# Geometry of distributions and F-Gordon equation

Author Affiliations

1 Faculty of Mathematics, School of Mathematics, Iran University of Science and Technology, Narmak, Tehran, 16846-13114, Iran

2 Department of Basic Sciences, Eslamshahr Branch, Islamic Azad University, Eslamshahr, 3314853186, Iran

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Mathematical Sciences 2012, 6:49  doi:10.1186/2251-7456-6-49

 Received: 29 May 2012 Accepted: 14 July 2012 Published: 9 October 2012

This is an Open Access article distributed under the terms of the Creative Commons Attribution License ( http://creativecommons.org/licenses/by/2.0), which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited.

### Abstract

In this paper, we describe the geometry of distributions by their symmetries and present a simplified proof of the Frobenius theorem and some related corollaries. Then, we study the geometry of solutions of the F-Gordon equation, a PDE which appears in differential geometry and relativistic field theory.

##### Keywords:
Distribution; Lie symmetry; Contact geometry; Klein-Gordon equation

### Introduction

We begin this paper with the geometry of distributions. The main idea here is the various notions of symmetry and their use in solving a given differential equation. In the ‘Tangent and cotangent distribution’ section, we introduce the basic notions and definitions.

In the ‘Integral manifolds and maximal integral manifolds’ section, we describe the relation between differential equations and distributions. In the ‘Symmetries’ section, we present the geometry of distributions by their symmetries and find out the symmetries of the F-Gordon equation by this machinery. In the ‘A proof of the Frobenius theorem’ section, we introduce a simplified proof of the Frobenius theorem and some related corollaries. In the ‘Symmetries and solutions’ section, we describe the relations between symmetries and solutions of a distribution.

In all steps, we study the F-Gordon equation as an application and also a partial differential equation which appears in differential geometry and relativistic field theory. It is a generalized form of the Klein-Gordon equation utt uxx + u = 0 as well as a relativistic version of the Schrodinger equation, which is used to describe spinless particles. It was named after Walter Gordon and Oskar Klein [1,2].

### Tangent and cotangent distribution

Throughout this paper, M denotes an (m + n)-dimensional smooth manifold.

#### Definition 2.1

A map D:M TM is called an m-dimensional tangent distribution on M, or briefly Tanm-distribution, if

is an m-dimensional subspace of TxM. The smoothness of D means that for each x M, there exists an open neighborhood U of x and smooth vector fields X1,⋯,Xm such that

#### Definition 2.2

A map D : M TM is called an n-dimension cotangent distribution onM, or briefly Cotn-distribution, if

is an n-dimensional subspace of . The smoothness of D means that for each x M, there exists an open neighborhood U of x and smooth 1-forms ω1,⋯ωn such that

In the sequel, without loss of generality, we can assume that these definitions are globally satisfied.

There is a correspondence between these two types of distributions. For Tanm-distribution D, there exist nowhere zero smooth vector fields X1,⋯,Xm on M such that D = 〈X1,⋯,Xm〉, and similarly, for Cotn-distribution D, there exist global smooth 1-forms ω1,⋯,ωn on M such that D = 〈ω1,⋯,ωn〉.

#### Example 2.3

(Cartan distribution) Let M = Rk + 1. Denote the coordinates in M by x,p0,p1,..,pk, and given a function f(x,p0,⋯,pk−1), consider the following differential 1-forms

and the distribution D = 〈ω0,⋯,ωk−1〉. This is the 1-dimensional distribution, called the Cartan distribution. This distribution can also be described by a single vector field X, D = 〈X〉, where

#### Example 2.4

(F-Gordon equation) Let F : R5 R be a differentiable function. The corresponding F-Gordon PDE is uxy = F(x,y,u,ux,uy). We construct 7-dimensional sub-manifold M defined by s = F(x,y,u,p,q), of

Consider the 1-forms

This distribution can also be described by the following vector fields:

#### Definition 2.5

Let D : M TM be a Tanm-distribution and set

It is clear that dimAnnDx = n. A 1-form ω ∈ Ω1(M)annihilates D on a subset N M, if and only if ωx ∈ AnnDx for all x M.

The set of all differential 1-forms on M which annihilates D, is called annihilator of D and denoted by AnnD.

Therefore, for each Tanm-distribution,

we can construct a Cotn-distribution

and vice versa. In the other words, for each Tanm-distribution D = 〈X1,⋯,Xm〉, we can construct a Cotn-distribution D = AnnD =〈ω1,⋯,ωn〉, and vice versa.

#### Theorem 2.6

(a) D and its annihilator are modules over C(M).

(b) Let X be a smooth vector field on M and ω ∈ AnnD, then

#### Proof

(a) is clear, and for (b), if Y belongs to D, then ω(Y) = 0 and

### Integral manifolds and maximal integral manifolds

#### Definition 3.1

Let D be a distribution. A bijective immersed sub-manifold N M is called an integral manifold of D if one of the following equivalence conditions is satisfied:

(1) TxN Dx, for all x N.

(2) .

Moreover, N M is called maximal integral manifold if for each x N, there exists an open neighborhood U of x such that there is no integral manifold N containing N U.

It is clear that the dimension of maximal integral manifold does not exceed the dimension of the distribution.

#### Definition 3.2

D is called a completely integrable distribution, or briefly CID, if for all maximal integral manifold N, one of the following equivalence conditions is satisfied:

(1) dim N = dim D.

(2) TxN = Dx for all x N

(3) , and if Nbe an integral manifold with N N, then N N.

In the sequel, the set of all maximal integral manifolds is denoted by N.

#### Theorem 3.3

; that is ωi|N = 0 for i = 1,⋯,n.

#### Example 3.4

(Continuation of Example 2.3) If N is an integral curve of the distribution, then x can be chosen as a coordinate on N, and therefore,

Conditions ω0|N = 0,⋯,ωk−1|N = 0 imply that , , or that

for some function h : R R.

The last equation ωk−1|N = 0 gives us an ordinary differential equation h(k)(x) = f(x,h(x),h(x), ⋯,h(k−1)(x)).

The existence theorem shows us once more that the integral curves do exist, and therefore, the Cartan distribution is a CID.

#### Example 3.5

(Continuation of Example 2.4) This distribution in not a CID because there is no 4-dimensional integral manifold, and dim D = 4. For, if N be a 4-dimensinal integral manifold of the distribution, then (x,y,u,p) can be chosen as coordinates on N, and therefore,

Condition ω1|N = 0 implies that −pdxh(x,y,u,p)dy + du = 0, which is impossible.

By the same reason, we conclude that there is no 3-dimensional integral manifold.

Now, if N be a 2-dimensinal integral manifold of the distribution, then (x,y) can be chosen as coordinates on N, and therefore,

Conditions ω1|N = 0 and ω2|N = 0 imply that l = hx, m = hy, n = lx = hxx and o = my = hyy.

The last equation ω3|N = 0 implies that hxy = F(x,y,h,hx,hy). This distribution is not a CID.

### Symmetries

In this section, we consider a distribution D = 〈X1,⋯, Xm〉 = 〈ω1,⋯,ωn〉 on manifold Mn + m.

#### Definition 4.1

A diffeomorphism F : M M is called a symmetry of D if FDx = DF(x) for all x M.

Therefore,we have the following theorem.

#### Theorem 4.2

The following conditions are equivalent:

(1) F is a symmetry of D;

(2) Fωis determine the same distribution D; that is D = 〈Fω1,⋯,Fωn〉;

(3) Fωi∧⋯∧ωn = 0 for i = 1,⋯,n;

(4) , where aij C(M);

(5) (FXi|x) ∈ DF(x) for all x M and i = 1,⋯,n; and

(6) , where bij C(M).

#### Theorem 4.3

If F be a symmetry of D and N be an integral manifold, then F(N) is an integral manifold.

#### Proof

F is a diffeomorphism; therefore, F(N) is a sub-manifold of M. From other hand, if x N, then ωi|F(x) = (Fωi)|x = 0 for all i = 1,⋯,n; therefore, F(N) = {F(x) | x N} is an integral manifold. □

#### Theorem 4.4

Let N be the set of all maximal integral manifolds and F : M M be a symmetry, then F(N) = N.

#### Proof

If x N, then ωi|F(x) = (Fωi)|x = 0 for all i = 1,⋯,n; therefore, F(x) ∈ N and F(N) ⊂ N. □

Now, if y N, then there exists x M such that F(x) = y, since F is a diffeomorphism. Therefore, (Fωi)|x = ωi|F(x) = ωi|y = 0 for all i = 1,⋯,n; thus, x N and N F(N).

#### Definition 4.5

A vector field X on M is called an infinitesimal symmetry of distribution D, or briefly a symmetry of D, if the flow of X be a symmetry of D for all t.

#### Theorem 4.6

A vector field X X(M) is a symmetry if and only if

#### Proof

Let X be a symmetry. If Ω = ω1∧⋯∧ωn, then {(FlX)ωi}∧ Ω = 0, by condition (3) in Theorem 4.2. Moreover, by the definition , one gets

Therefore LXωi|D = 0.

In converse, let LXωi|D = 0 or for i = 1,⋯,n and bij C(M). Now, if , then

(1)

and

where and

(2)

Therefore, γ = (γ1⋯,γn) is a solution of the linear homogeneous system of ODEs (2) with the initial conditions (1), and γ must be identically zero.

#### Theorem 4.7

X is symmetry if and only if for all Y D, then [X,Y] ∈ D.

#### Proof

By the above theorem, X is a symmetry if and only if for all ω ∈ AnnD, then LX ω ∈Ann D.

The Theorem comes from the Theorem 2.6 (b): LX ω = −ωLX on D. In other words, (LXω)Y = −ω[X,Y] for all Y D. □

Denote by SymD the set of all symmetries of a distribution D.

#### Example 4.8

(Continuation of Example 3.4) Let k = 2. A vector field is an infinitesimal symmetry of D if and only if LYωi ≡ 0 modD, for i = 1,2. These give two equations:

#### Example 4.9

(Continuation of Example 3.5) We consider the point infinitesimal transformation:

Then, Z is an infinitesimal symmetry of D if and only if LZωi ≡ 0 modD, for i = 1,2,3. These give ten equations:

Complicated computations using Maple show that

and X = X(x,uqy), Y = Y(y,upx), and U(x,y,u) must satisfy in PDE:

### A proof of the Frobenius theorem

#### Theorem 5.1

Let X ∈ SymD D and N be maximal integral manifold. Then, X is tangent to N.

#### Proof

Let X(x) ∉ Tx N. Then, there exists an open set U of x and sufficiently small ϵ such that is a smooth sub-manifold of M.

Since X D, So is an integral manifold.

Since X ∈ SymD, so tangent to belongs to D, for all −ϵ < t < ϵ.

On the other hand, tangent spaces to are sums of tangent spaces to and the 1-dimensional subspace generated by X, but both of them belong to D, and their means are . □

#### Theorem 5.2

If X D ∩ SymD and N be a maximal integral manifold, then for all t.

#### Theorem 5.3 (Frobenious)

A distribution D is completely integrable, if and only if it is closed under Lie bracket. In other words, [X,Y] ∈ Dfor each X,Y D.

#### Proof

Let N be a maximal integral manifold with Tx N = Dx. Therefore, for all X,Y D, X and Y are tangent to N, and so [X,Y] is also tangent to N.

On the other hand, let for all X,Y D, their [X,Y] ∈ D. By the Theorem, all X D is a symmetry too, and so all X D is tangent to N, and this means Tx N = Dx, for all x N. □

#### Theorem 5.4

A distribution D is completely integrable if and only if D ⊂ SymD.

#### Theorem 5.5

Let D = 〈ω1,⋯,ωn〉 be a completely integrable distribution and X D. Then, the differential 1-forms vanish on D for all t.

#### Proof

If D is completely integrable, then X is a symmetry. Hence,

### Symmetries and solutions

#### Definition 6.1

If an (infinitesimal) symmetry X belongs to the distribution D, then it is called a characteristic symmetry. Denote by Char(D) := SD D the set of all characteristic symmetries [3,4].

It is shown that Char(D) is an ideal of the Lie algebra SD and is a module on C(M). Thus, we can define the quotient Lie algebra

#### Definition 6.2

Elements of Shuf(D) are called shuffling symmetries of D.

Any symmetry X ∈ SymD generates a flow on N (the set of all maximal integral manifolds of D), and, in fact, the characteristic symmetries generate trivial flows. In other words, classes X mod Char(D) mix or ‘shuffle’ the set of all maximal manifolds.

#### Example 6.3

(Continuation of Example 4.8) Let k = 2. In this case,

Therefore, Shuf(D) is spanned by , where

#### Example 6.4

(Continuation of Example 4.9) In this case, we have

in Shuf(D). Therefore, Shuf(D) is spanned by

where

and X = X(x,uqy), Y = Y(y,upx), and U(x,y,u) must satisfy in PDE:

(3)

#### Example 6.5

(Quasilinear Klein-Gordon Equation) In this example, we find the shuffling symmetries of the quasilinear Klein-Gordon equation

as an application of the previous example, where α, β, and γ are real constants. The equation can be transformed by defining and . Then, by the chain rule, we obtain α2uξη + γ2 u = βu3. This equation reduces to

(4)

by t = y, a = −(γ/α)2, and b = β/α2.

By solving the PDE (3), we conclude that Shuf(D) is spanned by the three following vector fields:

For example, we have

and if u = h(x,y) be a solution of (4), then is also a new solution of (4), for sufficiently small s R.

### Competing interests

The authors declare that they have no competing interests.

### Authors’ contributions

In this paper, RA and MN described the geometry of distributions by their symmetries. Also, a simplified proof of the Frobenius theorem as well as some related corollaries are presented. Moreover, MN and RA studied the geometry of solutions of the F-Gordon equation. Both authors read and approved the final manuscript.

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3. Alekseevskij, DV, Lychagin, VV, Vinogradov, AM: Basic Ideas and Concepts of Differential Geometry, Springer, New York

4. Kushner, A, Lychagin, VV, Rubtsov, V: Contact Geometry and Non-linear Differential Equations, Cambridge University Press, Cambridge