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 FGordon equation, a PDE which appears in differential geometry and relativistic field theory.
Keywords:
Distribution; Lie symmetry; Contact geometry; KleinGordon equationIntroduction
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 FGordon 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 FGordon 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 KleinGordon equation u_{tt }− u_{xx} + 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 mdimensional tangent distribution on M, or briefly Tan^{m}distribution, if
is an mdimensional subspace of T_{x}M. The smoothness of D means that for each x ∈ M, there exists an open neighborhood U of x and smooth vector fields X_{1},⋯,X_{m} such that
Definition 2.2
A map D : M → T^{∗}M is called an ndimension cotangent distribution onM, or briefly Cot^{n}distribution, if
is an ndimensional subspace of . The smoothness of D means that for each x ∈ M, there exists an open neighborhood U of x and smooth 1forms ω^{1},⋯ω^{n }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 Tan^{m}distribution D, there exist nowhere zero smooth vector fields X_{1},⋯,X_{m} on M such that D = 〈X_{1},⋯,X_{m}〉, and similarly, for Cot^{n}distribution D, there exist global smooth 1forms ω^{1},⋯,ω^{n }on M such that D = 〈ω^{1},⋯,ω^{n}〉.
Example 2.3
(Cartan distribution) Let M = R^{k + 1}. Denote the coordinates in M by x,p_{0},p_{1},..,p_{k}, and given a function f(x,p_{0},⋯,p_{k−1}), consider the following differential 1forms
and the distribution D = 〈ω^{0},⋯,ω^{k−1}〉. This is the 1dimensional distribution, called the Cartan distribution. This distribution can also be described by a single vector field X, D = 〈X〉, where
Example 2.4
(FGordon equation) Let F : R^{5 }→ R be a differentiable function. The corresponding FGordon PDE is u_{xy }= F(x,y,u,u_{x},u_{y}). We construct 7dimensional submanifold M defined by s = F(x,y,u,p,q), of
Consider the 1forms
This distribution can also be described by the following vector fields:
Definition 2.5
Let D : M → TM be a Tan^{m}distribution and set
It is clear that dimAnnD_{x }= n. A 1form ω ∈ Ω^{1}(M)annihilates D on a subset N ⊂ M, if and only if ω_{x }∈ AnnD_{x }for all x ∈ M.
The set of all differential 1forms on M which annihilates D, is called annihilator of D and denoted by AnnD.
Therefore, for each Tan^{m}distribution,
we can construct a Cot^{n}distribution
and vice versa. In the other words, for each Tan^{m}distribution D = 〈X_{1},⋯,X_{m}〉, we can construct a Cot^{n}distribution D = AnnD =〈ω^{1},⋯,ω^{n}〉, 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 submanifold N ⊂ M is called an integral manifold of D if one of the following equivalence conditions is satisfied:
(1) T_{x}N ⊆ D_{x}, for all x ∈ N.
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) T_{x}N = D_{x }for all x ∈ N
(3) , and if N^{′}be 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 ω^{i}_{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 ω^{0}_{N }= 0,⋯,ω^{k−1}_{N }= 0 imply that , , or that
for some function h : R → R.
The last equation ω^{k−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 4dimensional integral manifold, and dim D = 4. For, if N be a 4dimensinal integral manifold of the distribution, then (x,y,u,p) can be chosen as coordinates on N, and therefore,
Condition ω^{1}_{N }= 0 implies that −pdx−h(x,y,u,p)dy + du = 0, which is impossible.
By the same reason, we conclude that there is no 3dimensional integral manifold.
Now, if N be a 2dimensinal integral manifold of the distribution, then (x,y) can be chosen as coordinates on N, and therefore,
Conditions ω^{1}_{N }= 0 and ω^{2}_{N }= 0 imply that l = h_{x}, m = h_{y}, n = l_{x }= h_{xx} and o = m_{y }= h_{yy}.
The last equation ω^{3}_{N }= 0 implies that h_{xy }= F(x,y,h,h_{x},h_{y}). This distribution is not a CID.
Symmetries
In this section, we consider a distribution D = 〈X_{1},⋯, X_{m}〉 = 〈ω^{1},⋯,ω^{n}〉 on manifold M^{n + m}.
Definition 4.1
A diffeomorphism F : M → M is called a symmetry of D if F_{∗}D_{x }= D_{F(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) F^{∗}ω^{i}s determine the same distribution D; that is D = 〈F^{∗}ω^{1},⋯,F^{∗}ω^{n}〉;
(3) F^{∗}ω^{i}∧⋯∧ω^{n }= 0 for i = 1,⋯,n;
(4) , where a_{ij }∈ C^{∞}(M);
(5) (F_{∗}X_{i}_{x}) ∈ D_{F(x) }for all x ∈ M and i = 1,⋯,n; and
(6) , where b_{ij }∈ 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 submanifold of M. From other hand, if x ∈ N, then ω^{i}_{F(x) }= (F^{∗}ω^{i})_{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 ω^{i}_{F(x) }= (F^{∗}ω^{i})_{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, (F^{∗}ω^{i})_{x }= ω^{i}_{F(x) }= ω^{i}_{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 = ω^{1}∧⋯∧ω^{n}, then {(Fl^{X})^{∗}ω^{i}}∧ Ω = 0, by condition (3) in Theorem 4.2. Moreover, by the definition , one gets
□
Therefore L_{X}ω^{i}_{D }= 0.
In converse, let L_{X}ω^{i}_{D }= 0 or for i = 1,⋯,n and b_{ij }∈ C^{∞}(M). Now, if , then
and
Therefore, γ = (γ_{1}⋯,γ_{n}) 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 L_{X }ω ∈Ann D.
The Theorem comes from the Theorem 2.6 (b): L_{X }ω = −ω∘L_{X }on D. In other words, (L_{X}ω)Y = −ω[X,Y] for all Y ∈ D. □
Denote by Sym_{D} 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 L_{Y}ω^{i }≡ 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 L_{Z}ω^{i }≡ 0 modD, for i = 1,2,3. These give ten equations:
Complicated computations using Maple show that
and X = X(x,u−qy), Y = Y(y,u−px), and U(x,y,u) must satisfy in PDE:
A proof of the Frobenius theorem
Theorem 5.1
Let X ∈ Sym_{D }∩ D and N be maximal integral manifold. Then, X is tangent to N.
Proof
Let X(x) ∉ T_{x }N. Then, there exists an open set U of x and sufficiently small ϵ such that is a smooth submanifold of M.
Since X ∈ D, So is an integral manifold.
Since X ∈ Sym_{D}, 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 1dimensional subspace generated by X, but both of them belong to D, and their means are . □
Theorem 5.2
If X ∈ D ∩ Sym_{D }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 T_{x }N = D_{x}. 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 T_{x }N = D_{x}, for all x ∈ N. □
Theorem 5.4
A distribution D is completely integrable if and only if D ⊂ Sym_{D}.
Theorem 5.5
Let D = 〈ω^{1},⋯,ω^{n}〉 be a completely integrable distribution and X ∈ D. Then, the differential 1forms 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) := S_{D }∩ D the set of all characteristic symmetries [3,4].
It is shown that Char(D) is an ideal of the Lie algebra S_{D }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 ∈ Sym_{D} 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,u−qy), Y = Y(y,u−px), and U(x,y,u) must satisfy in PDE:
Example 6.5
(Quasilinear KleinGordon Equation) In this example, we find the shuffling symmetries of the quasilinear KleinGordon 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 α^{2}u_{ξη} + γ^{2 }u = βu^{3}. This equation reduces to
by t = y, a = −(γ/α)^{2}, and b = β/α^{2}.
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 FGordon equation. Both authors read and approved the final manuscript.
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