Geometry of distributions and F-Gordon equation

  • Mehdi Nadjafikhah1 and

    Affiliated with

    • Reza Aghayan2Email author

      Affiliated with

      Mathematical Sciences20126:49

      DOI: 10.1186/2251-7456-6-49

      Received: 29 May 2012

      Accepted: 14 July 2012

      Published: 9 October 2012

      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 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:MTM is called an m-dimensional tangent distribution on M, or briefly Tan m -distribution, if
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equa_HTML.gif
      is an m-dimensional subspace of T x M. The smoothness of D means that for each xM, there exists an open neighborhood U of x and smooth vector fields X 1,⋯,X m such that
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equb_HTML.gif

      Definition 2.2

      A map D : MT M is called an n-dimension cotangent distribution on M, or briefly Cot n -distribution, if
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equc_HTML.gif
      is an n-dimensional subspace of http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq1_HTML.gif . The smoothness of D means that for each xM, there exists an open neighborhood U of x and smooth 1-forms ω 1,⋯ω n such that
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equd_HTML.gif

      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 1-forms 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 1-forms
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Eque_HTML.gif
      and the distribution D = 〈ω 0,⋯,ω k−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
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equf_HTML.gif

      Example 2.4

      (F -Gordon equation) Let F : R 5R be a differentiable function. The corresponding F-Gordon PDE is u xy = F(x,y,u,u x ,u y ). We construct 7-dimensional sub-manifold M defined by s = F(x,y,u,p,q), of
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equg_HTML.gif
      Consider the 1-forms
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equh_HTML.gif
      This distribution can also be described by the following vector fields:
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equi_HTML.gif

      Definition 2.5

      Let D : MTM be a Tan m -distribution and set
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equj_HTML.gif

      It is clear that dimAnn D x = n. A 1-form ω ∈ Ω1(M)annihilates D on a subset NM, if and only if ω x ∈ Ann D x for all xM.

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

      Therefore, for each Tan m -distribution,
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equk_HTML.gif
      we can construct a Cot n -distribution
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equl_HTML.gif

      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 = Ann D =〈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 ∈ Ann D, then
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equm_HTML.gif

      Proof

      (a) is clear, and for (b), if Y belongs to D, then ω(Y) = 0 and
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equn_HTML.gif

      Integral manifolds and maximal integral manifolds

      Definition 3.1

      Let D be a distribution. A bijective immersed sub-manifold NM is called an integral manifold of D if one of the following equivalence conditions is satisfied:
      1. (1)

        T x ND x , for all xN.

         
      2. (2)

        http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq2_HTML.gif .

         

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

      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. (1)

        dim N = dim D.

         
      2. (2)

        T x N = D x for all xN

         
      3. (3)

        http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq3_HTML.gif , and if N be an integral manifold with NN , then N N.

         

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

      Theorem 3.3

      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq4_HTML.gif ; 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,
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equo_HTML.gif
      Conditions ω 0| N = 0,⋯,ω k−1| N = 0 imply that http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq5_HTML.gif , http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq6_HTML.gif , or that
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equp_HTML.gif

      for some function h : RR.

      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 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,
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equq_HTML.gif

      Condition ω 1| N = 0 implies that −p dxh(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,
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equr_HTML.gif

      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 : MM is called a symmetry of D if F D x = D F(x)for all xM.

      Therefore,we have the following theorem.

      Theorem 4.2

      The following conditions are equivalent:
      1. (1)

        F is a symmetry of D;

         
      2. (2)

        F ω i s determine the same distribution D; that is D = 〈F ω 1,⋯,F ω n 〉;

         
      3. (3)

        F ω i ∧⋯∧ω n = 0 for i = 1,⋯,n;

         
      4. (4)

        http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq7_HTML.gif , where a ij C (M);

         
      5. (5)

        (F X i | x ) ∈ D F(x)for all xM and i = 1,⋯,n; and

         
      6. (6)

        http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq8_HTML.gif , 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 sub-manifold of M. From other hand, if xN, then ω i | F(x)= (F ω i )| x = 0 for all i = 1,⋯,n; therefore, F(N) = {F(x) | xN} is an integral manifold. □

      Theorem 4.4

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

      Proof

      If xN, 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 yN, then there exists xM 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, xN and NF(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 http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq9_HTML.gif of X be a symmetry of D for all t.

      Theorem 4.6

      A vector field XX(M) is a symmetry if and only if
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equs_HTML.gif

      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 http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq10_HTML.gif , one gets
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equt_HTML.gif

      Therefore L X ω i | D = 0.

      In converse, let L X ω i | D = 0 or http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq11_HTML.gif for i = 1,⋯,n and b ij C (M). Now, if http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq12_HTML.gif , then
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equ1_HTML.gif
      (1)
      and
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equu_HTML.gif
      where http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq13_HTML.gif and
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equ2_HTML.gif
      (2)

      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 YD, then [X,Y] ∈ D.

      Proof

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

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

      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 http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq14_HTML.gif is an infinitesimal symmetry of D if and only if L Y ω i ≡ 0 mod D, for i = 1,2. These give two equations:
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equv_HTML.gif

      Example 4.9

      (Continuation of Example 3.5) We consider the point infinitesimal transformation:
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equw_HTML.gif
      Then, Z is an infinitesimal symmetry of D if and only if L Z ω i ≡ 0 mod D, for i = 1,2,3. These give ten equations:
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equx_HTML.gif
      Complicated computations using Maple show that
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equy_HTML.gif
      and X = X(x,uqy), Y = Y(y,upx), and U(x,y,u) must satisfy in PDE:
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equz_HTML.gif

      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 http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq15_HTML.gif is a smooth sub-manifold of M.

      Since XD, So http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq16_HTML.gif is an integral manifold.

      Since X ∈ Sym D , so tangent to http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq17_HTML.gif belongs to D, for all −ϵ < t < ϵ.

      On the other hand, tangent spaces to http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq18_HTML.gif are sums of tangent spaces to http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq19_HTML.gif and the 1-dimensional subspace generated by X, but both of them belong to D, and their means are http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq20_HTML.gif . □

      Theorem 5.2

      If XD ∩ Sym D and N be a maximal integral manifold, then http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq21_HTML.gif 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,YD.

      Proof

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

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

      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 XD. Then, the differential 1-forms http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq22_HTML.gif vanish on D for all t.

      Proof

      If D is completely integrable, then X is a symmetry. Hence,
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equaa_HTML.gif

      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
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equab_HTML.gif

      Definition 6.2

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

      Any symmetry X ∈ Sy m 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,
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equac_HTML.gif
      Therefore, Shuf(D) is spanned by http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq23_HTML.gif , where
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equad_HTML.gif

      Example 6.4

      (Continuation of Example 4.9) In this case, we have
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equae_HTML.gif
      in Shuf(D). Therefore, Shuf(D) is spanned by
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equaf_HTML.gif
      where
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equag_HTML.gif
      and X = X(x,uqy), Y = Y(y,upx), and U(x,y,u) must satisfy in PDE:
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equ3_HTML.gif
      (3)

      Example 6.5

      (Quasilinear Klein-Gordon Equation) In this example, we find the shuffling symmetries of the quasilinear Klein-Gordon equation
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equah_HTML.gif
      as an application of the previous example, where α, β, and γ are real constants. The equation can be transformed by defining http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq24_HTML.gif and http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq25_HTML.gif . Then, by the chain rule, we obtain α 2 u ξη + γ 2 u = β u 3. This equation reduces to
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equ4_HTML.gif
      (4)

      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:
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equai_HTML.gif
      For example, we have
      http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_Equaj_HTML.gif

      and if u = h(x,y) be a solution of (4), then http://static-content.springer.com/image/art%3A10.1186%2F2251-7456-6-49/MediaObjects/40096_2012_59_IEq26_HTML.gif is also a new solution of (4), for sufficiently small sR.

      Declarations

      Authors’ Affiliations

      (1)
      Faculty of Mathematics, School of Mathematics, Iran University of Science and Technology
      (2)
      Department of Basic Sciences, Eslamshahr Branch, Islamic Azad University

      References

      1. Barone A, Esposito F, Magee CJ, Scott AC: Theory and applications of the Sine-Gordon equation. Riv. Nuovo Cim 1:227–267 1971.View Article
      2. Kragh H: Equation with the many fathers. The Klein-Gordon equation in 1926. Am. J. Phys 52(11):1024–1033 1984.MathSciNetView Article
      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;

      Copyright

      © Nadjafikhah and Aghayan; licensee Springer. 2012

      This article is published under license to BioMed Central Ltd. 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.