 1} ∧ ω ^{p − 1} ] ≡ 0 (mod. ω ⁰ , . . . , ω ^p ) ; (iii) dω ^{k − 1} + T (ω ^{− 1} ∧ ω ^k ) ≡ 0 (mod. ω ^{− 1} ∧ ω ^{− 1} , ω ⁰ , . . . , ω ^{k − 1} ).

We have thus proved that for each local differential equation of order k there exist a differentiable mapping T : R k → Hom(V ⁻ 1 × V _k ⁰ : V k ) and an absolute parallelism ω = ω ^{− 1} + ω ⁰ + · · · ω ^k satisfying the above conditions (C 1 ), (C 2 ).

Theorem 4.1 Let V _k ⁰ be a subspace of V k and put W _k ⁰ = V ⁻ 1 ⊕ V 0 ⊕ · · · ⊕ V _k ⁰ (direct sum). Let R k be a differentiable manifold with dim R k = dim W _k ⁰ . If there exists a W _k ⁰ -valued absolute parallelism ω = ω ^{− 1} + ω ⁰ + · · · ω ^k and a differentiable mapping T : R k → Hom(V ⁻ 1 × V _k ⁰ : V k ) satisfying the conditions (C 1 ) and (C 2 ), then R k can be locally embedded into W k (V ⁻ 1 , V ₀ ) as a local differential equation or order k.

Proof. Since ω gives rise to an isomorphism ω x : T x (R k ) → ^∼ W _k ⁰ (x ∈ R k ) one can consider the inverse mapping of ω x , say τ x : W _k ⁰ → ^∼ T x (R k ). τ has the property: ω ^p ((τ (X q )) = δ _q ^p X q , X _q ∈ V _q (−1 ≤ p, q ≤ k). By the condition (i) of C 2 , we can find differentiable mappings v ^{− 1} : R k → V ⁻ ₁ and A ⁻ ₁ : R k → GL(V ⁻ ₁ ) such that ω ^{− 1} = A ⁻ ₁ (dv ^{− 1} ). Since the system ω ^{− 1} = ω ⁰ = 0 is completely integrable, there exists a differentiable mapping v ⁰ : R k → V 0

such that the system dv ^{− 1} = dv ⁰ = 0 is equivalent to the system ω ^{− 1} = ω ⁰ = 0. Therefore, ω ⁰ can be written as

ω ⁰ = A 0 (dv ⁰ − v ¹ dv ^{− 1} ) ,

where A ₀ denotes a differentiable mapping A ₀ : R _k → GL(V 0 ) and v ¹ denotes a differen- tiable mapping v ¹ : R k → Hom(V ⁻ 1 , V ₀ ) = V ₁ . If k = 1 the argument comes to an end.

Let k ≥ 2. Consider the following proposition (P j ) for 1 ≤ j ≤ k:

(P j ) There exist differentiable mappings v ^{− p} : R k → V p and A p : R k → GL(V p ), p =

−1, 0, 1, . . . j, such that

(1) dv ^{− 1} , dv ⁰ , . . . , dv ^{j − 1} are linearly independent;

(2) ω ^{− 1} = A ⁻ 1 (dv ^{− 1} ), ω ⁰ = A 0 (dv ⁰ − v ¹ dv ^{− 1} ) and for p = 1, 2, . . . , j − 1,

ω ^p = A p (dv ^p − v ^{p − 1} dv ^{− 1} ) (mod. ω ⁰ , . . . , ω ^{p − 1} ).

We have proved (P 1 ). For an integer 1 ≤ j ≤ k − 1, assume that (P j ) is established. From the inequality 0 ≤ j − 1 ≤ k − 2 and condition (ii) of (C 2 ) we have

dω ^{j − 1} ≡ −[ω ^{− 1} ∧ ω ^j ] (mod. ω ⁰ , . . . , ω ^{j − 1} ) . From (2) of (P j ) we have

dω ^{j − 1} ≡ −A j − 1 (dvj ∧ dv ^{− 1} ) (mod. ω ⁰ , . . . , ω ^{j − 1} ) . These two equations yield

− A j − 1 (dvj ∧ dv ^{− 1} ) ≡ −[ω ^{− 1} ∧ ω ^j ] (mod. ω ⁰ , . . . , ω ^{j − 1} ) . (4.1) Substituting τ (X p ) ∧ τ (X ⁻ 1 ), X ⁻ ₁ ∈ V ⁻ ₁ , X p ∈ V p (j + 1 ≤ p ≤ k) to this equation, we obtain

dv ^j (τ(X p ))

A ⁻ − ¹ 1 (X ⁻ 1 )

= 0 and hence

dv ^j (τ(X p )) = 0 (X p ∈ V p : p = j + 1, . . . , k) . (4.2) Substituting τ(X j ) ∧ τ (X ⁻ 1 ), X ⁻ 1 ∈ V ⁻ 1 , X j ∈ V j to equation (4.1), we have

A j − 1 dv ^j (τ(X j ))

A ⁻ − ¹ 1 (X ⁻ 1 )

= −[X ⁻ 1 , X j ] and hence

dv ^j (τ (X j )) = A ⁻ _j − ¹ 1 ◦ X _j ◦ A − 1 . (4.3) The equation (4.2) implies that dv ^j is expressed as

dv ^j = B _j (ω ^j ) (mod. ω ^{− 1} , ω ⁰ , . . . , ω ^{j − 1} ) (4.4) and the equation (4.3) implies that the differentiable mapping B j : R k → Hom(V j , V j ) is given by the formula

B j (X j ) = A ⁻ _j − ¹ 1 ◦ X j ◦ A ⁻ ₁ (X j ∈ V j ) ,

so that B _j (x) is non-singular for any x ∈ R _k . Therefore one can see that dv ^{− 1} , dv ⁰ , . . . , dv ^j are linearly independent. By equation (4.4), ω ^j can be written as

ω ^j = A j (dv ^j − v ^j+1 dv ^{− 1} ) mod. ω ⁰ , . . . , ω ^{j − 1} ) .

where A j = B _j ^{− 1} and v ^j+1 denotes a differentiable mapping v ^j+1 : R k → Hom(V ⁻ 1 , V j ).

Substituting τ(X ⁻ 1 ), X ⁻ 1 ∈ V ⁻ 1 , to this equation, we have dv ^j (τ (X ⁻ 1 )) = v ^j+1 (A ⁻ − ¹ 1 (X ⁻ 1 )) .

Substituting τ(X ⁻ 1 ) ∧ τ(Y ⁻ 1 ), X ⁻ 1 , Y ⁻ 1 ∈ V ⁻ 1 , to equation (4.1), we obtain

dv ^j (τ (X ⁻ 1 ))(A ⁻ − ¹ 1 (Y ⁻ 1 )) = dv ^j (τ (Y ⁻ 1 ))(A ⁻ − ¹ 1 (X ⁻ 1 )) .

These two equations imply

v ^j+1 (X ⁻ ₁ )(Y ⁻ ₁ ) = v ^j+1 (Y ⁻ ₁ )(X ⁻ ₁ )

for any X − 1 , Y − 1 ∈ V − 1 , so that v ^j+1 (x) lies in V _j+1 for any x ∈ R _k . Thus we can establish by induction the proposition (P _j+1 ) and hence (P k ). Define the differentiable mapping F : R k → W k (V ⁻ 1 , V 0 ) by the formula

u ^p ◦ F = v ^p (−1 ≤ p ≤ k) and put

dv ^k ≡ v _k ^k (ω ^k ) (mod. ω ^{− 1} , ω ⁰ , . . . , ω ^{k − 1} ) .

If v _k ^k (x) ∈ Hom(V _k ⁰ , V k ) is injective for any x ∈ R k , F is an immersion and determines locally an embedding. From the proposition (P k ) we have

dω ^{k − 1} ≡ −A k − 1 (dv ^k ∧ A ⁻ − ¹ 1 (ω ^{− 1} )) (mod. ω ⁰ , . . . , ω ^{k − 1} )

≡ −A k − 1 (v _k ^k (ω ^k ) ∧ A ⁻ − ¹ 1 (ω ^{− 1} )) (mod. ω ^{− 1} ∧ ω ^{− 1} , ω ⁰ , . . . , ω ^{k − 1} ) . By the condition (iii) of (C 2 ) we obtain

A k − 1 (v ^k _k (ω ^k ) ∧ A ⁻ − ¹ 1 (ω ^{− 1} )) ≡ T (ω ^{− 1} ∧ ω ^k ) (mod. ω ^{− 1} ∧ ω ^{− 1} , ω ⁰ , . . . , ω ^{k − 1} ) . Substituting τ(X ⁻ 1 ) ∧ τ(X k ), X − 1 ∈ V − 1 , X k ∈ V _k ⁰ to this equation, we have

A k − 1 (v ^k _k (X k ) ∧ A ⁻ − ¹ 1 (X ⁻ 1 )) = T (X ⁻ 1 , X k ) .

If v ^k _k (X k ) = 0, then T (X ⁻ 1 , X k ) = 0 for any X ⁻ ₁ ∈ V ⁻ ₁ . From condition (C 1 ) we obtain X k = 0. Hence v _k ^k (x) ∈ Hom(V _k ⁰ , V k ) is injective for any x ∈ R k . Set ρ e = ρ ^k ◦ F . By the definition of F we have

u ^p ◦ ρ e = u ^p ◦ ρ ^k ◦ F = u ^p ◦ F = v ^p (−1 ≤ p ≤ k − 1) .

This relation and (1) of the proposition (P k ) imply that ρ e is a submersion. Therefore F determines locally a local differential equation of order k.

Remark 4.1 By virtue of this theorem, a system (R k , W _k ⁰ , T, ω) satisfying the conditions stated in the theorem may be also called a local differential equation of order k.

5 Equivalence

Let V _k ⁰ be a subspace of V k (k ≥ 1). We set W _k ⁰ = V ⁻ ₁ ⊕ V ₀ ⊕ · · · ⊕ V _k ⁰ and D ^p = V p ⊕V p − 1 ⊕· · ·⊕V _k ⁰ , p = 0, 1, . . . , k. We define the Lie subgroup G(W _k ⁰ ) of GL(W k (V ⁻ 1 , V 0 )) as follows:

G(W _k ⁰ ) = n

g ∈ GL(W k (V ⁻ 1 , V 0 ))

g(V ⁻ 1 ⊕ V _k ⁰ ) = V ⁻ 1 ⊕ V _k ⁰ , g(D ^p ) = D ^p (0 ≤ p ≤ k) o

.

Definition 5.1 We shall say that two local differential equations (R k , W _k ⁰ , T, ω ) and (R k , W _k ⁰ , T ^′ , ω ^′ ) are structurally equivalent if there exists a differentiable mapping A : R k → G(W _k ⁰ ) such

that ω ^′ = A(ω).

Definition 5.2 A local differential equations (R k , W _k ⁰ , T, ω ) will be said of type W _k ⁰ if T x (X ⁻ 1 , X k ) = [X ⁻ 1 , X k ] 0 for all x ∈ R k , X ⁻ ₁ ∈ V ⁻ ₁ , X k ∈ V _k ⁰ and if ω satisfies the condition:

(C ₂ ^′ ) (i) dω ^{− 1} ≡ 0 (mod. ω ^{− 1} );

(ii) For p = 0, 1, . . . , k − 1,

dω ^p + [ω ^{− 1} ∧ ω ^{p − 1} ] ≡ 0 (mod. ω ⁰ , . . . , ω ^p ) , where [·, ·] ₀ denotes the natural bracket operation W _k ⁰ (cf. Proposition 4.1).

Remark 5.1 Almost all local differential equations which admit a lot of solutions turn out to be structurally equivalent to a local differential equation of type W _k ⁰ for some V _k ⁰ . Example 5.1 Take the system of second order differential equations

∂ ² z

∂x ² = 0 , ∂ ² z

∂x ∂y = z . (5.1)

Putting on R ⁶ = {(x, y, z, p, q, t)} ω ¹ = dx, ω ² = dy, ω ³ = dz − p dx −q dy , ω ⁴ = dp −z dy, ω ⁵ = dq − z dx − t dy and ω ⁶ = dt, we have

 

 

 

 

 



dω ¹ = 0 , dω ² ≡ 0 , dω ³ = ω ¹ ∧ ω ⁴ + ω ² ∧ ω ⁵ , dω ⁴ = ω ² ∧ ω ³ + p ω ¹ ∧ ω ² ,

dω ⁵ = ω ² ∧ ω ⁶ − ω ¹ ∧ ω ³ − q ω ¹ ∧ ω ² , dω ⁶ = 0 .

Since one can not remove the terms pω ¹ ∧ ω ² and qω ¹ ∧ ω ² , ω is not of type W ₂ ⁰ for any V ₂ ⁰ ⊂ V ₂ . On the other hand, the given system (5.1) has no solutions except z = 0.

Proposition 5.1 Let P be a differentiable manifold with dim P ≥ dim W _k ⁰ . Suppose there exists a W _k ⁰ -valued 1-form ω = ω ^{− 1} +ω ⁰ +· · · +ω ^k such that ω p : T p (P ) → W _k ⁰ is surjective for any p ∈ P and dω ≡ 0 (mod. ω). If ω satisfies the condition (C ₂ ^′ ) in Definition 5.2, then (P, W _k ⁰ , ω) determines locally a local differential equation of type W _k ⁰ .

Proof. Since ω = 0 is completely integrable, there exists, for each p ∈ P , an open

neighborhood U of p, a differentiable manifold R k with dim R k = dim W _k ⁰ and a fibering

π : U → R k such that each fiber is a maximal integral manifold of ω| U = 0. Let

σ : R k → U be a differentiable cross section of π and put ω = σ ^∗ ω. Then it is clear that

(R k , W _k ⁰ , ω) is a local differential equation of type W _k ⁰ . In general, the obtained system

depends on the choice of cross sections.

For a subspace V _k ⁰ of V k we set (V _k ⁰ ) ⁽¹⁾ = n

X ∈ Hom(V ⁻ 1 , V _k ⁰ )

X(u)(v) = X(v)(u) , u, v ∈ V ⁻ 1

o . For a subspace U of V ⁻ ₁ , we set

V _k ⁰ (U ) = n

X k ∈ V _k ⁰

X k (u) = 0 , u ∈ U o .

Definition 5.3 A subspace V _k ⁰ of V k is said involutive if there exists a series of subspaces (0) ⊂ U 0 ⊂ U 1 ⊂ · · · ⊂ U n − 1 ⊂ U n = V ⁻ 1 with dim U i = i such that

dim(V _k ⁰ ) ⁽¹⁾ = X n

i=0

dim V _k ⁰ (U i ) .

A local differential equation (R k , W _k ⁰ , ω) of type W _k ⁰ is said involutive if V _k ⁰ is involutive.

Example 5.2 We use the notation of Example 4.1. By the symbol U = (X 1 , X 2 , . . . , X n ) we mean that the vector space U is spanned by the basis X ₁ , X ₂ , . . . , X _n .

(1) dim V ⁻ ₁ = 1, dim V ₀ = 1.

(i) k = 1. Then W ₁ ⁰ = V ⁻ ₁ ⊕ V ₀ = (X ₁ , X ₂ ) with [X ₁ , X ₂ ] = 0. A W ₁ ⁰ -valued 1-form ω = ω ¹ X 1 + ω ² X 2 is of type W ₁ ⁰ if it satisfies

dω ¹ ≡ 0 (mod. ω ¹ ) , dω ² ≡ 0 (mod. ω ² ) .

(ii) k = 2. Then W ₂ ⁰ = V ⁻ ₁ ⊕ V ₀ ⊕ V ₁ = (X 1 , X ₂ , X ₃ ) with [X 1 , X ₃ ] = −X 2 and otherwise [X i , X j ] = 0. ω = ω ¹ X 1 + ω ² X 2 + ω ³ X 3 is of type W ₂ ⁰ if it satisfies

 

 

 



dω ¹ ≡ 0 (mod. ω ¹ ) , dω ² = ω ¹ ∧ ω ³ (mod. ω ² ) dω ³ = 0 (mod. ω ² , ω ³ )

.

The Pfaffian system Σ = hω ² , ω ³ i determines a family of second-ordeer ordinary differential equations.

(2) dim V − 1 = 2, dim V ₀ = 1.

(i) k = 1. Let V ₁ ⁰ be a 1-dimensional subspace of V ₁ .

The we can choose a basis X ₁ , X ₂ , X ₃ , X ₄ of W ₁ ⁰ such that X ₁ , X ₂ ∈ V − 1 and [X 2 , X ₄ ] = −X 3 and otherwise [X i , X j ] = 0. ω = P 4

i=1 ω ⁱ X i is of type W ₁ ⁰ if it

satisfies

dω ¹ ≡ 0 , dω ² ≡ 0 (mod. ω ¹ , ω ² ) , dω ³ ≡ ω ² ∧ ω ⁴ (mod. ω ³ ) .

We have already seen this structure equation in Section 3.

(ii) k = 2. Let V ₂ ⁰ be a 2-dimensional subspace of V 2 . Then the Lie algebra W ₂ ⁰ is isomorphic to the following three Lie algebras.

(a) W ₂ ⁰ = (X 1 , X ₂ , X ₃ , X ₄ , X ₅ , X ₆ , X ₇ ) with

[X ₁ , X ₄ ] = −X ₃ , [X ₂ , X ₅ ] = −X ₃ , [X ₁ , X ₆ ] = −X ₄ [X ₁ , X ₇ ] = −X ₅ , [X ₂ , X ₇ ] = −X ₄ , [X ₂ , X ₆ ] = X ₅ and otherwise [X i , X j ] = 0; ω = P 7

i=1 ω ⁱ X i is of type W ₂ ⁰ if it satisfies

 

 

 



dω ¹ ≡ 0 , dω ² ≡ 0 (mod. ω ¹ , ω ² ) , dω ³ ≡ ω ¹ ∧ ω ⁴ + ω ² ∧ ω ⁵ (mod. ω ³ ) ,

dω ⁴ ≡ ω ¹ ∧ ω ⁶ + ω ² ∧ ω ⁷ (mod. ω ³ , ω ⁴ , ω ⁵ ) , dω ⁵ ≡ ω ¹ ∧ ω ⁷ − ω ² ∧ ω ⁶ (mod. ω ³ , ω ⁴ , ω ⁵ ) . (b) W ₂ ⁰ = (X 1 , X ₂ , X ₃ , X ₄ , X ₅ , X ₆ , X ₇ ) with

[X 1 , X 4 ] = −X 3 , [X 2 , X 5 ] = −X 3

[X 1 , X 6 ] = −X 4 , [X 2 , X 7 ] = −X 5

and otherwise [X i , X _j ] = 0; ω = P 7

i=1 ω ⁱ X _i is of type W ₂ ⁰ if it satisfies

 

 

 



dω ¹ ≡ 0 , dω ² ≡ 0 (mod. ω ¹ , ω ² ) , dω ³ ≡ ω ¹ ∧ ω ⁴ + ω ² ∧ ω ⁵ (mod. ω ³ ) , dω ⁴ ≡ ω ¹ ∧ ω ⁶ (mod. ω ³ , ω ⁴ , ω ⁵ ) , dω ⁵ ≡ ω ² ∧ ω ⁷ (mod. ω ³ , ω ⁴ , ω ⁵ ) .

Let V ₂ ⁰ be a 1-dimensional involutive subspace of V ₂ . Then there is only one case up to isomorphic algebra.

(c) W ₂ ⁰ = (X 1 , X 2 , X 3 , X 4 , X 5 , X 6 ) with

[X 1 , X ₄ ] = −X 3 , [X 2 , X ₅ ] = −X 3 , [X 2 , X ₆ ] = −X 5

and otherwise [X i , X j ] = 0; ω = P 6

i=1 ω ⁱ X i is of type W ₂ ⁰ if it satisfies

 

 

 



dω ¹ ≡ 0 , dω ² ≡ 0 (mod. ω ¹ , ω ² ) , dω ³ ≡ ω ¹ ∧ ω ⁴ + ω ² ∧ ω ⁵ (mod. ω ³ ) , dω ⁴ ≡ 0 (mod. ω ³ , ω ⁴ , ω ⁵ ) ,

dω ⁵ ≡ ω ² ∧ ω ⁶ (mod. ω ³ , ω ⁴ , ω ⁵ ) . We have already discussed this case in Section 3.

6 Lie algebraic aspects of differential equations

In this section we shall consider the Lie-algebraic aspect of local differential equations.

Let V _k ⁰ be a subspace of V k . We set

W _k ⁰ (V _k ⁰ ) = V − 1 ⊕ V ₀ ⊕ · · · ⊕ V _k ⁰ , D ^p = V p ⊕ V p − 1 ⊕ · · · ⊕ V _k ⁰ (0 ≤ p ≤ k) .

We define the Lie algebra J (W _k ⁰ (V _k ⁰ )) as follows:

J k (W _k ⁰ (V _k ⁰ )) = n

X ∈ gl(W _k ⁰ (V _k ⁰ ))

X (V ⁻ 1 ⊕V _k ⁰ ) ⊂ V ⁻ 1 ⊕V _k ⁰ , X(D ^p ) ⊂ D ^p (0 ≤ p ≤ k) o . Every element of J k (W _k ⁰ (V _k ⁰ )) is of the form:

V ⁻ ₁ V ₀ V ₁ · · · V k − 1 V _k ⁰



 

 

 

 

∗ 0 0 · · · 0 0 0 ∗ 0 · · · 0 0 0 ∗ ∗ · · · 0 0 ... ... ... ... ...

0 ∗ ∗ · · · ∗ 0

∗ ∗ · · · ∗ ∗



 

 

 

  V ⁻ 1

V ₀ V ₁ ...

V k − 1

V _k ⁰

where ∗ denotes a certain matrix. We define a mapping ∂ : Hom(W _k ⁰ , J k (W _k ⁰ )) → Hom(W _k ⁰ , ∧W _k ⁰ : W _k ⁰ ) by the formula

(∂S)(X ∧ Y ) = S(X)(Y ) − S(Y )(X) ,

for all S ∈ Hom(W _k ⁰ , J k (W _k ⁰ )) and all X, Y ∈ W _k ⁰ , where we put W _k ⁰ = W _k ⁰ (V _k ⁰ ).

Definition 6.1 Let J be a Lie algebra. A system (J , M, J 0 ) will be called a differential Lie algebra if it satisfies the following conditions:

(1) J 0 is a Lie subalgebra of J and M is a vector subspace of J such that J = M ⊕ J 0

(direct sum);

(2) For some subspaces V − 1 , V ₀ and V _k ⁰ of M, we have M = W _k ⁰ (V _k ⁰ );

(3) The linear isotropy representation ρ : J 0 → gl(M) (ρ(X 0 )(X) = the m-component of [X 0 , X ], X 0 ∈ J 0 , X ∈ M) has its values in J k (W _k ⁰ (V _k ⁰ ));

(4) Let α : M ∧ M → M be the linear mapping defined by the formula α(X ∧ Y ) = the m-component of [X, Y ] , X, Y ∈ M . There exists an element S ∈ Hom(M, J k (W _k ⁰ )) such that

α(X ∧ Y ) = [X, Y ] 0 + (∂S)(X ∧ Y ) ,

where [ , ] 0 denotes the natural bracket operation on W _k ⁰ (V _k ⁰ ). If J ₀ = (0), J (= M) will be called fundamental.

Theorem 6.1 Let (J , M, J 0 ) be a differential Lie algebra. Let G be a Lie group corre-

sponding to the Lie algebra J and let θ be the Maurer-Cartan form of G. Then the M

component ω of θ with respect to the decomposition J = M ⊕ J 0 determines locally a

local differential equation.

Proof. Let M = W _k ⁰ (V _k ⁰ ) for the subspaces V ⁻ 1 , V 0 and V _k ⁰ of M. We set θ = ω + ω 0 , ω ₀ being the J 0 -component of θ. Then from the structure equation of Maurer-Cartan we obtain

dω + 1

2 α(ω ∧ ω) − ρ(ω ₀ ) ∧ ω = 0 and hence

dω + 1

2 [ω ∧ ω] 0 + 1

2 (∂S)(ω ∧ ω) − ρ(ω 0 ) ∧ ω = 0 . Since S(X) and ρ(X ₀ ) (X ∈ M, X ₀ ∈ J ₀ ) lie in J k (W _k ⁰ ), we have

dω ^{− 1} ≡ 0 (mod. ω ^{− 1} and for p = 0, 1, . . . , k − 1

dω ^p + [ω ^{− 1} ∧ ω ^{p − 1} ] 0 ≡ 0 (mod. ω ⁰ , ω ¹ , . . . , ω ^p ) . Therefore the theorem follows from Proposition 5.1.

Example 6.1 We define the 6-dimensional Lie algebra J = (X 1 , X ₂ , X ₃ , X ₄ , X ₅ , X ₆ ) by the following bracket operations:

[X 1 , X 4 ] = −X 2 , [X 1 , X 5 ] = −X 1 , [X 2 , X 4 ] = −X 3 , [X 2 , X 5 ] = −X 2

[X 2 , X 6 ] = −X 2 , [X 3 , X 5 ] = −X 3 , [X 3 , X 6 ] = −2X 3 , [X 4 , X 6 ] = −X 4

and otherwise [X i , X j ] = 0. Set M = (X 1 , X 2 , X 3 , X 4 ) and J 0 = (X 5 , X 6 ). Then (J , M, J ₀ ) is a differential Lie algebra of order 1. Let G be a connected Lie group corresponding to the Lie algebra J and let ω be the Maurer-Cartan form of G. Setting ω = P 6

i=1 ω ⁱ X _i we have

 

 

 

 

 

 

 



dω ¹ = ω ¹ ∧ ω ⁵ ,

dω ² = ω ¹ ∧ ω ⁴ + ω ² ∧ (ω ⁵ + ω ⁶ ) , dω ³ = ω ² ∧ ω ⁴ + ω ³ ∧ (ω ⁵ + 2ω ⁶ ) , dω ⁴ = ω ⁴ ∧ ω ⁶ ,

dω ⁵ = 0 , dω ⁶ = 0 .

The differential equation (3.1) considered in Section 3 belongs to this homogeneous case.

The differential equation is left invariant by the Lie group, which can be considered as a subgroup of the contact transformation group.

Theorem 6.2 If J = J ⁻ 1 + J 0 + J 1 (direct sum) is a semi-simple graded Lie algebra,

i.e. [J i , J j ] ⊂ J i+j (i, j = 0, ±1, ±2, . . .), where we put J p = (0) for p ≤ −2 and p ≥ 2,

then J has a structure of fundamental differential Lie algebra.

Proof. Let B be the Killinng-Cartan form of J . The linear endomorphism s of J defined by

s(X ⁻ 1 + X 0 + X 1 ) = −X ⁻ 1 + X 0 − X 1 (X ⁻ 1 ∈ J ⁻ 1 , X 0 ∈ J 0 , X 1 ∈ J 1 ) is an involutive automorphism of J . Hence

B(X ₁ , X ₀ ) = B(s(X ₁ ), s(X ₀ )) = B(−X ₁ , X ₀ ) = −B(X ₁ , X ₀ ) (X ₀ ∈ J ₀ , X ₁ ∈ J ₁ ) . Therefore we have

B(J 1 , J 0 ) = 0 . (6.1)

Let X 1 ∈ J 1 be an element satisfying [X 1 , J ⁻ 1 ] = (0). For Y ⁻ 1 ∈ J ⁻ 1 and Z i ∈ J i

(i = −1, 0, 1), we have ad(X 1 ) ◦ ad(Y ⁻ 1 )(Z ⁻ 1 ) = 0,

ad(X 1 ) ◦ ad(Y ⁻ 1 )(Z 0 ) = [X 1 , [Y ⁻ 1 , Z 0 ]] = 0 and

ad(X 1 ) ◦ ad(Y ⁻ 1 )(Z 1 ) = −[Y 1 , [Z 1 , X ₁ ]] − [Z 1 , [X 1 , Y − 1 ]] = 0 . Hence

B(X 1 , J ⁻ 1 ) = 0 . (6.2)

For Y 1 ∈ J 1 and Z i ∈ J i (i = −1, 0, 1), we have

ad(X 1 ) ◦ ad(Y 1 )(Z ⁻ 1 ) = −[Y 1 , [Z ⁻ 1 , X ₁ ]] − [Z ⁻ 1 , [X 1 , Y ₁ ]] = 0 , ad(X 1 ) ◦ ad(Y 1 )(Z 0 ) = 0 and ad(X 1 ) ◦ ad(Y 1 )(Z 1 ) = 0. Hence

B(X 1 , J 0 ) = 0 . (6.3)

These three relations (6.1), (6.2) and (6.3) yield B(X ₁ , J ) = 0. Since J is semi-simple, this implies X 1 = 0. Therefore J 1 can be considered a subspace of Hom(J ⁻ 1 , J 0 ) by the mapping h : J 1 → Hom(J ⁻ 1 , J 0 ) defined by h(X 1 )(X ⁻ 1 ) = [X 1 , X − 1 ], X ₁ ∈ J 1 , X ⁻ ₁ ∈ J ⁻ ₁ . Next, we define the element s ∈ Hom(J , J ₁ (W ₁ ⁰ (J ₁ ))) by the formulas:

S(X ⁻ 1 ) = 0 , S(X 1 ) = 0 , S(X 0 )(Y ⁻ 1 ) = [X 0 , Y − 1 ] , S(X 0 )(Y 0 ) = 1

2 [X 0 , Y ₀ ] , S(X 0 )(Y 1 ) = [X 0 , Y ₁ ] , X ⁻ 1 , Y ⁻ 1 ∈ J ⁻ 1 , X 0 , Y 0 ∈ J 0 , X 1 , Y 1 ∈ J 1 . Then we have

[X, Y ] = [X, Y ] 0 + (∂S)(X ∧ Y ) for all X, Y ∈ J .

The simple real Lie algebras having the structure stated in Theorem 6.2 are classified

in S. Kobayashi and T. Nagano [11]. Among these simple Lie algebras sl(2 : R ) is the

simplest example. Moreover sl(2 : R ) has the structure of a differential Lie algebra which is not fundamental. Set

M =

0 a

b 0

a, b ∈ R

, J o =

u 0

0 −u

u ∈ R

.

(sl(2 : R ), M, J ₀ ) is a differential Lie algebra of order 1. Let ω be the Maurer-Cartan form of SL(2 : R ) and set

ω = ω ¹

0 1 0 0

+ ω ²

0 0 1 0

− ω ³

0 1 0 −1

. We have the structure equation

 



dω ¹ = −2ω ¹ ∧ ω ³ , dω ² = 2ω ² ∧ ω ³ , dω ³ = ω ¹ ∧ ω ² . For example, we can choose ω ¹ , ω ² , ω ³ as follows:

 



ω ¹ = e ^2z dx ,

ω ² = e ^{− 2z} (dy − (y ² + a(x)y + b(x))dx) , ω ³ = dz − (y + ¹ ₂ a(x))dx ,

where a(x) and b(x) denote two arbitrary differentiable functions of the variable x. There- fore we can see that SL(2 : R ) corresponds to the family of ordinary differential equations of Riccati type.

7 Cartan example

Now we can consider the involutive system of second-order differential equations which is one of the typical examples in Cartn’s paper [6]:

∂ ² z

∂x ² = 1 3

∂ ² z

∂y ² 3

, ∂ ² z

∂x ∂y = 1 2

∂ ² z

∂y ² 2

. (7.1)

Setting on R ⁶ = {(x, y, z, p, q, t)} ω ¹ = dx, ω ² = dy + t dx, ω ³ = dz − p dx − q dy, ω ⁴ = dp − t dq + ¹ ₆ t ³ dx + ¹ ₂ t ² dy, ω ⁵ = dq − ¹ ₂ t ² dx − t dy and ω ⁶ = dt, we have the structure equations of the system: 

 

 

 



 

 

 



dω ¹ = 0 ,

dω ² = −ω ¹ ∧ ω ⁶ ,

dω ³ = ω ¹ ∧ ω ⁴ + ω ² ∧ ω ⁵ , dω ⁴ = ω ⁵ ∧ ω ⁶ ,

dω ⁵ = ω ² ∧ ω ⁶ , dω ⁶ = 0 .

(7.2)

which is of type W ₂ ⁰ inn Example 5.2 (2)(ii)(c). Th integration of the system is deeply

related to the structure equation (7.2). It is easy to see that (ω ⁶ , ω ² , ω ⁵ , ω ⁴ , ω ³ ) forms

a solvable system of the Pfaffian system Σ = hω ³ , ω ⁴ , ω ⁵ i. Therefore we can obtain by quadrature five independent first integrals of the Cauchy characteristic system of Σ:

u ₁ = z − xp + xqt + 1

6 x ² t ³ , u ₂ = p − qt + 1

6 xt ³ + 1 2 yt ² u ₃ = q − 1

2 yt ² , u ₄ = y + xt , u ₅ = t , so that the system Σ is expressed as

 



ω ³ − xω ⁴ = du 1 − (u 3 + u 4 u 5 ) du 4 , ω ⁴ = du 2 + u 3 du 5 ,

ω ⁵ = du ₃ + u ₄ du ₅ .

By this expression, we can construct the general integral surfaces of the given system (see [6, §38]).

Let G be the Lie subgroup of GL(6 : R ) consisting of matrices of the form



 

 

 



∗ ∗ ∗ ∗ ∗ 0

∗ ∗ ∗ ∗ ∗ 0 0 0 ∗ 0 0 0 0 0 ∗ ∗ ∗ 0 0 0 ∗ ∗ ∗ 0

∗ ∗ ∗ ∗ ∗ ∗



 

 

 

 ,

where ∗ is an element of R . Let B G be the G-structure defined by the dual frame of (ω ¹ , ω ² , . . . , ω ⁶ ). Then a contact transformation leaving the equations (7.2) invariant induces an automorphism of this G-structure and vice versa. The structure group G can be reduced to the Lie subgroup G 7 whose Lie algebra J 7 is given as follows:

J 7 =

 

 

 

 

 

 

 





 

 

 



a ₁ − a ₂ −a 3 a ₄ 0 0 0

0 a ₁ a ₅ 0 ⁴ ₃ a ₃ 0

0 0 2a 1 + a 2 0 0 0

0 0 a ₆ a ₁ + 2a 2 0 0

0 0 a ₇ a ₃ a ₁ + a ₂ 0

0 −a 6 0 −a 5 4

3 a 7 a 2



 

 

 



, a i ∈ R , i = 1, 2, . . . , 7

 

 

 

 

 

 

 

 .

The usual prolongations of J 7 satisfy dim J ₇ ⁽¹⁾ = 1 and J ₇ ^{(P )} = {0} (p ≥ 2). Therefore the local automorphism group of the given system with respect to the group G is of finite type.

The structure equation (7.2) determines the Lie algebra M = (X 1 , X ₂ , X ₃ , X ₄ , X ₅ , X ₆ ) with the bracket operation

[X 1 , X ₄ ] = −X 3 , [X 1 , X ₆ ] = −X 2 , [X 2 , X ₅ ] = −X 3

[X 2 , X 6 ] = −X 5 , [X 5 , X 6 ] = −X 4

and otherwise [X i , X j ] = 0. This Lie algebra has a structure of fundamental graded Lie

algebra in the sense of N. Tanaka [16]. We set J ⁻ 1 = (X 1 , X ₆ ), J ⁻ 2 = (X 2 ), J ⁻ 3 = (X 5 ),

J ⁻ 4 = (X 4 ), J ⁻ 5 = (X 3 ). Then M = J ⁻ 5 + J ⁻ 4 + J ⁻ 3 + J ⁻ 2 + J ⁻ 1 (direct sum) is a fundamental graded algebra of the 5th kind. The structure of the automorphism group with respect to the group G is given by Tanaka’s prolongation method [16]. We can verify that the graded Lie algebra J prolonged from M with respect to the Lie algebra of G has the following structure:

(1) J = P 5

p= − 5 J p (direct sum);

(2) J 0 is a Cartan subalgebra of J ;

(3) dim J 0 = 2, dim J ^± 1 = 2 and dim J ^± p = p for p = 2, 3, 4, 5;

(4) J is isomorphic to the exceptional simple real Lie algebra of dimension 14.

In order to transform the involutive systems which admit this 14-dimensional simple Lie group of contact transformations to the standard form (7.1), we need the integration of a system of differential equations associated with the simple group.

References

[1] R. Bryant, S.S. Chern, R. Gardner, H. Goldshmidt, and P. Griffiths. Exterior differ- ential systems. Springer-Verlag, N.Y., 1991.

[2] E. Cartan. Sur la r´eduction `a sa forme canonique de la structure d’un groupe de transformations fini et continu. Amer. J. Math., 18:1–61, 1896.

[3] E. Cartan. Sur l’int´egration des syst`emes d’´equations aux diff´erentielles totales. Ann.

Ec. Norm., 18:241–3111, 1901.

[4] E. Cartan. Sur l’int´egration des syst`emes diff´erentiels compl`etement int´egrables. C.

R. Acad. Sc., 134:1415–1418, 1902.

[5] E. Cartan. Les sous-groupes des groupes continus de transformations. Ann. Ec.

Norm., 25:57–194, 1908.

[6] E. Cartan. Les syst`emes de Pfaff `a cinq variables et les ´equations aux d´eriv´ees partielles du second ordre. Ann. Ec. Norm., 27:109–192, 1910.

[7] E. Cartan. Les syst`emes diff´erentiels ext´erieurs et leurs applications g´eom´etriques.

Hermann, Paris, 1945.

[8] F. Estrabrook and Wahlquist H. Prolongation structures of nonlinear evolution equa- tions. J. Math. Phys., 16:1–7, 1975.

[9] P. Griffiths. On Cartan method of Lie group and moving frame as applied to uniquenes and existence questions in differential geometry. Duke Math. J., 41:775–

814, 1974.

[10] Th. Ivey and J.M. Landesberg. Cartan for beginners: differential geometry via moving frames and exterior differential systems. A.M.S.,Graduate Studies,61, 2004.

[11] S. Kobayashi and S. Nagano. On filtred Lie algebras and geometric structures I. J.

Math. Mech., 13:875–907, 1964.

[12] R. Montgomery. A tour of subriemannian geometries, their geodesics and applica- tions., volume 91. A.M.S.,Math.Survey and Monographs, 2002.

[13] P. Olver. Symmetry, invariants and equivalence. Springer-Verlag, N.Y., 1995.

[14] S. Sternberg. Lectures on differential geometry. Prentice-Hall., 1964.

[15] O. Stomark. Lie’s structural approach to PDE systems. Cambridge University Press., 2000.

[16] N. Tanaka. On differential systems, graded Lie algebra and pseudogroups. J. Math.

Kyoto Univ., 10:1–82, 1970.

[17] I. Zelenko. Fundamental form and Cartan’s tensor of (2,5)-distributions. Preprint,

SISSA-Trieste., 2005.