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Solutions of Exterior Differential Equations by Quadratures
Journal of Mathematical Analysis and Applications 249, 592᎐613 Ž2000. doi:10.1006rjmaa.2000.6915, available online at http:rrwww.idealibrary.com on
Solutions of Exterior Differential Equations by Quadratures Dominic G. B. Edelen 3503 A¨ enue P, Gal¨ eston, Texas 77550 Submitted by William F. Ames Received April 11, 2000
Homotopy operators are used to obtain unconstrained local solutions of systems of exterior differential equations by quadratures. An analysis is given of when given systems of constraints on the solutions can be realized. The results are applied to several classes of classical problems. A collection of superconductor singular solutions of the free gauge field equations for matrix Lie groups is also obtained, and the general forms of gauge covariant constant fields are given. 䊚 2000 Academic Press
1. INTRODUCTION Poincare ´ defined, in effect, a linear homotopy operator, H, on any star shaped region of a differentiable manifold, and showed that it verified the identity s dH q H d for any smooth differential form of positive degree; whence, the Poincare ´ Lemma: any closed form Ž d s 0. is locally exact Ž s dH .. There the matter sat, for all intent and purposes. This was because the differential geometry community centered most of its efforts on determining the obstructions to global extensions of the Poincare ´ Lemma, for such obstructions were show to contain essential intrinsic information about the global nature of a differentiable manifold. There is, however, a significant amount of additional local information that can be obtained from the Poincare ´ identity, as shown in w1᎐5x. Since H is a Žlocal. linear operator, it has a well defined kernel. If is any exterior differential form in the kernel of H Ž H s 0., then the Poincare ´ identity gives s H d ; that is, the linear homotopy operator H in¨ erts the exterior deri¨ ati¨ e operator d for any exterior differential form in kerŽ H .. Thus, H is the exterior calculus analog of the indefinite integral of the 592 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
EXTERIOR DIFFERENTIAL EQUATIONS
593
ordinary calculus, albeit H is only defined locally. Since s dH q H d , elements in kerŽ H . are referred to as antiexact differential forms. The ability to invert the operator d provides access to procedures for solving systems of exterior differential equations by quadratures and to related matters. The analysis given here concentrates on the structure of the solutions of exterior differential equations, in contrast to the submanifolds on which such solutions vanish, the latter being the topic of primary interest in differential geometry. A terse proof of the Poincare ´ identity and an analysis of the properties of kerŽ H . are given in Section 2. General quadrature solutions of local differential systems, as developed in w1᎐3x, and an important corollary are given in Section 3. Differential systems with explicitly given constraints are analyzed in Section 4. Some classical problems, of importance in the study of the geometry of solutions of nonlinear partial differential equations and in the calculus of variations, are analyzed in Section 5. The use of antiexact forms in general gauge theories is presented in Section 6, where a class of ‘‘super conductor singular’’ solutions of any system of free gauge field equations is constructed. Let M be an n-dimensional differentiable manifold and let ⌳ be the graded module of exterior differential forms over M. The term ‘‘module’’ is taken to mean the module over the algebra ⌳0 of forms of degree zero. The usual case is where ⌳0 is the algebra of C⬁ functions, and this will be assumed unless explicitly stated to the contrary. If ⌳0 is taken to be the algebra of C k functions, then ⌳ is said to have a C k structure as a graded module. The submodule of ⌳ of exterior differential forms of degree k is denoted by ⌳k . The subspace of elements of ⌳k with coefficients that are homogeneous functions of degree h is denoted by hoŽ h, k .. Finally, let GLŽ m, alg. be the group of nonsingular m = m matrices with values in the associative, commutative algebra ‘‘alg,’’ the usual cases being GLŽ m, ⺢. and GLŽ m, ⌳0 .. The group GLŽ m, alg. is obtained by the exponential map of all m = m matrices with values in alg.
2. HOMOTOPY OPERATORS AND THE ALGEBRAIC DECOMPOSITION OF ⌳ If p is any point in M, then there exists a neighborhood UŽ p . of p and a system of local coordinate functions x i ¬ 1 F i F n4 such that each of the coordinate functions has the value 0 at p. Let S be a starshaped subset of UŽ p . with respect to the homotopy h : S ª S ¬ y i s x i ,
0 F F 1.
Ž 2.1.
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DOMINIC G. B. EDELEN
The discussion will be restricted, from now on, to the manifold S, so that all results will be local results with respect to M, but global with respect to S. In particular, ⌳ will now be taken to be the module of exterior differential forms over S. A generalized ‘‘radius’’ vector field on S is defined by X s x ⭸i i
Ž 2.2.
with respect to the coordinate cover x i 4 , where ⭸ i ¬ 1 F i F n4 is a basis for the module of derivations of ⌳0 and the summation convention is assumed. A linear homotopy operator H is defined on ⌳ by H s X @
1
H0
Ž hU .
d
,
Ž 2.3.
where X @  stands for the inner multiplication of a differential form  by a vector field X. The integral in Ž2.3. is a Riemann᎐Graves integral, and hence it is well defined for any form of positive degree in a C 0 structure. Noting that X @ H01 Ž hU . d s H01 X @Ž hU . d s 0 for g ⌳0 , it follows that H is well defined on a C 0 structure. In fact, since hU Ž dy i . s dx i and hU Ž f Ž y j .. s f Ž x j . s r f Ž x j . for any f g hoŽ r, 0., by Euler’s theorem for homogeneous functions, a direct evaluation gives H s
1 kqr
X @  g ho Ž r q 1, k y 1 .
᭙ g ho Ž r , k . .
Ž 2.4.
Let ⌽ be a diffeomorphism of M. Then ⌽ induces a homotopy operator H⌽ on the image of S under ⌽ in M by ⌽U H⌽ s H ⌽U .
Ž 2.5.
A combination of Ž2.3. and Ž2.5. yields the evaluation H⌽ s Ž ⌽# X . @
y1 U
1 H0 Ž ⌽ ( h ( ⌽ .
d
Ž 2.6.
on the image of S in M under the action of ⌽. Restricting ⌽ to a diffeomorphism of S to S, Ž2.6. shows that a homotopy operator is defined for any coordinate cover of S that reduces to H for the coordinate cover x i 4 . Accordingly, properties established for H with the coordinate co¨ er x i 4 continue to hold for all coordinate co¨ ers of S with the homotopy operators gi¨ en by Ž2.6.. There is thus an abundance of linear homotopy operators that can be defined on ⌳Ž S .. It is, however, sufficient to restrict attention to the linear homotopy operator H that is defined by Ž2.3..
EXTERIOR DIFFERENTIAL EQUATIONS
595
The parameter transformation s e applied to the homotopy Ž2.1. gives the homotopy TX Ž . : S ª S ¬ y i s e x i ,
y⬁ F F 0,
Ž 2.7.
where TX Ž . is the flow operator on the orbits of the vector field X. It is then a simple matter to see that hUexpŽ . s expŽ £ X . , y⬁ F F 0, and hence Ž2.3. gives H s X @
⬁
0
Hy⬁ expŽ £
X
. d s X @ H exp Ž y £ X . d .
Ž 2.8.
0
These equivalent evaluations of H will be useful in the proof of the Poincare ´ identity. LEMMA 2.1. The homotopy operator H is a linear map from ⌳k to ⌳ky 1 that ¨ erifies the identity
s dH q H d
Ž 2.9.
for all positi¨ e k, while Hf s 0 and f Ž x i . s H df q f Ž 0 .
Ž 2.10.
for k s 0. Proof. That H is a map from ⌳k to ⌳ky1 follows directly from Ž2.3. and the fact that the operator V @ has this property for any vector field V. Use of the definition Ž2.3. gives H d s X @ H01 Ž hU d . d s X @ dH01 Ž hU . d , and hence dH q H d s £ X
1
H0
Ž hU .
d
,
Ž 2.11.
where £ is the symbol for Lie differentiation. When the alternative 0 evaluation H01 Ž hU . d s Hy⬁ expŽ £ X . d is used, Ž2.11. gives dH q H d s
0
d
Hy⬁ d Žexp Ž £
s y lim
ªy⬁
X
. . d
Ž exp Ž £ X . . .
Ž 2.12.
For any form of positive degree, the limit vanishes, while for any f g ⌳0 , it evaluates to f Ž0.. That Hf s 0 is true follows directly from rewriting Ž2.3.
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DOMINIC G. B. EDELEN
in the equivalent form H s
1
H0
X @Ž hU .
d
and noting that X @ applied to any element of ⌳0 evaluates to zero. The definition Ž2.3. of H and Lemma 2.1 can be used to show that the operator H exhibits the following properties Žsee w3, p. 177x.: HH ' 0, H dH s H , H dH d s Hd, X @ H ' 0,
Ž 2.13.
dH d s d,
Ž 2.14.
dH dH s dH,
Ž 2.15.
HX @ ' 0.
Ž 2.16.
Thus, H is nilpotent, and dH and Hd are projection operators. Since H is a linear operator on ⌳, it has a well defined kernel, A s g ⌳ ¬ H s 0 4 ,
Ž 2.17.
and hence A is the direct sum of its homogeneous subspaces A k ¬ 0 F k F n4 . Noting that H⌳0 ' 0, it follows that A 0 s ⌳0 . It is then an easy matter to show that Ž2.3. implies A s g ⌳ ¬ X @ s 04 ,
Ž 2.18.
and that A n s 0. Further, since X @ is an antiderivation on ⌳, Ž2.8. shows that A p n A q ; A pq q, A k q A k ; A k , and hence A is a submodule of ⌳ over ⌳0 . Thus, the characterization of A is strictly algebraic! Since HH ' 0, H gi¨ es rise to the algebraically generated exact sequence i
H
H
H
H
H
0 ª ⌳n ª ⌳ny1 ª ⭈⭈⭈ ª ⌳1 ª ⌳0 ª 0,
Ž 2.19.
where i stands for the natural inclusion. Let E be the subspace of ⌳ consisting of exact exterior differential forms. The identity Ž2.9. has the equivalent statement ⌳k s E k q A k ,
Ž 2.20.
with the projection relations E s dH ⌳ ,
A s H d⌳ ,
Ž 2.21.
and the identification E n s ⌳n. Accordingly, any element of A is referred to as an antiexact differential form.
EXTERIOR DIFFERENTIAL EQUATIONS
597
THEOREM 2.1. The additi¨ e splitting of ⌳ gi¨ en by Ž2.20. is strict in any C p-structure, for p G 0, and hence ⌳ s E [ A.
Ž 2.22.
In addition, H is the in¨ erse of d for any antiexact differential form, and E can be defined algebraically as kokerŽ H .. Proof. Noting that A 0 s ⌳0 , E 0 s 0, A n s 0, E n s ⌳n, it is only necessary to establish that the splitting Ž2.20. is strict for forms of degree k with 1 F k - n. Assume that g E k . Then s d for some g ⌳ky 1. However, s e q a where e s dH is the exact part of and a s H d is the antiexact part of . Accordingly, s da with a g A ky 1. In order that the belong to A, it is necessary that 0 s X @ s X @ da. However, £ X A ' X @ d A q dŽ X @ A . ' X @ d A, and hence we have the requirement £ X a s 0. This shows that the requirements
s da,
£ X a s 0, a g A ky 1 ,
Ž 2.23.
are satisfied by any element in E k l A k . Hence, a necessary and sufficient condition for the additive decomposition Ž2.20. to be strict is that Ž2.23. is satisfied only by the zero element of A k . Now, £ X is a derivation, and hence it is only necessary to compute the value of £ X on all elements of A 0 s ⌳0 and A 1. For k s 1, a s f Ž x i . g ⌳0 , and hence £ X a s X @ da s 0 is satisfied by all a g hoŽ0, 0., by Euler’s theorem. However, the only bounded, single-valued elements of hoŽ0, 0. are the constant functions, and hence s da s 0 for the case k s 1. For k s 2, a s f i Ž x k . dx i g A 1 for x i f i Ž x k . s 0, and hence 0 s £ X a s Ž f i q X @ df i . dx i yields the requirements X @ df i q f i s 0, 1 F i F n. Euler’s theorem on homogeneous functions shows that these conditions are satisfied only when f i Ž x k . g hoŽy1, 0., 1 F i F n. Since the only bounded element of hoŽy1, 0. is the zero function, it follows that a s 0 and hence s da s 0. An examination of the part of the above proof after Ž2.23. shows that £ X A k s 0 only for elements of hoŽyk, k ., and hence the only bounded, single-valued elements of A that satisfy £ X s 0 in a C p-structure with p G 0 are the constant elements of ⌳0 s A 0 . The constant elements of ⌳0 are, however, precisely those elements of ⌳0 that are closed but not exact. A complete account of the structure of E k l A k is only possible in the wider class of forms whose coefficients have singularities, E k l A k s d Ž X @ ho Ž yk, k . . ; ho Ž yk, k . . For example, ␣ s xy2 Ž x dy y y dx . s dŽ xy . ; hoŽy1, 1. is in E 1 l A 1 , but is singular on x s 0, while  s 2 zy3 Ž z dx n dy y dz n Ž x dy y y dx .. s dŽ zy2 Ž x dy y y dx .. ; hoŽy2, 2. is in E 2 l A 2 but is singular on z s 0.
598
DOMINIC G. B. EDELEN
3. GENERAL QUADRATURE SOLUTIONS OF LOCAL DIFFERENTIAL SYSTEMS Let ⍀ a g ⌳k ¬ 1 F a F m, k ) 04 be m k-forms on M Žnot necessarily independent.. They generate an associated ideal Ik s I ⍀ 1 , . . . , ⍀ m 4 of ⌳ over M. If the ⍀’s are known k-forms, then there is little left to do, other than to obtain a basis for Ik ; that is, to find a minimal set of k-forms that generates the same ideal. Differential systems come into play when the ⍀’s are to be determined by solving a system of exterior differential equations d⍀ a q ⌫ba n ⍀ b s ⌺ a , where the ⌫’s are 1-forms and the ⌺’s are Ž k q 1.-forms on M. For k s 1, ⍀ a s ia dx i, ⌫ba s ␥ bai dx i, 2 ⌺ a s i aj dx i n dx j, i aj s yjia, 1 F a, b F m, 1 F i, j F n, this is the system of partial differential equations
⭸ ja ⭸ xi
y
⭸ ia ⭸xj
q ␥ bai jb y ␥ baj ib s i aj .
The importance of differential systems was made evident by the pioneering work of E. Cartan w6x and continues to this day w7, 8x. Now, there may or may not be m independent solutions of the given system of exterior differential equations, and hence the generators of the associated ideal Ik can not be assumed to be independent a priori. EXAMPLE 1. The differential system d a q Ž x dy q y dx .G ba n b s 0 on ⺢ 5, with k s 1, m s 3, ŽŽ G ba .. a nonsingular, constant-valued 3 = 3 matrix, and coordinate cover x, y, z, u, ¨ 4 , only has solutions of the form
a s d a Ž x, y . y G ba H Ž Ž x dy y y dx . n d b Ž x, y . . , for any choice of the a Ž x, y . g ⌳0 Žsee w9x.. Thus, the associated ideal I a ¬ 1 F a F 34 can have at most two independent generators. The arguments to follow are significantly simplified by use of matrix notation. Let ⍀ denote the column matrix whose entries are the k-forms ⍀ a ¬ 1 F a F m4 . In order to retain the fact that ⍀ is a column matrix with m entries, ⍀ will be referred to as an element of length m. Let ⌫ s ŽŽ ⌫ba .. be an m = m matrix of 1-forms, let ⌺ be a column matrix of Ž k q 1.-forms of length m, and let ⌫ n ⍀ s ŽŽ ⌫ba n ⍀ b .. Ži.e., matrix multiplication with the exterior product.. The differential system can then be written as a matrix system of degree k and length m: d⍀ q ⌫ n ⍀ s ⌺. The associated ideal is written Ik s I ⍀ 4 .
Ž 3.1.
EXTERIOR DIFFERENTIAL EQUATIONS
599
Although the generators of Ik are not necessarily independent, there is a natural equivalence relation for the generators. Let A g GLŽ m, ⌳0 .. Then Ik s I ⍀ 4 s I A⍀ 4 , because Ik is a module over ⌳0 . However, GLŽ m, ⺢. is a subgroup of GLŽ m, ⌳0 ., and any element of GLŽ m, ⺢. factors through Ž3.1. in the obvious manner Ži.e., dK ⍀ s K d⍀ for any K g GLŽ m, ⺢... Let p be a point in M and let GL p Ž m, ⌳0 . be the subgroup of GLŽ m, ⌳0 . whose elements evaluate to the identity matrix E at p. There is a natural projection P: GLŽ m, ⌳0 . ª GL p Ž m, ⌳0 . ¬ B˜ s B Ž p .y1 B such that P ŽGLŽ m, ⺢.. s E and GLŽ m, ⌳0 . s GLŽ m, ⺢. = GL p Ž m, ⌳0 .. Thus, as far as the differential system Ž3.1. is concerned, only elements of GL p Ž m, ⌳0 . can change things Ži.e., A g GL 0 Ž m, ⌳0 . and ˜ will be said to be ideal dA s 0 implies A s E .. Two elements ⍀ and ⍀ ˜ s A⍀ for some A g GL p Ž m, ⌳0 ., in which case equi¨ alent if and only if ⍀ they are necessarily equivalent for any B g GLŽ m, ⌳0 .. The notion of ideal equivalence is of particular importance in the study of differential systems that generate ideals of ⌳. The differential system Ž3.1. obviously entails integrability conditions. Exterior differentiation and elimination in the standard manner leads to the complete differential system of length m and degree k,
Ž 夹.
d⍀ q ⌫ n ⍀ s ⌺, d⌫ q ⌫ n ⌫ s ⌰,
d⌺ q ⌫ n ⌺ s ⌰ n ⍀ , d⌰ q ⌫ n ⌰ s ⌰ n ⌫.
Ž 3.2. Ž 3.3.
By analogy with the Cartan equations of structure, ⌫ is referred to as the matrix of connection 1-forms, ⌰ is referred to as the matrix of cur¨ ature 2-forms, and ⌺ is referred to as the matrix of torsion Ž k q 1.-forms of the differential system Ž夹.. These connection, curvature, and torsion forms are not necessarily those of a differentiable manifold. Note that the structure of the connection-curvature equations Ž3.3. is universal for all differential systems of length m, for they always involve forms of degrees one and two only. An elementary calculation shows that the ideal equivalence transformation of a differential system induced by A g GL 0 Ž m, ⌳0 . is given by ⍀ s A⍀ ,
⌺ s A⌺,
⌰ s A⌰ Ay1 ,
⌫ s Ž A⌫ y dA . Ay1 ,
in which case the ‘‘barred’’ quantities satisfy the complete differential system d⍀ q ⌫ n ⍀ s ⌺,
d⌺ q ⌫ n ⌺ s ⌰ n ⍀ ,
d⌫ q ⌫ n ⌫ s ⌰,
d⌰ q ⌫ n ⌰ s ⌰ n ⌫.
600
DOMINIC G. B. EDELEN
The original complete differential system Ž夹. and the new one have exactly the same structure, which is a property that will prove to be essential in what follows. A system of relations is said to be ideal equi¨ alence co¨ ariant if and only if its structure is invariant under all ideal equivalence transformations. Thus, any complete differential system is ideal equi¨ alence co¨ ariant. Constraints such as ⌰ s 0 or ⌺ s 0 are thus ideal equivalence covariant systems of conditions, while no explicit evaluation of ⌫ is ideal equivalence covariant due to the presence of the inhomogeneous terms Ž dA. Ay1 . Now, everything depends on which quantities in a differential system are given and which are to be calculated from that given system. Suppose that a differential system can be solved for all of the quantities ⍀, ⌺, ⌫, ⌰4 simultaneously. All such solutions will be referred to as unconstrained solutions. The question of the existence of solutions of the differential system with given data, and the construction of such solutions, can then be treated by considering the given data as a system of constraints that the unconstrained solutions must satisfy. This is the approach taken here. Any point p of a differentiable manifold M has a neighborhood UŽ p . and a system of coordinate functions x i ¬ 1 F i F n4 such that the neighborhood S ; UŽ p . is starshaped with respect to the homotopy h discussed in the previous section. Consideration will now be restricted to S, so that all results will be local. In particular, ⌳ will now be taken to mean the exterior algebra of differential forms over S. The direct sum decomposition ⌳ s E [ A and H d ␣ s ␣ for all ␣ g A established in the previous section can be used to construct all unconstrained solutions of a complete differential system Ž夹. by quadratures. The classic terminology is adopted here; namely, a solution is obtained by quadratures if and only if it can be written in terms of known functions or the integrals of known functions ŽRiemann᎐Graves integrals in this case.. The proof of the following theorem is given in w3, Chap. 5x. Žcf. Theorem 5-15.1.. THEOREM 3.1 ŽLocal Quadrature Theorem.. The general, unconstrained solution of a differential system Ž夹. of length m and degree k, on a starshaped neighborhood S with homotopy operator H, is gi¨ en by ⍀ s A d q y H Ž n d . 4 ,
Ž 3.4.
⌺ s A d q n q Hd Ž n d . y n H Ž n d . 4 , y1
⌫ s A y dA4 A
,
y1
⌰ s A d q n 4 A
Ž 3.5. Ž 3.6.
for any choice of A g GL 0 Ž m, ⌳ ., for any choices of the column matrices of antiexact forms , of length m of appropriate degrees, and for any choice of the m = m matrix of antiexact 1-forms . Under these circumstances, the
601
EXTERIOR DIFFERENTIAL EQUATIONS
following relations are satisfied throughout S,
s H Ž Ay1 ⍀ . ,
s H Ž Ay1 ⌺ . ,
s H Ž Ay1 ⌰ A . ,
Ž 3.7.
and A satisfies the system of matrix Riemann᎐Gra¨ es integral equations A s E y H Ž ⌫A . .
Ž 3.8.
This is all that can be done if consideration is restricted solely to the differential system Ž夹.. If Ik is the ideal of ⌳ that is generated by ⍀, then ideal equivalence can be used to remove the occurrences of the attitude matrix A. The previously noted relations that are induced by an ideal equivalence transformation ⍀ s A⍀, and the fact that A s E y H Ž ⌫A. has the solution A s E for any ⌫ with antiexact entries, lead to the following useful result. COROLLARY 3.1. E¨ ery differential system is ideal equi¨ alent to a differential system with a connection matrix of antiexact 1-forms. If Ik is the ideal of ⌳ that is generated by solutions ⍀ of the differential system Ž夹., then Ik is ˜ with generated by ⍀
˜ s d q y H Ž n d . , ⍀
Ž 3.9.
˜⌺ s d q n q Hd Ž n d . y n H Ž n d . ,
Ž 3.10.
˜s, ⌫
˜ s d q n , ⌰
Ž 3.11.
for the same choice of the matrices of , , of antiexact forms. The evaluations ⍀ s A d q y H Ž n d .4 , s H Ž Ay1 ⌺ ., ⌰ s A d q n 4 Ay1 show that accounts for the presence of nontrivial torsion Ž k q 1.-forms while H Ž n d . accounts for the presence of nontrivial curvature 2-forms. This evaluation of ⍀ shows that the situation in which s H Ž n d . is thus of interest because ⍀ then has the simplest possible form ⍀ s A d . The easiest way to analyze this problem is to use Corollary 3.1 to transform to an ideal equivalent differential ˜ has only antiexact entries. Use of Ž3.9. through Ž3.11. system for which ⌫ ˜ s d and ˜⌺ s HdŽ n d . q shows that this is the case only when ⍀ ˜ n ⍀, ˜ that is, when ˜⌺ s 0 mod Ik . For k s 1, dH Ž n d . s n d s ⌫ this is the converse of the Frobenius theorem, although there is no restriction on the value of k in the discussion just given. THEOREM 3.2 ŽFrobenius Theorem for k-Forms.. Let Ik be an ideal of ⌳ that is generated by m F Ž Nk . k-forms. If Ik is the closed ideal and the differential system generated by Ik has ¨ anishing cur¨ ature 2-forms, then Ik is generated by m exact k-forms.
602
DOMINIC G. B. EDELEN
Proof. Let ⍀ be a column matrix of generators of Ik of length m. Since Ik is closed, by hypothesis, ⍀ satisfies a differential system of the form d⍀ q ⌫ n ⍀ s 0. Now, the conditions d Ik ; Ik and ⌰ s 0 are ideal equivalence covariant conditions. A transformation to the ideal equivalent system indicated in Corollary 3.1 thus yields the simplified ˜ qn⍀ ˜ s 0. The associated curvature 2-forms are ⌰ ˜s conditions d⍀ ˜ s 0 implies s 0. Accordingly, d q n , and hence the hypothesis ⌰ Ž3.10. shows that the evaluation of ˜ ⌺s0 ⌺ reduces to ˜ ⌺ s d , and hence ˜ ˜ can always be achieved because s H ⌺ s 0. The evaluation Ž3.9. thus ˜ s d , and hence all of the entries of ⍀ ˜ are exact k-forms. Since yields ⍀ there are no further constraints that the must satisfy, the entries of d can be chosen to be independent because m F Ž Nk .. Example 1, with k s 1 and ⌰ / 0, shows that the ideal equivalence covariant conditions ⌰ s 0 are necessary, even in the case k s 1. However, an elementary calculation and Cartan’s lemma show that in the case k s 1, the requirement Žconstraint . that the entries of d be independent implies ⌰ s 0. That this is not the case for k ) 1 can be seen as follows. LEMMA 3.1. The relations H Ž n d . s 0 can be satisfied only if s 0 or the entries of satisfy the constraints n £ X s 0. Proof. The relation H Ž n d . s 0 can hold only if n d belongs to kerŽ H .. This is the case only when 0 s X @Ž n d . s y n X @ d s y n £ X , because the entries of are antiexact forms. Calculation shows that £ X is an automorphism of A k . The problem thus becomes an algebraic one; namely, find all column matrices  g A ky 1 of length m such n  s 0 and then solve the quasilinear partial differential equations £ X s  with the same principal part such that X @ s 0.
4. CONSTRAINED SOLUTIONS OF LOCAL DIFFERENTIAL SYSTEMS The effects of constraints that obtain by specification of connection or curvature forms will first be investigated. Let ⌫ be a given m = m matrix of 1-forms on S. In this event A can be determined Žpossibly on a smaller open subset of S . by solving the Riemann᎐Graves integral equations A s E y H Ž ⌫A. Žsimply iterate and check that the sequence is dominated by a corresponding exponential series.. Once A is known, the first of Ž3.6. gives s Ay1 Ž ⌫A q dA.. It thus remains to check that is antiexact. This is the case if and only if 0 s X @ . Accordingly, the relation X @Ž ⌫A q dA. s 0 must be satisfied. Exterior differentiation of A s E y H Ž ⌫A.
603
EXTERIOR DIFFERENTIAL EQUATIONS
yields dA s ydH Ž ⌫A., and hence ⌫A q dA s ⌫A y dH Ž ⌫A. s HdŽ ⌫A. because dH q Hd s identity. A combination of these relations shows that X @Ž ⌫A q dA. s 0, and thus is antiexact. In this event, the second part of Ž3.6. yields ⌰ s A d q n 4 Ay1 . THEOREM 4.1. If the matrix of connection 1-forms ⌫ is assigned, then the attitude matrix A g GL 0 Ž m, ⌳ . and the matrix of antiexact 1-forms are uniquely determined by A s E y H Ž ⌫A . ,
s Ay1 Ž ⌫A q dA . ,
in which case ⌰ s d⌫ q ⌫ n ⌫ ser¨ es to determine the matrix of cur¨ ature 2-forms. Next, assume that ⌰ is a given m = m matrix of 2-forms on S. Not every matrix of 2-forms is a possible choice because ⌰ must satisfy the cur¨ ature equations d⌰ q ⌫ n ⌰ s ⌰ n ⌫ for some connection matrix ⌫. In fact, Theorem 3.1 shows that the attitude matrix A and s H Ž Ay1 ⌰ A. serve to determine an admissible matrix of curvature 2-forms uniquely through the relations ⌰ s AŽ d q n . Ay1 . Since the attitude matrix A is not determined by a matrix of curvature 2-forms, we are free to choose any A g GL 0 Ž m, ⌳0 .. With each such choice of A, the m = m matrix A s H Ž Ay1 ⌰ A. g A 1 is well defined. An admissible matrix of curvature 2-forms is thus defined by ⌰A s AŽ dA q A n A . Ay1 with the property that H Ž Ay1 ⌰ A. s H Ž Ay1 ⌰A A.; that is, Ay1 ⌰ A and Ay1 ⌰A A have the same exact part dA . The subscript A has been attached in order to point out that all such quantities are dependent on the choice of the attitude matrix A. A direct calculation shows that ⌰A s ⌰ y A Hd Ž Ay1 ⌰ A . y H Ž Ay1 ⌰ A . n H Ž Ay1 ⌰ A . 4 Ay1 . THEOREM 4.2. Let ⌰ be a gi¨ en m = m matrix of 2-forms on S. For each choice of the attitude matrix A g GL 0 Ž m, ⌳ ., there is an associated matrix of Ž compatible. cur¨ ature 2-forms ⌰A s A dH Ž Ay1 ⌰ A . q H Ž Ay1 ⌰ A . n H Ž Ay1 ⌰ A . 4 Ay1
Ž 4.1.
for which Ay1 ⌰ A and Ay1 ⌰A A ha¨ e the same exact part, and
A s H Ž Ay1 ⌰ A . s H Ž Ay1 ⌰A A . ,
⌫A s AA y dA4 Ay1 . Ž 4.2.
In particular, if ⌰ s 0, then A s 0 and there are infinitely many associated matrices of connection 1-forms ⌫A s yŽ dA. Ay1 , for all A g GL 0 Ž m, ⌳ .. What constraints are imposed by specification of the torsion Ž k q 1. forms? Clearly, constraints do arise because any ⌺ must satisfy the torsion
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equations d⌺ q ⌫ n ⌺ s ⌰ n ⍀. Let ⌺ be a given column matrix of Ž k q 1.-forms of length m. Then is evaluated by s H Ž Ay1 ⌺ ., and hence Ž3.5. gives the requirements Ay1 ⌺ s dH Ž Ay1 ⌺ . q n H Ž Ay1 ⌺ . q Hd Ž n d . y n H Ž n d . . THEOREM 4.3. If the torsion Ž k q 1.-forms are specified, then the matrices of antiexact quantities A, , and must satisfy the Ž essentially nonlinear . system of m constraints Ay1 ⌺ s dH Ž Ay1 ⌺ . q n H Ž Ay1 ⌺ . q Hd Ž n d . y n H Ž n d . .
Ž 4.3.
These nonlinear constraints are what often make problems with specified torsion very difficult. If the further constraints of vanishing curvature are imposed, then s 0 and Ž4.3. reduces to the requirements Ay1 ⌺ s dH Ž Ay1 ⌺ .; that is, if the cur¨ ature 2-forms of a differential system all ¨ anish, then the only torsion Ž k q 1.-forms that can be specified are of the form ⌺ s B d for some B g GLŽ m, ⌳ . and some g ⌳ky 1 , in which case A s B Ž0.y1 B. The following result is also immediate from Ž4.3.. If the constraints ⌺ s 0 are imposed, then and must satisfy the additional constraints HdŽ n d . s n H Ž l d .. LEMMA 4.1. The general solution of the relations H d s n H
Ž 4.4.
is gi¨ en by
s d q n ,
gA
Ž 4.5.
˜ n . in which case will necessarily satisfy the conditions d q n s ⌰ Proof. The homotopy decomposition gives s d q a , where and a are antiexact. Thus, H d s H d a s n H s n . The result then follows on noting that H d a s a for any antiexact a . The integrability ˜ n . conditions for Ž4.5. are d q n s ⌰ THEOREM 4.4. If ⌰ / 0, then the constraints of ¨ anishing torsion can be realized only when n d satisfy the constraints
n d s d q n
Ž 4.6.
for some matrix of antiexact k-forms of length m, in which case
˜ s d y , ⍀
˜ n Ž d y . s 0. ⌰
Ž 4.7.
EXTERIOR DIFFERENTIAL EQUATIONS
605
Proof. The remarks following Theorem 4.3 show that the constraints ⌺ s 0 can be satisfied only when and satisfy the constraints HdŽ n d . s n H Ž n d .. Use of Lemma 4.1 with s n d gives the requirements Ž4.6.. When these requirements are met, s H ˜ ⌺ s 0, H Ž n d . s H Ž d q n . s H d s because both and are column matrices of antiexact forms. A substitution of these evaluations into Ž3.9. gives Ž4.7.. The following theorem is of use for problems in which the torsion Ž k q 1.-forms do not vanish. THEOREM 4.5 ŽTorsion Transformation Theorem.. Let be a column matrix of Ž k y 1.-forms of length m that transforms under ideal equi¨ alence transformations ⍀ s A⍀ of Ik by s A . Then the torsion transformation
˜ q d q ⌫ n ⍀s⍀
Ž 4.8.
maps the complete differential system Ž夹. into the ideal equi¨ alence co¨ ariant differential system
˜ q⌫n⍀ ˜ s ˜⌺, d⍀
˜ q ⌫ l ˜⌺ s ⌰ n ⍀ ˜, d⌺
Ž 4.9.
d⌫ q ⌫ n ⌫ s ⌰,
d⌰ q ⌫ l ⌰ s ⌰ n ⌫,
Ž 4.10.
with the same connection and cur¨ ature forms, while the new torsion forms are gi¨ en by
˜⌺ s ⌺ y ⌰ n .
Ž 4.11.
Proof. The relations Ž4.9. through Ž4.11. follow directly by substitution of the relations Ž4.8. into the differential system Ž夹.. Since the original system was ideal equivalence covariant and d q ⌫ n is ideal equivalence covariant under the hypothesis s A , the resulting differential system Ž4.9., Ž4.10. is also ideal equivalence covariant. If ⌺ has the ideal equivalence covariant evaluation ⌺ s ⌰ n for some column matrix of Ž k y 1.-forms of length m, then the complete ˜ q d q ⌫ n for any ⍀ ˜ that differential system Ž夹. is satisfied by ⍀ s ⍀ ˜ q⌫n⍀ ˜ s 0. Thus, satisfies the complete differential system generated by d⍀ if ⌰ is assigned, the only generality left in the solutions comes from all ˜ of the corresponding complete differential system with given solutions ⍀ curvature 2-forms and vanishing torsion k-forms. This representation is ˜ s 0 is always a solution. not vacuous because ⍀
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DOMINIC G. B. EDELEN
5. SOME CLASSICAL PROBLEMS The question of when does a differentiable manifold admit a contact structure, ⍀ a s dq a y ria d i g ⌳1 , 1 F a F m, summation on i from 1 through r 4 , was solved by Bryant et al. w7x for m / 2 by analytic methods. Contact structures are important, for they provide a natural setting for problems in nonlinear differential equations and in the calculus of variations w2; 3, Chaps. 5 and 6; 7; 8; 10x. Since q a, ria, i 4 all belong to ⌳0 and ⌳0 s A 0 , the restriction to forms of degree one tends to confuse the issue as far as the differential structure is concerned. The problem will thus be analyzed for forms of degree k G 1, and then cut down to k s 1 later. Let q, ˜ ˜ri ¬ 1 F i F r 4 be column matrices of Ž k y 1.-forms of length m ˜ s dq˜ y ˜ri and define a column matrix of contact k-forms of length m by ⍀ n d i, where i g ⌳0 ¬ 1 F i F r 4 such that d 1 n ⭈⭈⭈ n d r / 0. The ˜ 4 is thus well defined. Any other collection of contact ideal Ck s I ⍀ ˜ s AŽ dq˜ y ˜ri n d i . / dq y ri n generators of Ck is given by ⍀ s A⍀ i d , which shows that the structure dq y ri n d i is not ideal equivalence covariant. In order to overcome this difficulty, set q s Aq, ˜ ri s Ar˜i and observe that ⍀ satisfies the exterior differential equations d⍀ q ⌫ n ⍀ s y Ž dri q ⌫ n ri . n d i , ⌰ s d⌫ q ⌫ n ⌫ s 0.
⌫ s y Ž dA . Ay1 ,
Ž 5.1.
If we set R i s dri q ⌫ n ri , then these relations are ideal equivalence covariant and they satisfy the exterior differential equations dR i q ⌫ n R i s ⌰ n ri s 0. All information is thus encoded in the ideal equivalence covariant form d⍀ q ⌫ n ⍀ s yR i n d i ,
dR i q ⌫ n R i s 0,
d⌫ q ⌫ n ⌫ s ⌰ s 0.
Ž 5.2. Ž 5.3.
The definition of the torsion Ž k q 1.-forms gives ⌺ s yR i n d i, and the R i equations show that the torsion equations d⌺ q ⌫ n ⌺ s ⌰ n ⍀ are identically satisfied. The system Ž5.2., Ž5.3. is a complete exterior differential system with ¨ anishing cur¨ ature and the ideal Ik s I ⍀, R i ¬ 1 F i F r 4 is a closed ideal. The classic approach can thus be followed in the case k s 1 by using Cartan’s theorem to the effect that the Cauchy characteristic subspace of T Ž M . of a closed ideal is a Lie module over ⌳0 . The problem can be solved for all values of k by the techniques developed here. Since the differential system is ideal equivalence covariant, Corollary 3.1 allows the transformation to an equivalent differential system with antiexact connection 1-forms. However, the given differential system has vanishing curvature and hence
EXTERIOR DIFFERENTIAL EQUATIONS
607
s 0. The exterior differential system Ž5.2., Ž5.3. is thus ideal equivalent to the system
˜ s yR˜i n d i , d⍀
˜i s 0, s 0, ⌰ ˜ s 0, dR
Ž 5.4. ˜ and the Poincare ´ lemma thus yields the desired solutions ⍀ s d y ˜ri n d i, R˜i s dr˜i . The same result can be obtained by use of the general ˜ s d q s d y HdŽ ˜ri solution generated by the operator H; that is, ⍀ i. n d , where and are related by dŽ y q H Ž ˜ ri n d i .. s 0 since i ˜ ˜ s 0. Conversely, if ⍀ s d y ˜ ri n d , then d⍀ s ydr˜i n d i, and ˜ s 0. These soluhence ⌫ s 0 implies the ideal equivalence conditions ⌰ tions have been obtained without constraints of independence or other˜ R˜i , can be chosen to be wise. Questions of whether the entries of ⍀, independent can be answered by computing the codimensions of the Cauchy characteristic modules of the ideals Ck and Ik . The classical analytic approach would start with the statements d⍀ ' yR i n d i mod Ck , and hence the connection 1-forms are not directly accessible. It is then a lengthy analytic task to try to show that the usual independence conditions imply ⌰ s 0, which happens to fail in the case k s 1, m s 2. The procedure given above obtains the conditions ⌰ s 0 directly from the requirement of ideal equivalence covariance for the governing differential system for the generators of the ideal Ck , and then the answer is almost trivial. Indeed, the conditions ⌰ s 0 are exactly the additional conditions that are required in the case k s 1, m s 2 in order to restrict the solutions of the differential system to yield contact structures. The analysis given above shows that the contact problem is character˜ s 0, ⌺ s R i n d i. An obvious generalization ized by the requirements ⌰ ˜ s 0 and allowing ⌺ s R i n ␣ i is obtained by relaxing the conditions ⌰ i 1 with ␣ g ⌳ . Now, ⌺ has the structure R i n ␣ i and the ␣ ’s are independent if and only if s ␣ 1 n ⭈⭈⭈ n ␣ r / 0. Accordingly, we obtain the requirements ⌺ n s 0. The generalized contact problem is to solve d⍀ ' ⌺ mod Ck s I ⍀ 4 subject to the constraints ⌺ n s 0. For k s r s 1, m s 2 this is the famous problem d 1 ' ␣ n R1 ,
d 2 ' ␣ n R 2 mod I 1 , 2 4
solved by Cartan w11x under the further constraint 1 n 2 n ␣ n R1 n R 2 / 0 on ⺢ 5. Some of the solutions given by Cartan of this problem are not contact 1-forms. THEOREM 5.1. Let ⍀ be a column matrix of k-forms and ⌺ be a column matrix of Ž k q 1.-forms, both of length m, set Ck s I ⍀ 4 , and let ␣ i g ⌳1 ¬ 1 F i F r 4 satisfy
s ␣ 1 n ⭈⭈⭈ n ␣ r / 0.
Ž 5.5.
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DOMINIC G. B. EDELEN
The general solution of the generalized contact problem d⍀ ' ⌺
mod Ck ,
⌺ns0
Ž 5.6.
is gi¨ en by
˜, ⍀ s A⍀
˜ ⌺ s A⌺,
⌫ s Ž A y dA . Ay1 ,
˜ Ay1 Ž 5.7. ⌰ s A⌰
for all A g GL 0 Ž m, ⌳0 ., where
˜ s d q y H Ž n d . , ⍀
Ž 5.8.
˜⌺ s d q n q Hd Ž n d . y n H Ž n d . ,
Ž 5.9.
˜s, ⌫
˜ s d q n ⌰
Ž 5.10.
for all choices of matrices of antiexact forms , , 4 that satisfy the constraints ˜ ⌺ n s 0. Proof. The relations Ž5.6. are satisfied if and only if d⍀ q ⌫ n ⍀ s ⌺ for some matrix of connection 1-forms ⌫. Completing these equations to a complete differential system yields the system Ž夹. which is ideal equivalance covariant. Accordingly, Corollary 3.1 shows that it can be trans˜ s is a matrix of antiexact formed to an equivalent system such that ⌫ 1-forms. The general solution of this new system is given by Ž5.8. through Ž5.10., and hence all solutions of the original system are given by Ž5.7. for all A g GL 0 Ž m, ⌳0 . and for all choices of the antiexact quantities , , 4 . It then only remains to choose the antiexact quantities , , 4 so that the constraints ˜ ⌺ n s 0 are satisfied. The explicit form of the constraints, namely R i n ␣ i s d q n q Hd Ž n d . y n H Ž n d . Ž 5.11. for some choice of the column matrices R i of k-forms of length m, is obtained by substituting Ž5.9. into ⌺ s R i n ␣ i. The obvious nonlinearity of these relations is what makes this a difficult problem. If we choose s 0, then R i n ␣ i s d and hence the choice s H Ž R i n ␣ i . will work. However, d s R i n ␣ i requires dŽ R i n ␣ i . s 0 in this case, but there are no restrictions placed on the choice of the matrix of antiexact Ž k y 1.-forms . In the general case, application of H to both sides of Ž5.11. leads to the evaluation s H Ž R i n ␣ i .. When this evaluation of is put back into Ž5.11., the residual constraints HdŽ n d y R i n ␣ i . s n H Ž n d y R i n ␣ i . are obtained. Lemma 4.1 can thus be used to obtain the equivalent requirements n d s R i n ␣ i q d q n .
EXTERIOR DIFFERENTIAL EQUATIONS
609
These latter representations of the constraints are easier to deal with since they uncover the restrictions that n d must satisfy. EXAMPLE 2. The effects of non-vanishing curvature can be seen by considering the following exterior differential system in ⺢ 5 : Ž x, y, z, u, ¨ ., d 1 q n 2 s 2 a dx n dy,
d 2 s 2 b dz n dy,
where Ž a, b . are constants such that ab / 0 and s z dy y y dz g A 1. Here ␣ 1 s dy, R11 s 2 a dx, R12 s 2 b dz, so that R11 n R12 n ␣ 1 / 0, and the matrix of connection 1-forms ⌫ss
0 0
g A1 , 0
has one non-zero curvature 2-form entry d s 2 dz n dy. The results of Theorem 5.1 shows that the general solution of this problem is given by A s E and
1 s dp q a  y H Ž n dq Ž y, z . . ,
2 s dq Ž y, z . q b ,
where  s x dy y y dx g A 1, p is an arbitrary element of ⌳0 Ž⺢ 5 ., while q is an arbitrary function of only the two variables y and z. Clearly, 1 n 2 n ␣ 1 / 0 and
1 n d 1 s 1 n 2 a dx q Ž X @ dq . dz 4 n dy / 0, but 2 n d 2 s 0, so that 2 is not a contact 1-form. In fact, 2 is only a 1-form on the 2-dimensional space ⺢ 2 : Ž y, z ., and this result is forced by the presence of the non-zero curvature 2-form d . Thus, the requirement ⌰ s 0 is necessary in order to obtain a contact structure. Gauge Theory Gauge theory has a natural setting in this context. Let Gr Ž m, ⺢. ; GLŽ m, ⺢. be an r-parameter Lie subgroup of m = m matrices; that is, the Lie algebra of Gr is generated by the r matrices ␥␣ ¬ 1 F ␣ F r 4 such that ␥␣ ␥ y ␥ ␥␣ s C␣ ␥ , where C␣ are the structure constants. An element K g Gr acts on a representation space of column matrices ⍀ of length m ´ s K ⍀, where the elements of ⍀ can be k-forms on a manifold M. by ⍀ This action is called global action. If different elements of Gr Ž m, ⺢. are used at different points of M, then Gr Ž m, ⺢. is replaced by the r-parameter Lie gauge group Gr Ž m, ⌳0 . of m = m matrices with values in ⌳0 . This is achieved by maps from M into the group space of Gr Ž M, ⺢. given by K Ž k . s expŽ k ␣␥␣ . ª AŽ f . s expŽ f ␣␥␣ . ᭙ f ␣ g ⌳0 ¬ 1 F ␣ F r 4 , where the
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DOMINIC G. B. EDELEN
ranges of the f ’s are contained in the group space of Gr Ž m, ⺢.. The action of elements of Gr Ž m, ⌳0 . is called local action. The discussion given above shows that there is a natural decomposition of Gr Ž m, ⌳0 . that is given by Gr Ž m, ⌳0 . s Gr Ž m, ⺢. = Gr 0 Ž m, ⌳0 ., where all elements of Gr 0 Ž m, ⌳0 . evaluate to the identity matrix at a point p of M. Thus, Gr 0 Ž m, ⌳0 . carries all information of how the elements of Gr Ž m, ⌳0 . vary from point to point. ´ s dŽ K ⍀ . s K d⍀. For local action of Gr , For global action of Gr , d⍀ ´ Ž . Ž . d⍀ s d A⍀ s A d⍀ q dA n ⍀. This difficulty is overcome by introducing an m = m matrix of gauge connection 1-forms ⌫ s W ␣␥␣ , where W␣ g ⌳1 ¬ 1 F ␣ F r 4 are the compensating 1-forms for the local action of ´ s Ž A⌫ y dA. Ay1 then gives d⍀ ´ q⌫ ´n⍀ ´s Gr . The requirement ⌫ 0. Ž . Ž A d⍀ q ⌫ n ⍀ for all A g Gr m, ⌳ , and hence A again factors on the left. The matrix of curvature 2-forms is defined by ␣ ⌰ s d⌫ q ⌫ n ⌫ s Ž dW ␣ q 12 C W  n W . ␥␣ .
Ž 5.12.
The conditions ⌰ ' 0 are both necessary and sufficient that there exist a gauge transformation generated by an element of Gr 0 Ž m, ⌳0 . such that ´ s 0; that is, W´ ␣ s 0. ⌫ The exterior differential system d⌫ q ⌫ n ⌫ s ⌰,
d⌰ q ⌫ n ⌰ s ⌰ n ⌫
Ž 5.13.
is covariant under the action of the gauge group Gr 0 Ž m, ⌳0 ., and hence Corollary 3.1 shows that there exists a gauge transformation that makes ⌫ antiexact, ⌫ s Ž A y dA . Ay1 ,
X @ s 0,
s w ␣␥␣ ,
X @ w ␣ s 0.
Ž 5.14. The matrix that accomplishes this is the solution of the Riemann᎐Graves integral equation A s E y H Ž W ␣␥␣ A .
Ž 5.15.
and is necessarily an element of Gr 0 Ž m, ⌳0 . because ␥␣ ¬ 1 F ␣ F r 4 spans the Lie algebra of Gr Ž m, ⺢.. The curvature 2-forms are then evaluated in this antiexact gauge by
˜ Ay1 , ⌰ s A⌰
␣ ˜ s d q n s Ž dw ␣ q 12 C ⌰ w  n w . ␥␣ s ˜␣␥␣ .
Ž 5.16.
EXTERIOR DIFFERENTIAL EQUATIONS
611
˜ by the r This has the effect of replacing the m2 2-form entries of ⌰ 2-forms ˜␣ ¬ 1 F ␣ F r 4 . The results given above hold for a C 1-structure, rather than the usual ⬁ C -structure. Thus the only solutions of the equations ˜␣ s 0 in a C 1structure such that X @ w ␣ s 0 are w ␣ s 0; that is, all antiexact gauge connection 1-forms with ¨ anishing cur¨ ature 2-forms in the C 1-structure are tri¨ ial. The importance of this result is as follows. The ‘‘free gauge field equations’’ of any gauge theory Ži.e., the field equations that obtain in the absence of matter fields. are homogeneous differential equations in the entries of the gauge curvature 2-forms. Accordingly, the free gauge field equations are always satisfied by ⌰ s ␣␥␣ s 0, and hence the gauge connection 1-forms can be transformed away for all such solutions in a C p , p G 1 structure. Physicists, however, rely heavily on ‘‘singular’’ solutions of field equations Ži.e., f s 1rr for ⵜ 2 f s 0.. It is therefore useful to construct all super conductor singular solutions of the equations ˜␣ s 0. The term ‘‘super conductor’’ is used because the corresponding situation in electromagnetic field theory has both the electric and the magnetic fields confined to the singularities of the solution. The symbol , will be used to denote equality at all regular points. THEOREM 5.2. equations
All super conductor singular solutions of the gauge field
⌰ s d⌫ q ⌫ n ⌫ , 0, Ž 5.17. 0 0 for an r-parameter Lie gauge group Gr Ž m, ⌳ . ; GLŽ m, ⌳ ., are gi¨ en by
˜ y dA . Ay1 , ⌫ s Ž A⌫
˜ s y␣ Ž h . dh  ␥␣ g ho Ž y1, 1 . Ž 5.18. ⌫
for some A g Gr Ž m, ⺢. = Gr 0 Ž m, ⌳0 . and for some elements h ␣ ¬ 1 F ␣ F r 4 of hoŽ0, 0. such that the range of the h’s is contained in the group space of Gr Ž m, ⺢., where at least one of the h’s is not a constant function and ␣ Ž k . dk  ¬ 1 F ␣ F r 4 are the Maurer᎐Cartan 1-forms for Gr Ž m, ⺢.. Proof. The representation ⌫ s Ž A y dA. Ay1 in any C p structure shows that ⌫ s Ž dA. Ay1 are the only connection 1-forms for which ⌰ s 0. Now, the group equations for Gr Ž m, ⺢. w12, p. 89x, namely dK Ž k . s ␣ Ž k .␥␣ K Ž k ., pull back to the equations dAŽ f . s ␣ Ž f .␥a A under any map from S into the group space of Gr Ž m, ⺢. given by k ␣ s f ␣ Ž x i . ¬ 1 F ␣ F r 4 , and hence Ž dA. Ay1 s ␣ Ž f .␥␣ . This shows that all matrices of connection 1-forms in any C p structure with vanishing curvature matrices are given by ⌫ s yŽ dA. Ay1 s y ␣ Ž f .␥␣ , all of which can be transformed away by appropriately chosen gauge transformations. Since the Maurer᎐Cartan 1-forms ␣ 4 are unique for any given Gr Ž m, ⺢., ⌫ s y ␣ Ž f .␥␣ are the only matrices of connection 1-forms for which the associated curvature 2-forms vanish throughout S Ži.e., ␣ s yd ␣ ␣ q 12 C␥  n ␥ s 0 in view of the Maurer᎐Cartan equations w12x.. In
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DOMINIC G. B. EDELEN
order to construct singular solutions, the assumption of a C p structure has to be relaxed because the only connections that cannot be transformed away by a gauge transformation are those that satisfy 0 s X @ ⌫ s y␣ Ž f . X @ df  ␥ ; that is, X @ df  s 0. The results obtained in Section 2 show that this is the case if and only if f  g hoŽ0, 0. ¬ 1 F  F r 4 . Now, f  g hoŽ0, 0. can always be chosen so that the f ’s have their range in the group space of Gr Ž m, ⺢. because the composition of any smooth function with an element of hoŽ0, 0. is an element of hoŽ0, 0. Ži.e., simply compose hoŽ0, 0. with a function from ⺢ to ⺢ with bounded range, such as arctanŽ⭈. or tanhŽ⭈. with an appropriately chosen constant.. This ˜ s y␣ Ž h. dh  ␥␣ , with h ␣ g hoŽ0, 0. ¬ 1 argument gives the evaluation ⌫ F  F r 4 , where at least one of the h’s is not constant. Now, dh  g hoŽy1, 1. and the fact that hoŽy1, 1. is a module over hoŽ0, 0. shows that ˜ g hoŽy1, 1.. It follows that ⌰ s 0 at all regular points of ⌫, ˜ by noting ⌫ the validity of the Maurer᎐Cartan equations, and hence ⌰ , 0 over S.
˜ s 1␣ Ž h. dh1 ␥␣ with h1 s An interesting example is given by ⌫ 2 2 arctan ln < R q ct < y ln < R y ct <4 , R s x q y 2 q z 2 , on ⺢ 3 = ⺢ with coordinate functions x, y, z, t 4 , and the remaining h’s all constant. This gives super conductor singular solutions that are singular on the entire null cone of Minkowski 4-space with vertex at the origin! More elementary examples are obtained with h1 s arctan Ž yrx . or h1 s tanhŽ yrx ., while outrageous properties are associated with h1 s sinŽ yrx .. A classification of all super conductor singular solutions can be obtained in terms of loop ˜ integrals of ⌫. Let ⍀ be a column matrix of k-forms that forms a representation space for the action of Gr 0 Ž m, ⌳0 ., and let ⍀ be constrained by the requirement that its entries satisfy the differential system d⍀ q ⌫ n ⍀ s ⌺, where ⌫ is a matrix of gauge compensating 1-forms for the local action of Gr 0 Ž m, ⌳0 .. Accordingly, ⌫ has the evaluation ⌫ s Ž Aw ␣␥␣ y dA. Ay1 , w ␣ g A 1 ¬ 1 F ␣ F r 4 , and Theorem 3.1 shows that all such solvable systems have the representation ⍀ s A d q y H Ž w ␣␥␣ n d . 4 ,
Ž 5.19.
⌺ s A d q w ␣␥␣ n q Hd Ž w ␣␥␣ n d . y w ␥ n H Ž w ␣␥␣ n d . 4
Ž 5.20. for any choice of A g Gr 0 Ž m, ⌳0 . and any choice of the antiexact quantities , 4 . If the torsion forms, ⌺, all vanish, then the entries of ⍀ are said to be gauge co¨ ariant constant Ži.e., d⍀ q ⌫ n ⍀ s 0.. In this event s H ⌺ s 0, and the available choices of d are limited by the con-
EXTERIOR DIFFERENTIAL EQUATIONS
613
straints Hd Ž w ␣␥␣ n d . s w ␥ l H Ž w ␣␥␣ n d . .
Ž 5.21.
REFERENCES 1. D. G. B. Edelen, ‘‘Lagrangian Mechanics of Nonconservative Nonlolonomic Systems,’’ Noordhoff, Leyden, 1977. 2. D. G. B. Edelen, ‘‘Isovector Methods for Equations of Balance,’’ Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands, 1980. 3. D. G. B. Edelen, ‘‘Applied Exterior Calculus,’’ Wiley, New York, 1985. 4. D. G. B. Edelen, A metric free electrodynamics with electric and magnetic charges, Ann. Phys. Ž N.Y.. 122 Ž1978., 366᎐400. 5. D. G. B. Edelen and D. C. Lagoudas, ‘‘Gauge Theory and Defects in Solids,’’ North-Holland, Amsterdam, 1988. 6. E. Cartan, ‘‘Les systemes differentiels exterieurs et leurs applications geometriques,’’ Hermann, Paris, 1945. 7. R. L. Bryant et al., ‘‘Exterior Differential Systems,’’ Springer-Verlag, Berlin, 1991. 8. P. A. Griffiths, ‘‘Exterior Differential Systems and the Calculus of Variations,’’ Birkhauser, ¨ Boston, 1983. 9. D. G. B. Edelen, Explicit solutions of a class of exterior differential systems, Quaestiones Math. 20 Ž1997., 1᎐15. 10. D. G. B. Edelen and J. Wang, ‘‘Transformation Methods for Nonlinear Partial Differential Equations,’’ World Scientific, Singapore, 1992. 11. E. Cartan, Les systems ` de Pfaff `a cinq variables et les ´equations aux derivees ´ ´ partielles du ´ second ordre, Ann. Sci. Ecole Norm. Sup. Ž 3 . 27 Ž1910., 109᎐192. 12. F. W. Warner, ‘‘Foundations of Differentiable Manifolds and Lie Groups,’’ SpringerVerlag, Berlin, 1983.
Solutions of Exterior Differential Equations by Quadratures Dominic G. B. Edelen 3503 A¨ enue P, Gal¨ eston, Texas 77550 Submitted by William F. Ames Received April 11, 2000
Homotopy operators are used to obtain unconstrained local solutions of systems of exterior differential equations by quadratures. An analysis is given of when given systems of constraints on the solutions can be realized. The results are applied to several classes of classical problems. A collection of superconductor singular solutions of the free gauge field equations for matrix Lie groups is also obtained, and the general forms of gauge covariant constant fields are given. 䊚 2000 Academic Press
1. INTRODUCTION Poincare ´ defined, in effect, a linear homotopy operator, H, on any star shaped region of a differentiable manifold, and showed that it verified the identity s dH q H d for any smooth differential form of positive degree; whence, the Poincare ´ Lemma: any closed form Ž d s 0. is locally exact Ž s dH .. There the matter sat, for all intent and purposes. This was because the differential geometry community centered most of its efforts on determining the obstructions to global extensions of the Poincare ´ Lemma, for such obstructions were show to contain essential intrinsic information about the global nature of a differentiable manifold. There is, however, a significant amount of additional local information that can be obtained from the Poincare ´ identity, as shown in w1᎐5x. Since H is a Žlocal. linear operator, it has a well defined kernel. If is any exterior differential form in the kernel of H Ž H s 0., then the Poincare ´ identity gives s H d ; that is, the linear homotopy operator H in¨ erts the exterior deri¨ ati¨ e operator d for any exterior differential form in kerŽ H .. Thus, H is the exterior calculus analog of the indefinite integral of the 592 0022-247Xr00 $35.00 Copyright 䊚 2000 by Academic Press All rights of reproduction in any form reserved.
EXTERIOR DIFFERENTIAL EQUATIONS
593
ordinary calculus, albeit H is only defined locally. Since s dH q H d , elements in kerŽ H . are referred to as antiexact differential forms. The ability to invert the operator d provides access to procedures for solving systems of exterior differential equations by quadratures and to related matters. The analysis given here concentrates on the structure of the solutions of exterior differential equations, in contrast to the submanifolds on which such solutions vanish, the latter being the topic of primary interest in differential geometry. A terse proof of the Poincare ´ identity and an analysis of the properties of kerŽ H . are given in Section 2. General quadrature solutions of local differential systems, as developed in w1᎐3x, and an important corollary are given in Section 3. Differential systems with explicitly given constraints are analyzed in Section 4. Some classical problems, of importance in the study of the geometry of solutions of nonlinear partial differential equations and in the calculus of variations, are analyzed in Section 5. The use of antiexact forms in general gauge theories is presented in Section 6, where a class of ‘‘super conductor singular’’ solutions of any system of free gauge field equations is constructed. Let M be an n-dimensional differentiable manifold and let ⌳ be the graded module of exterior differential forms over M. The term ‘‘module’’ is taken to mean the module over the algebra ⌳0 of forms of degree zero. The usual case is where ⌳0 is the algebra of C⬁ functions, and this will be assumed unless explicitly stated to the contrary. If ⌳0 is taken to be the algebra of C k functions, then ⌳ is said to have a C k structure as a graded module. The submodule of ⌳ of exterior differential forms of degree k is denoted by ⌳k . The subspace of elements of ⌳k with coefficients that are homogeneous functions of degree h is denoted by hoŽ h, k .. Finally, let GLŽ m, alg. be the group of nonsingular m = m matrices with values in the associative, commutative algebra ‘‘alg,’’ the usual cases being GLŽ m, ⺢. and GLŽ m, ⌳0 .. The group GLŽ m, alg. is obtained by the exponential map of all m = m matrices with values in alg.
2. HOMOTOPY OPERATORS AND THE ALGEBRAIC DECOMPOSITION OF ⌳ If p is any point in M, then there exists a neighborhood UŽ p . of p and a system of local coordinate functions x i ¬ 1 F i F n4 such that each of the coordinate functions has the value 0 at p. Let S be a starshaped subset of UŽ p . with respect to the homotopy h : S ª S ¬ y i s x i ,
0 F F 1.
Ž 2.1.
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DOMINIC G. B. EDELEN
The discussion will be restricted, from now on, to the manifold S, so that all results will be local results with respect to M, but global with respect to S. In particular, ⌳ will now be taken to be the module of exterior differential forms over S. A generalized ‘‘radius’’ vector field on S is defined by X s x ⭸i i
Ž 2.2.
with respect to the coordinate cover x i 4 , where ⭸ i ¬ 1 F i F n4 is a basis for the module of derivations of ⌳0 and the summation convention is assumed. A linear homotopy operator H is defined on ⌳ by H s X @
1
H0
Ž hU .
d
,
Ž 2.3.
where X @  stands for the inner multiplication of a differential form  by a vector field X. The integral in Ž2.3. is a Riemann᎐Graves integral, and hence it is well defined for any form of positive degree in a C 0 structure. Noting that X @ H01 Ž hU . d s H01 X @Ž hU . d s 0 for g ⌳0 , it follows that H is well defined on a C 0 structure. In fact, since hU Ž dy i . s dx i and hU Ž f Ž y j .. s f Ž x j . s r f Ž x j . for any f g hoŽ r, 0., by Euler’s theorem for homogeneous functions, a direct evaluation gives H s
1 kqr
X @  g ho Ž r q 1, k y 1 .
᭙ g ho Ž r , k . .
Ž 2.4.
Let ⌽ be a diffeomorphism of M. Then ⌽ induces a homotopy operator H⌽ on the image of S under ⌽ in M by ⌽U H⌽ s H ⌽U .
Ž 2.5.
A combination of Ž2.3. and Ž2.5. yields the evaluation H⌽ s Ž ⌽# X . @
y1 U
1 H0 Ž ⌽ ( h ( ⌽ .
d
Ž 2.6.
on the image of S in M under the action of ⌽. Restricting ⌽ to a diffeomorphism of S to S, Ž2.6. shows that a homotopy operator is defined for any coordinate cover of S that reduces to H for the coordinate cover x i 4 . Accordingly, properties established for H with the coordinate co¨ er x i 4 continue to hold for all coordinate co¨ ers of S with the homotopy operators gi¨ en by Ž2.6.. There is thus an abundance of linear homotopy operators that can be defined on ⌳Ž S .. It is, however, sufficient to restrict attention to the linear homotopy operator H that is defined by Ž2.3..
EXTERIOR DIFFERENTIAL EQUATIONS
595
The parameter transformation s e applied to the homotopy Ž2.1. gives the homotopy TX Ž . : S ª S ¬ y i s e x i ,
y⬁ F F 0,
Ž 2.7.
where TX Ž . is the flow operator on the orbits of the vector field X. It is then a simple matter to see that hUexpŽ . s expŽ £ X . , y⬁ F F 0, and hence Ž2.3. gives H s X @
⬁
0
Hy⬁ expŽ £
X
. d s X @ H exp Ž y £ X . d .
Ž 2.8.
0
These equivalent evaluations of H will be useful in the proof of the Poincare ´ identity. LEMMA 2.1. The homotopy operator H is a linear map from ⌳k to ⌳ky 1 that ¨ erifies the identity
s dH q H d
Ž 2.9.
for all positi¨ e k, while Hf s 0 and f Ž x i . s H df q f Ž 0 .
Ž 2.10.
for k s 0. Proof. That H is a map from ⌳k to ⌳ky1 follows directly from Ž2.3. and the fact that the operator V @ has this property for any vector field V. Use of the definition Ž2.3. gives H d s X @ H01 Ž hU d . d s X @ dH01 Ž hU . d , and hence dH q H d s £ X
1
H0
Ž hU .
d
,
Ž 2.11.
where £ is the symbol for Lie differentiation. When the alternative 0 evaluation H01 Ž hU . d s Hy⬁ expŽ £ X . d is used, Ž2.11. gives dH q H d s
0
d
Hy⬁ d Žexp Ž £
s y lim
ªy⬁
X
. . d
Ž exp Ž £ X . . .
Ž 2.12.
For any form of positive degree, the limit vanishes, while for any f g ⌳0 , it evaluates to f Ž0.. That Hf s 0 is true follows directly from rewriting Ž2.3.
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DOMINIC G. B. EDELEN
in the equivalent form H s
1
H0
X @Ž hU .
d
and noting that X @ applied to any element of ⌳0 evaluates to zero. The definition Ž2.3. of H and Lemma 2.1 can be used to show that the operator H exhibits the following properties Žsee w3, p. 177x.: HH ' 0, H dH s H , H dH d s Hd, X @ H ' 0,
Ž 2.13.
dH d s d,
Ž 2.14.
dH dH s dH,
Ž 2.15.
HX @ ' 0.
Ž 2.16.
Thus, H is nilpotent, and dH and Hd are projection operators. Since H is a linear operator on ⌳, it has a well defined kernel, A s g ⌳ ¬ H s 0 4 ,
Ž 2.17.
and hence A is the direct sum of its homogeneous subspaces A k ¬ 0 F k F n4 . Noting that H⌳0 ' 0, it follows that A 0 s ⌳0 . It is then an easy matter to show that Ž2.3. implies A s g ⌳ ¬ X @ s 04 ,
Ž 2.18.
and that A n s 0. Further, since X @ is an antiderivation on ⌳, Ž2.8. shows that A p n A q ; A pq q, A k q A k ; A k , and hence A is a submodule of ⌳ over ⌳0 . Thus, the characterization of A is strictly algebraic! Since HH ' 0, H gi¨ es rise to the algebraically generated exact sequence i
H
H
H
H
H
0 ª ⌳n ª ⌳ny1 ª ⭈⭈⭈ ª ⌳1 ª ⌳0 ª 0,
Ž 2.19.
where i stands for the natural inclusion. Let E be the subspace of ⌳ consisting of exact exterior differential forms. The identity Ž2.9. has the equivalent statement ⌳k s E k q A k ,
Ž 2.20.
with the projection relations E s dH ⌳ ,
A s H d⌳ ,
Ž 2.21.
and the identification E n s ⌳n. Accordingly, any element of A is referred to as an antiexact differential form.
EXTERIOR DIFFERENTIAL EQUATIONS
597
THEOREM 2.1. The additi¨ e splitting of ⌳ gi¨ en by Ž2.20. is strict in any C p-structure, for p G 0, and hence ⌳ s E [ A.
Ž 2.22.
In addition, H is the in¨ erse of d for any antiexact differential form, and E can be defined algebraically as kokerŽ H .. Proof. Noting that A 0 s ⌳0 , E 0 s 0, A n s 0, E n s ⌳n, it is only necessary to establish that the splitting Ž2.20. is strict for forms of degree k with 1 F k - n. Assume that g E k . Then s d for some g ⌳ky 1. However, s e q a where e s dH is the exact part of and a s H d is the antiexact part of . Accordingly, s da with a g A ky 1. In order that the belong to A, it is necessary that 0 s X @ s X @ da. However, £ X A ' X @ d A q dŽ X @ A . ' X @ d A, and hence we have the requirement £ X a s 0. This shows that the requirements
s da,
£ X a s 0, a g A ky 1 ,
Ž 2.23.
are satisfied by any element in E k l A k . Hence, a necessary and sufficient condition for the additive decomposition Ž2.20. to be strict is that Ž2.23. is satisfied only by the zero element of A k . Now, £ X is a derivation, and hence it is only necessary to compute the value of £ X on all elements of A 0 s ⌳0 and A 1. For k s 1, a s f Ž x i . g ⌳0 , and hence £ X a s X @ da s 0 is satisfied by all a g hoŽ0, 0., by Euler’s theorem. However, the only bounded, single-valued elements of hoŽ0, 0. are the constant functions, and hence s da s 0 for the case k s 1. For k s 2, a s f i Ž x k . dx i g A 1 for x i f i Ž x k . s 0, and hence 0 s £ X a s Ž f i q X @ df i . dx i yields the requirements X @ df i q f i s 0, 1 F i F n. Euler’s theorem on homogeneous functions shows that these conditions are satisfied only when f i Ž x k . g hoŽy1, 0., 1 F i F n. Since the only bounded element of hoŽy1, 0. is the zero function, it follows that a s 0 and hence s da s 0. An examination of the part of the above proof after Ž2.23. shows that £ X A k s 0 only for elements of hoŽyk, k ., and hence the only bounded, single-valued elements of A that satisfy £ X s 0 in a C p-structure with p G 0 are the constant elements of ⌳0 s A 0 . The constant elements of ⌳0 are, however, precisely those elements of ⌳0 that are closed but not exact. A complete account of the structure of E k l A k is only possible in the wider class of forms whose coefficients have singularities, E k l A k s d Ž X @ ho Ž yk, k . . ; ho Ž yk, k . . For example, ␣ s xy2 Ž x dy y y dx . s dŽ xy . ; hoŽy1, 1. is in E 1 l A 1 , but is singular on x s 0, while  s 2 zy3 Ž z dx n dy y dz n Ž x dy y y dx .. s dŽ zy2 Ž x dy y y dx .. ; hoŽy2, 2. is in E 2 l A 2 but is singular on z s 0.
598
DOMINIC G. B. EDELEN
3. GENERAL QUADRATURE SOLUTIONS OF LOCAL DIFFERENTIAL SYSTEMS Let ⍀ a g ⌳k ¬ 1 F a F m, k ) 04 be m k-forms on M Žnot necessarily independent.. They generate an associated ideal Ik s I ⍀ 1 , . . . , ⍀ m 4 of ⌳ over M. If the ⍀’s are known k-forms, then there is little left to do, other than to obtain a basis for Ik ; that is, to find a minimal set of k-forms that generates the same ideal. Differential systems come into play when the ⍀’s are to be determined by solving a system of exterior differential equations d⍀ a q ⌫ba n ⍀ b s ⌺ a , where the ⌫’s are 1-forms and the ⌺’s are Ž k q 1.-forms on M. For k s 1, ⍀ a s ia dx i, ⌫ba s ␥ bai dx i, 2 ⌺ a s i aj dx i n dx j, i aj s yjia, 1 F a, b F m, 1 F i, j F n, this is the system of partial differential equations
⭸ ja ⭸ xi
y
⭸ ia ⭸xj
q ␥ bai jb y ␥ baj ib s i aj .
The importance of differential systems was made evident by the pioneering work of E. Cartan w6x and continues to this day w7, 8x. Now, there may or may not be m independent solutions of the given system of exterior differential equations, and hence the generators of the associated ideal Ik can not be assumed to be independent a priori. EXAMPLE 1. The differential system d a q Ž x dy q y dx .G ba n b s 0 on ⺢ 5, with k s 1, m s 3, ŽŽ G ba .. a nonsingular, constant-valued 3 = 3 matrix, and coordinate cover x, y, z, u, ¨ 4 , only has solutions of the form
a s d a Ž x, y . y G ba H Ž Ž x dy y y dx . n d b Ž x, y . . , for any choice of the a Ž x, y . g ⌳0 Žsee w9x.. Thus, the associated ideal I a ¬ 1 F a F 34 can have at most two independent generators. The arguments to follow are significantly simplified by use of matrix notation. Let ⍀ denote the column matrix whose entries are the k-forms ⍀ a ¬ 1 F a F m4 . In order to retain the fact that ⍀ is a column matrix with m entries, ⍀ will be referred to as an element of length m. Let ⌫ s ŽŽ ⌫ba .. be an m = m matrix of 1-forms, let ⌺ be a column matrix of Ž k q 1.-forms of length m, and let ⌫ n ⍀ s ŽŽ ⌫ba n ⍀ b .. Ži.e., matrix multiplication with the exterior product.. The differential system can then be written as a matrix system of degree k and length m: d⍀ q ⌫ n ⍀ s ⌺. The associated ideal is written Ik s I ⍀ 4 .
Ž 3.1.
EXTERIOR DIFFERENTIAL EQUATIONS
599
Although the generators of Ik are not necessarily independent, there is a natural equivalence relation for the generators. Let A g GLŽ m, ⌳0 .. Then Ik s I ⍀ 4 s I A⍀ 4 , because Ik is a module over ⌳0 . However, GLŽ m, ⺢. is a subgroup of GLŽ m, ⌳0 ., and any element of GLŽ m, ⺢. factors through Ž3.1. in the obvious manner Ži.e., dK ⍀ s K d⍀ for any K g GLŽ m, ⺢... Let p be a point in M and let GL p Ž m, ⌳0 . be the subgroup of GLŽ m, ⌳0 . whose elements evaluate to the identity matrix E at p. There is a natural projection P: GLŽ m, ⌳0 . ª GL p Ž m, ⌳0 . ¬ B˜ s B Ž p .y1 B such that P ŽGLŽ m, ⺢.. s E and GLŽ m, ⌳0 . s GLŽ m, ⺢. = GL p Ž m, ⌳0 .. Thus, as far as the differential system Ž3.1. is concerned, only elements of GL p Ž m, ⌳0 . can change things Ži.e., A g GL 0 Ž m, ⌳0 . and ˜ will be said to be ideal dA s 0 implies A s E .. Two elements ⍀ and ⍀ ˜ s A⍀ for some A g GL p Ž m, ⌳0 ., in which case equi¨ alent if and only if ⍀ they are necessarily equivalent for any B g GLŽ m, ⌳0 .. The notion of ideal equivalence is of particular importance in the study of differential systems that generate ideals of ⌳. The differential system Ž3.1. obviously entails integrability conditions. Exterior differentiation and elimination in the standard manner leads to the complete differential system of length m and degree k,
Ž 夹.
d⍀ q ⌫ n ⍀ s ⌺, d⌫ q ⌫ n ⌫ s ⌰,
d⌺ q ⌫ n ⌺ s ⌰ n ⍀ , d⌰ q ⌫ n ⌰ s ⌰ n ⌫.
Ž 3.2. Ž 3.3.
By analogy with the Cartan equations of structure, ⌫ is referred to as the matrix of connection 1-forms, ⌰ is referred to as the matrix of cur¨ ature 2-forms, and ⌺ is referred to as the matrix of torsion Ž k q 1.-forms of the differential system Ž夹.. These connection, curvature, and torsion forms are not necessarily those of a differentiable manifold. Note that the structure of the connection-curvature equations Ž3.3. is universal for all differential systems of length m, for they always involve forms of degrees one and two only. An elementary calculation shows that the ideal equivalence transformation of a differential system induced by A g GL 0 Ž m, ⌳0 . is given by ⍀ s A⍀ ,
⌺ s A⌺,
⌰ s A⌰ Ay1 ,
⌫ s Ž A⌫ y dA . Ay1 ,
in which case the ‘‘barred’’ quantities satisfy the complete differential system d⍀ q ⌫ n ⍀ s ⌺,
d⌺ q ⌫ n ⌺ s ⌰ n ⍀ ,
d⌫ q ⌫ n ⌫ s ⌰,
d⌰ q ⌫ n ⌰ s ⌰ n ⌫.
600
DOMINIC G. B. EDELEN
The original complete differential system Ž夹. and the new one have exactly the same structure, which is a property that will prove to be essential in what follows. A system of relations is said to be ideal equi¨ alence co¨ ariant if and only if its structure is invariant under all ideal equivalence transformations. Thus, any complete differential system is ideal equi¨ alence co¨ ariant. Constraints such as ⌰ s 0 or ⌺ s 0 are thus ideal equivalence covariant systems of conditions, while no explicit evaluation of ⌫ is ideal equivalence covariant due to the presence of the inhomogeneous terms Ž dA. Ay1 . Now, everything depends on which quantities in a differential system are given and which are to be calculated from that given system. Suppose that a differential system can be solved for all of the quantities ⍀, ⌺, ⌫, ⌰4 simultaneously. All such solutions will be referred to as unconstrained solutions. The question of the existence of solutions of the differential system with given data, and the construction of such solutions, can then be treated by considering the given data as a system of constraints that the unconstrained solutions must satisfy. This is the approach taken here. Any point p of a differentiable manifold M has a neighborhood UŽ p . and a system of coordinate functions x i ¬ 1 F i F n4 such that the neighborhood S ; UŽ p . is starshaped with respect to the homotopy h discussed in the previous section. Consideration will now be restricted to S, so that all results will be local. In particular, ⌳ will now be taken to mean the exterior algebra of differential forms over S. The direct sum decomposition ⌳ s E [ A and H d ␣ s ␣ for all ␣ g A established in the previous section can be used to construct all unconstrained solutions of a complete differential system Ž夹. by quadratures. The classic terminology is adopted here; namely, a solution is obtained by quadratures if and only if it can be written in terms of known functions or the integrals of known functions ŽRiemann᎐Graves integrals in this case.. The proof of the following theorem is given in w3, Chap. 5x. Žcf. Theorem 5-15.1.. THEOREM 3.1 ŽLocal Quadrature Theorem.. The general, unconstrained solution of a differential system Ž夹. of length m and degree k, on a starshaped neighborhood S with homotopy operator H, is gi¨ en by ⍀ s A d q y H Ž n d . 4 ,
Ž 3.4.
⌺ s A d q n q Hd Ž n d . y n H Ž n d . 4 , y1
⌫ s A y dA4 A
,
y1
⌰ s A d q n 4 A
Ž 3.5. Ž 3.6.
for any choice of A g GL 0 Ž m, ⌳ ., for any choices of the column matrices of antiexact forms , of length m of appropriate degrees, and for any choice of the m = m matrix of antiexact 1-forms . Under these circumstances, the
601
EXTERIOR DIFFERENTIAL EQUATIONS
following relations are satisfied throughout S,
s H Ž Ay1 ⍀ . ,
s H Ž Ay1 ⌺ . ,
s H Ž Ay1 ⌰ A . ,
Ž 3.7.
and A satisfies the system of matrix Riemann᎐Gra¨ es integral equations A s E y H Ž ⌫A . .
Ž 3.8.
This is all that can be done if consideration is restricted solely to the differential system Ž夹.. If Ik is the ideal of ⌳ that is generated by ⍀, then ideal equivalence can be used to remove the occurrences of the attitude matrix A. The previously noted relations that are induced by an ideal equivalence transformation ⍀ s A⍀, and the fact that A s E y H Ž ⌫A. has the solution A s E for any ⌫ with antiexact entries, lead to the following useful result. COROLLARY 3.1. E¨ ery differential system is ideal equi¨ alent to a differential system with a connection matrix of antiexact 1-forms. If Ik is the ideal of ⌳ that is generated by solutions ⍀ of the differential system Ž夹., then Ik is ˜ with generated by ⍀
˜ s d q y H Ž n d . , ⍀
Ž 3.9.
˜⌺ s d q n q Hd Ž n d . y n H Ž n d . ,
Ž 3.10.
˜s, ⌫
˜ s d q n , ⌰
Ž 3.11.
for the same choice of the matrices of , , of antiexact forms. The evaluations ⍀ s A d q y H Ž n d .4 , s H Ž Ay1 ⌺ ., ⌰ s A d q n 4 Ay1 show that accounts for the presence of nontrivial torsion Ž k q 1.-forms while H Ž n d . accounts for the presence of nontrivial curvature 2-forms. This evaluation of ⍀ shows that the situation in which s H Ž n d . is thus of interest because ⍀ then has the simplest possible form ⍀ s A d . The easiest way to analyze this problem is to use Corollary 3.1 to transform to an ideal equivalent differential ˜ has only antiexact entries. Use of Ž3.9. through Ž3.11. system for which ⌫ ˜ s d and ˜⌺ s HdŽ n d . q shows that this is the case only when ⍀ ˜ n ⍀, ˜ that is, when ˜⌺ s 0 mod Ik . For k s 1, dH Ž n d . s n d s ⌫ this is the converse of the Frobenius theorem, although there is no restriction on the value of k in the discussion just given. THEOREM 3.2 ŽFrobenius Theorem for k-Forms.. Let Ik be an ideal of ⌳ that is generated by m F Ž Nk . k-forms. If Ik is the closed ideal and the differential system generated by Ik has ¨ anishing cur¨ ature 2-forms, then Ik is generated by m exact k-forms.
602
DOMINIC G. B. EDELEN
Proof. Let ⍀ be a column matrix of generators of Ik of length m. Since Ik is closed, by hypothesis, ⍀ satisfies a differential system of the form d⍀ q ⌫ n ⍀ s 0. Now, the conditions d Ik ; Ik and ⌰ s 0 are ideal equivalence covariant conditions. A transformation to the ideal equivalent system indicated in Corollary 3.1 thus yields the simplified ˜ qn⍀ ˜ s 0. The associated curvature 2-forms are ⌰ ˜s conditions d⍀ ˜ s 0 implies s 0. Accordingly, d q n , and hence the hypothesis ⌰ Ž3.10. shows that the evaluation of ˜ ⌺s0 ⌺ reduces to ˜ ⌺ s d , and hence ˜ ˜ can always be achieved because s H ⌺ s 0. The evaluation Ž3.9. thus ˜ s d , and hence all of the entries of ⍀ ˜ are exact k-forms. Since yields ⍀ there are no further constraints that the must satisfy, the entries of d can be chosen to be independent because m F Ž Nk .. Example 1, with k s 1 and ⌰ / 0, shows that the ideal equivalence covariant conditions ⌰ s 0 are necessary, even in the case k s 1. However, an elementary calculation and Cartan’s lemma show that in the case k s 1, the requirement Žconstraint . that the entries of d be independent implies ⌰ s 0. That this is not the case for k ) 1 can be seen as follows. LEMMA 3.1. The relations H Ž n d . s 0 can be satisfied only if s 0 or the entries of satisfy the constraints n £ X s 0. Proof. The relation H Ž n d . s 0 can hold only if n d belongs to kerŽ H .. This is the case only when 0 s X @Ž n d . s y n X @ d s y n £ X , because the entries of are antiexact forms. Calculation shows that £ X is an automorphism of A k . The problem thus becomes an algebraic one; namely, find all column matrices  g A ky 1 of length m such n  s 0 and then solve the quasilinear partial differential equations £ X s  with the same principal part such that X @ s 0.
4. CONSTRAINED SOLUTIONS OF LOCAL DIFFERENTIAL SYSTEMS The effects of constraints that obtain by specification of connection or curvature forms will first be investigated. Let ⌫ be a given m = m matrix of 1-forms on S. In this event A can be determined Žpossibly on a smaller open subset of S . by solving the Riemann᎐Graves integral equations A s E y H Ž ⌫A. Žsimply iterate and check that the sequence is dominated by a corresponding exponential series.. Once A is known, the first of Ž3.6. gives s Ay1 Ž ⌫A q dA.. It thus remains to check that is antiexact. This is the case if and only if 0 s X @ . Accordingly, the relation X @Ž ⌫A q dA. s 0 must be satisfied. Exterior differentiation of A s E y H Ž ⌫A.
603
EXTERIOR DIFFERENTIAL EQUATIONS
yields dA s ydH Ž ⌫A., and hence ⌫A q dA s ⌫A y dH Ž ⌫A. s HdŽ ⌫A. because dH q Hd s identity. A combination of these relations shows that X @Ž ⌫A q dA. s 0, and thus is antiexact. In this event, the second part of Ž3.6. yields ⌰ s A d q n 4 Ay1 . THEOREM 4.1. If the matrix of connection 1-forms ⌫ is assigned, then the attitude matrix A g GL 0 Ž m, ⌳ . and the matrix of antiexact 1-forms are uniquely determined by A s E y H Ž ⌫A . ,
s Ay1 Ž ⌫A q dA . ,
in which case ⌰ s d⌫ q ⌫ n ⌫ ser¨ es to determine the matrix of cur¨ ature 2-forms. Next, assume that ⌰ is a given m = m matrix of 2-forms on S. Not every matrix of 2-forms is a possible choice because ⌰ must satisfy the cur¨ ature equations d⌰ q ⌫ n ⌰ s ⌰ n ⌫ for some connection matrix ⌫. In fact, Theorem 3.1 shows that the attitude matrix A and s H Ž Ay1 ⌰ A. serve to determine an admissible matrix of curvature 2-forms uniquely through the relations ⌰ s AŽ d q n . Ay1 . Since the attitude matrix A is not determined by a matrix of curvature 2-forms, we are free to choose any A g GL 0 Ž m, ⌳0 .. With each such choice of A, the m = m matrix A s H Ž Ay1 ⌰ A. g A 1 is well defined. An admissible matrix of curvature 2-forms is thus defined by ⌰A s AŽ dA q A n A . Ay1 with the property that H Ž Ay1 ⌰ A. s H Ž Ay1 ⌰A A.; that is, Ay1 ⌰ A and Ay1 ⌰A A have the same exact part dA . The subscript A has been attached in order to point out that all such quantities are dependent on the choice of the attitude matrix A. A direct calculation shows that ⌰A s ⌰ y A Hd Ž Ay1 ⌰ A . y H Ž Ay1 ⌰ A . n H Ž Ay1 ⌰ A . 4 Ay1 . THEOREM 4.2. Let ⌰ be a gi¨ en m = m matrix of 2-forms on S. For each choice of the attitude matrix A g GL 0 Ž m, ⌳ ., there is an associated matrix of Ž compatible. cur¨ ature 2-forms ⌰A s A dH Ž Ay1 ⌰ A . q H Ž Ay1 ⌰ A . n H Ž Ay1 ⌰ A . 4 Ay1
Ž 4.1.
for which Ay1 ⌰ A and Ay1 ⌰A A ha¨ e the same exact part, and
A s H Ž Ay1 ⌰ A . s H Ž Ay1 ⌰A A . ,
⌫A s AA y dA4 Ay1 . Ž 4.2.
In particular, if ⌰ s 0, then A s 0 and there are infinitely many associated matrices of connection 1-forms ⌫A s yŽ dA. Ay1 , for all A g GL 0 Ž m, ⌳ .. What constraints are imposed by specification of the torsion Ž k q 1. forms? Clearly, constraints do arise because any ⌺ must satisfy the torsion
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DOMINIC G. B. EDELEN
equations d⌺ q ⌫ n ⌺ s ⌰ n ⍀. Let ⌺ be a given column matrix of Ž k q 1.-forms of length m. Then is evaluated by s H Ž Ay1 ⌺ ., and hence Ž3.5. gives the requirements Ay1 ⌺ s dH Ž Ay1 ⌺ . q n H Ž Ay1 ⌺ . q Hd Ž n d . y n H Ž n d . . THEOREM 4.3. If the torsion Ž k q 1.-forms are specified, then the matrices of antiexact quantities A, , and must satisfy the Ž essentially nonlinear . system of m constraints Ay1 ⌺ s dH Ž Ay1 ⌺ . q n H Ž Ay1 ⌺ . q Hd Ž n d . y n H Ž n d . .
Ž 4.3.
These nonlinear constraints are what often make problems with specified torsion very difficult. If the further constraints of vanishing curvature are imposed, then s 0 and Ž4.3. reduces to the requirements Ay1 ⌺ s dH Ž Ay1 ⌺ .; that is, if the cur¨ ature 2-forms of a differential system all ¨ anish, then the only torsion Ž k q 1.-forms that can be specified are of the form ⌺ s B d for some B g GLŽ m, ⌳ . and some g ⌳ky 1 , in which case A s B Ž0.y1 B. The following result is also immediate from Ž4.3.. If the constraints ⌺ s 0 are imposed, then and must satisfy the additional constraints HdŽ n d . s n H Ž l d .. LEMMA 4.1. The general solution of the relations H d s n H
Ž 4.4.
is gi¨ en by
s d q n ,
gA
Ž 4.5.
˜ n . in which case will necessarily satisfy the conditions d q n s ⌰ Proof. The homotopy decomposition gives s d q a , where and a are antiexact. Thus, H d s H d a s n H s n . The result then follows on noting that H d a s a for any antiexact a . The integrability ˜ n . conditions for Ž4.5. are d q n s ⌰ THEOREM 4.4. If ⌰ / 0, then the constraints of ¨ anishing torsion can be realized only when n d satisfy the constraints
n d s d q n
Ž 4.6.
for some matrix of antiexact k-forms of length m, in which case
˜ s d y , ⍀
˜ n Ž d y . s 0. ⌰
Ž 4.7.
EXTERIOR DIFFERENTIAL EQUATIONS
605
Proof. The remarks following Theorem 4.3 show that the constraints ⌺ s 0 can be satisfied only when and satisfy the constraints HdŽ n d . s n H Ž n d .. Use of Lemma 4.1 with s n d gives the requirements Ž4.6.. When these requirements are met, s H ˜ ⌺ s 0, H Ž n d . s H Ž d q n . s H d s because both and are column matrices of antiexact forms. A substitution of these evaluations into Ž3.9. gives Ž4.7.. The following theorem is of use for problems in which the torsion Ž k q 1.-forms do not vanish. THEOREM 4.5 ŽTorsion Transformation Theorem.. Let be a column matrix of Ž k y 1.-forms of length m that transforms under ideal equi¨ alence transformations ⍀ s A⍀ of Ik by s A . Then the torsion transformation
˜ q d q ⌫ n ⍀s⍀
Ž 4.8.
maps the complete differential system Ž夹. into the ideal equi¨ alence co¨ ariant differential system
˜ q⌫n⍀ ˜ s ˜⌺, d⍀
˜ q ⌫ l ˜⌺ s ⌰ n ⍀ ˜, d⌺
Ž 4.9.
d⌫ q ⌫ n ⌫ s ⌰,
d⌰ q ⌫ l ⌰ s ⌰ n ⌫,
Ž 4.10.
with the same connection and cur¨ ature forms, while the new torsion forms are gi¨ en by
˜⌺ s ⌺ y ⌰ n .
Ž 4.11.
Proof. The relations Ž4.9. through Ž4.11. follow directly by substitution of the relations Ž4.8. into the differential system Ž夹.. Since the original system was ideal equivalence covariant and d q ⌫ n is ideal equivalence covariant under the hypothesis s A , the resulting differential system Ž4.9., Ž4.10. is also ideal equivalence covariant. If ⌺ has the ideal equivalence covariant evaluation ⌺ s ⌰ n for some column matrix of Ž k y 1.-forms of length m, then the complete ˜ q d q ⌫ n for any ⍀ ˜ that differential system Ž夹. is satisfied by ⍀ s ⍀ ˜ q⌫n⍀ ˜ s 0. Thus, satisfies the complete differential system generated by d⍀ if ⌰ is assigned, the only generality left in the solutions comes from all ˜ of the corresponding complete differential system with given solutions ⍀ curvature 2-forms and vanishing torsion k-forms. This representation is ˜ s 0 is always a solution. not vacuous because ⍀
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DOMINIC G. B. EDELEN
5. SOME CLASSICAL PROBLEMS The question of when does a differentiable manifold admit a contact structure, ⍀ a s dq a y ria d i g ⌳1 , 1 F a F m, summation on i from 1 through r 4 , was solved by Bryant et al. w7x for m / 2 by analytic methods. Contact structures are important, for they provide a natural setting for problems in nonlinear differential equations and in the calculus of variations w2; 3, Chaps. 5 and 6; 7; 8; 10x. Since q a, ria, i 4 all belong to ⌳0 and ⌳0 s A 0 , the restriction to forms of degree one tends to confuse the issue as far as the differential structure is concerned. The problem will thus be analyzed for forms of degree k G 1, and then cut down to k s 1 later. Let q, ˜ ˜ri ¬ 1 F i F r 4 be column matrices of Ž k y 1.-forms of length m ˜ s dq˜ y ˜ri and define a column matrix of contact k-forms of length m by ⍀ n d i, where i g ⌳0 ¬ 1 F i F r 4 such that d 1 n ⭈⭈⭈ n d r / 0. The ˜ 4 is thus well defined. Any other collection of contact ideal Ck s I ⍀ ˜ s AŽ dq˜ y ˜ri n d i . / dq y ri n generators of Ck is given by ⍀ s A⍀ i d , which shows that the structure dq y ri n d i is not ideal equivalence covariant. In order to overcome this difficulty, set q s Aq, ˜ ri s Ar˜i and observe that ⍀ satisfies the exterior differential equations d⍀ q ⌫ n ⍀ s y Ž dri q ⌫ n ri . n d i , ⌰ s d⌫ q ⌫ n ⌫ s 0.
⌫ s y Ž dA . Ay1 ,
Ž 5.1.
If we set R i s dri q ⌫ n ri , then these relations are ideal equivalence covariant and they satisfy the exterior differential equations dR i q ⌫ n R i s ⌰ n ri s 0. All information is thus encoded in the ideal equivalence covariant form d⍀ q ⌫ n ⍀ s yR i n d i ,
dR i q ⌫ n R i s 0,
d⌫ q ⌫ n ⌫ s ⌰ s 0.
Ž 5.2. Ž 5.3.
The definition of the torsion Ž k q 1.-forms gives ⌺ s yR i n d i, and the R i equations show that the torsion equations d⌺ q ⌫ n ⌺ s ⌰ n ⍀ are identically satisfied. The system Ž5.2., Ž5.3. is a complete exterior differential system with ¨ anishing cur¨ ature and the ideal Ik s I ⍀, R i ¬ 1 F i F r 4 is a closed ideal. The classic approach can thus be followed in the case k s 1 by using Cartan’s theorem to the effect that the Cauchy characteristic subspace of T Ž M . of a closed ideal is a Lie module over ⌳0 . The problem can be solved for all values of k by the techniques developed here. Since the differential system is ideal equivalence covariant, Corollary 3.1 allows the transformation to an equivalent differential system with antiexact connection 1-forms. However, the given differential system has vanishing curvature and hence
EXTERIOR DIFFERENTIAL EQUATIONS
607
s 0. The exterior differential system Ž5.2., Ž5.3. is thus ideal equivalent to the system
˜ s yR˜i n d i , d⍀
˜i s 0, s 0, ⌰ ˜ s 0, dR
Ž 5.4. ˜ and the Poincare ´ lemma thus yields the desired solutions ⍀ s d y ˜ri n d i, R˜i s dr˜i . The same result can be obtained by use of the general ˜ s d q s d y HdŽ ˜ri solution generated by the operator H; that is, ⍀ i. n d , where and are related by dŽ y q H Ž ˜ ri n d i .. s 0 since i ˜ ˜ s 0. Conversely, if ⍀ s d y ˜ ri n d , then d⍀ s ydr˜i n d i, and ˜ s 0. These soluhence ⌫ s 0 implies the ideal equivalence conditions ⌰ tions have been obtained without constraints of independence or other˜ R˜i , can be chosen to be wise. Questions of whether the entries of ⍀, independent can be answered by computing the codimensions of the Cauchy characteristic modules of the ideals Ck and Ik . The classical analytic approach would start with the statements d⍀ ' yR i n d i mod Ck , and hence the connection 1-forms are not directly accessible. It is then a lengthy analytic task to try to show that the usual independence conditions imply ⌰ s 0, which happens to fail in the case k s 1, m s 2. The procedure given above obtains the conditions ⌰ s 0 directly from the requirement of ideal equivalence covariance for the governing differential system for the generators of the ideal Ck , and then the answer is almost trivial. Indeed, the conditions ⌰ s 0 are exactly the additional conditions that are required in the case k s 1, m s 2 in order to restrict the solutions of the differential system to yield contact structures. The analysis given above shows that the contact problem is character˜ s 0, ⌺ s R i n d i. An obvious generalization ized by the requirements ⌰ ˜ s 0 and allowing ⌺ s R i n ␣ i is obtained by relaxing the conditions ⌰ i 1 with ␣ g ⌳ . Now, ⌺ has the structure R i n ␣ i and the ␣ ’s are independent if and only if s ␣ 1 n ⭈⭈⭈ n ␣ r / 0. Accordingly, we obtain the requirements ⌺ n s 0. The generalized contact problem is to solve d⍀ ' ⌺ mod Ck s I ⍀ 4 subject to the constraints ⌺ n s 0. For k s r s 1, m s 2 this is the famous problem d 1 ' ␣ n R1 ,
d 2 ' ␣ n R 2 mod I 1 , 2 4
solved by Cartan w11x under the further constraint 1 n 2 n ␣ n R1 n R 2 / 0 on ⺢ 5. Some of the solutions given by Cartan of this problem are not contact 1-forms. THEOREM 5.1. Let ⍀ be a column matrix of k-forms and ⌺ be a column matrix of Ž k q 1.-forms, both of length m, set Ck s I ⍀ 4 , and let ␣ i g ⌳1 ¬ 1 F i F r 4 satisfy
s ␣ 1 n ⭈⭈⭈ n ␣ r / 0.
Ž 5.5.
608
DOMINIC G. B. EDELEN
The general solution of the generalized contact problem d⍀ ' ⌺
mod Ck ,
⌺ns0
Ž 5.6.
is gi¨ en by
˜, ⍀ s A⍀
˜ ⌺ s A⌺,
⌫ s Ž A y dA . Ay1 ,
˜ Ay1 Ž 5.7. ⌰ s A⌰
for all A g GL 0 Ž m, ⌳0 ., where
˜ s d q y H Ž n d . , ⍀
Ž 5.8.
˜⌺ s d q n q Hd Ž n d . y n H Ž n d . ,
Ž 5.9.
˜s, ⌫
˜ s d q n ⌰
Ž 5.10.
for all choices of matrices of antiexact forms , , 4 that satisfy the constraints ˜ ⌺ n s 0. Proof. The relations Ž5.6. are satisfied if and only if d⍀ q ⌫ n ⍀ s ⌺ for some matrix of connection 1-forms ⌫. Completing these equations to a complete differential system yields the system Ž夹. which is ideal equivalance covariant. Accordingly, Corollary 3.1 shows that it can be trans˜ s is a matrix of antiexact formed to an equivalent system such that ⌫ 1-forms. The general solution of this new system is given by Ž5.8. through Ž5.10., and hence all solutions of the original system are given by Ž5.7. for all A g GL 0 Ž m, ⌳0 . and for all choices of the antiexact quantities , , 4 . It then only remains to choose the antiexact quantities , , 4 so that the constraints ˜ ⌺ n s 0 are satisfied. The explicit form of the constraints, namely R i n ␣ i s d q n q Hd Ž n d . y n H Ž n d . Ž 5.11. for some choice of the column matrices R i of k-forms of length m, is obtained by substituting Ž5.9. into ⌺ s R i n ␣ i. The obvious nonlinearity of these relations is what makes this a difficult problem. If we choose s 0, then R i n ␣ i s d and hence the choice s H Ž R i n ␣ i . will work. However, d s R i n ␣ i requires dŽ R i n ␣ i . s 0 in this case, but there are no restrictions placed on the choice of the matrix of antiexact Ž k y 1.-forms . In the general case, application of H to both sides of Ž5.11. leads to the evaluation s H Ž R i n ␣ i .. When this evaluation of is put back into Ž5.11., the residual constraints HdŽ n d y R i n ␣ i . s n H Ž n d y R i n ␣ i . are obtained. Lemma 4.1 can thus be used to obtain the equivalent requirements n d s R i n ␣ i q d q n .
EXTERIOR DIFFERENTIAL EQUATIONS
609
These latter representations of the constraints are easier to deal with since they uncover the restrictions that n d must satisfy. EXAMPLE 2. The effects of non-vanishing curvature can be seen by considering the following exterior differential system in ⺢ 5 : Ž x, y, z, u, ¨ ., d 1 q n 2 s 2 a dx n dy,
d 2 s 2 b dz n dy,
where Ž a, b . are constants such that ab / 0 and s z dy y y dz g A 1. Here ␣ 1 s dy, R11 s 2 a dx, R12 s 2 b dz, so that R11 n R12 n ␣ 1 / 0, and the matrix of connection 1-forms ⌫ss
0 0
g A1 , 0
has one non-zero curvature 2-form entry d s 2 dz n dy. The results of Theorem 5.1 shows that the general solution of this problem is given by A s E and
1 s dp q a  y H Ž n dq Ž y, z . . ,
2 s dq Ž y, z . q b ,
where  s x dy y y dx g A 1, p is an arbitrary element of ⌳0 Ž⺢ 5 ., while q is an arbitrary function of only the two variables y and z. Clearly, 1 n 2 n ␣ 1 / 0 and
1 n d 1 s 1 n 2 a dx q Ž X @ dq . dz 4 n dy / 0, but 2 n d 2 s 0, so that 2 is not a contact 1-form. In fact, 2 is only a 1-form on the 2-dimensional space ⺢ 2 : Ž y, z ., and this result is forced by the presence of the non-zero curvature 2-form d . Thus, the requirement ⌰ s 0 is necessary in order to obtain a contact structure. Gauge Theory Gauge theory has a natural setting in this context. Let Gr Ž m, ⺢. ; GLŽ m, ⺢. be an r-parameter Lie subgroup of m = m matrices; that is, the Lie algebra of Gr is generated by the r matrices ␥␣ ¬ 1 F ␣ F r 4 such that ␥␣ ␥ y ␥ ␥␣ s C␣ ␥ , where C␣ are the structure constants. An element K g Gr acts on a representation space of column matrices ⍀ of length m ´ s K ⍀, where the elements of ⍀ can be k-forms on a manifold M. by ⍀ This action is called global action. If different elements of Gr Ž m, ⺢. are used at different points of M, then Gr Ž m, ⺢. is replaced by the r-parameter Lie gauge group Gr Ž m, ⌳0 . of m = m matrices with values in ⌳0 . This is achieved by maps from M into the group space of Gr Ž M, ⺢. given by K Ž k . s expŽ k ␣␥␣ . ª AŽ f . s expŽ f ␣␥␣ . ᭙ f ␣ g ⌳0 ¬ 1 F ␣ F r 4 , where the
610
DOMINIC G. B. EDELEN
ranges of the f ’s are contained in the group space of Gr Ž m, ⺢.. The action of elements of Gr Ž m, ⌳0 . is called local action. The discussion given above shows that there is a natural decomposition of Gr Ž m, ⌳0 . that is given by Gr Ž m, ⌳0 . s Gr Ž m, ⺢. = Gr 0 Ž m, ⌳0 ., where all elements of Gr 0 Ž m, ⌳0 . evaluate to the identity matrix at a point p of M. Thus, Gr 0 Ž m, ⌳0 . carries all information of how the elements of Gr Ž m, ⌳0 . vary from point to point. ´ s dŽ K ⍀ . s K d⍀. For local action of Gr , For global action of Gr , d⍀ ´ Ž . Ž . d⍀ s d A⍀ s A d⍀ q dA n ⍀. This difficulty is overcome by introducing an m = m matrix of gauge connection 1-forms ⌫ s W ␣␥␣ , where W␣ g ⌳1 ¬ 1 F ␣ F r 4 are the compensating 1-forms for the local action of ´ s Ž A⌫ y dA. Ay1 then gives d⍀ ´ q⌫ ´n⍀ ´s Gr . The requirement ⌫ 0. Ž . Ž A d⍀ q ⌫ n ⍀ for all A g Gr m, ⌳ , and hence A again factors on the left. The matrix of curvature 2-forms is defined by ␣ ⌰ s d⌫ q ⌫ n ⌫ s Ž dW ␣ q 12 C W  n W . ␥␣ .
Ž 5.12.
The conditions ⌰ ' 0 are both necessary and sufficient that there exist a gauge transformation generated by an element of Gr 0 Ž m, ⌳0 . such that ´ s 0; that is, W´ ␣ s 0. ⌫ The exterior differential system d⌫ q ⌫ n ⌫ s ⌰,
d⌰ q ⌫ n ⌰ s ⌰ n ⌫
Ž 5.13.
is covariant under the action of the gauge group Gr 0 Ž m, ⌳0 ., and hence Corollary 3.1 shows that there exists a gauge transformation that makes ⌫ antiexact, ⌫ s Ž A y dA . Ay1 ,
X @ s 0,
s w ␣␥␣ ,
X @ w ␣ s 0.
Ž 5.14. The matrix that accomplishes this is the solution of the Riemann᎐Graves integral equation A s E y H Ž W ␣␥␣ A .
Ž 5.15.
and is necessarily an element of Gr 0 Ž m, ⌳0 . because ␥␣ ¬ 1 F ␣ F r 4 spans the Lie algebra of Gr Ž m, ⺢.. The curvature 2-forms are then evaluated in this antiexact gauge by
˜ Ay1 , ⌰ s A⌰
␣ ˜ s d q n s Ž dw ␣ q 12 C ⌰ w  n w . ␥␣ s ˜␣␥␣ .
Ž 5.16.
EXTERIOR DIFFERENTIAL EQUATIONS
611
˜ by the r This has the effect of replacing the m2 2-form entries of ⌰ 2-forms ˜␣ ¬ 1 F ␣ F r 4 . The results given above hold for a C 1-structure, rather than the usual ⬁ C -structure. Thus the only solutions of the equations ˜␣ s 0 in a C 1structure such that X @ w ␣ s 0 are w ␣ s 0; that is, all antiexact gauge connection 1-forms with ¨ anishing cur¨ ature 2-forms in the C 1-structure are tri¨ ial. The importance of this result is as follows. The ‘‘free gauge field equations’’ of any gauge theory Ži.e., the field equations that obtain in the absence of matter fields. are homogeneous differential equations in the entries of the gauge curvature 2-forms. Accordingly, the free gauge field equations are always satisfied by ⌰ s ␣␥␣ s 0, and hence the gauge connection 1-forms can be transformed away for all such solutions in a C p , p G 1 structure. Physicists, however, rely heavily on ‘‘singular’’ solutions of field equations Ži.e., f s 1rr for ⵜ 2 f s 0.. It is therefore useful to construct all super conductor singular solutions of the equations ˜␣ s 0. The term ‘‘super conductor’’ is used because the corresponding situation in electromagnetic field theory has both the electric and the magnetic fields confined to the singularities of the solution. The symbol , will be used to denote equality at all regular points. THEOREM 5.2. equations
All super conductor singular solutions of the gauge field
⌰ s d⌫ q ⌫ n ⌫ , 0, Ž 5.17. 0 0 for an r-parameter Lie gauge group Gr Ž m, ⌳ . ; GLŽ m, ⌳ ., are gi¨ en by
˜ y dA . Ay1 , ⌫ s Ž A⌫
˜ s y␣ Ž h . dh  ␥␣ g ho Ž y1, 1 . Ž 5.18. ⌫
for some A g Gr Ž m, ⺢. = Gr 0 Ž m, ⌳0 . and for some elements h ␣ ¬ 1 F ␣ F r 4 of hoŽ0, 0. such that the range of the h’s is contained in the group space of Gr Ž m, ⺢., where at least one of the h’s is not a constant function and ␣ Ž k . dk  ¬ 1 F ␣ F r 4 are the Maurer᎐Cartan 1-forms for Gr Ž m, ⺢.. Proof. The representation ⌫ s Ž A y dA. Ay1 in any C p structure shows that ⌫ s Ž dA. Ay1 are the only connection 1-forms for which ⌰ s 0. Now, the group equations for Gr Ž m, ⺢. w12, p. 89x, namely dK Ž k . s ␣ Ž k .␥␣ K Ž k ., pull back to the equations dAŽ f . s ␣ Ž f .␥a A under any map from S into the group space of Gr Ž m, ⺢. given by k ␣ s f ␣ Ž x i . ¬ 1 F ␣ F r 4 , and hence Ž dA. Ay1 s ␣ Ž f .␥␣ . This shows that all matrices of connection 1-forms in any C p structure with vanishing curvature matrices are given by ⌫ s yŽ dA. Ay1 s y ␣ Ž f .␥␣ , all of which can be transformed away by appropriately chosen gauge transformations. Since the Maurer᎐Cartan 1-forms ␣ 4 are unique for any given Gr Ž m, ⺢., ⌫ s y ␣ Ž f .␥␣ are the only matrices of connection 1-forms for which the associated curvature 2-forms vanish throughout S Ži.e., ␣ s yd ␣ ␣ q 12 C␥  n ␥ s 0 in view of the Maurer᎐Cartan equations w12x.. In
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DOMINIC G. B. EDELEN
order to construct singular solutions, the assumption of a C p structure has to be relaxed because the only connections that cannot be transformed away by a gauge transformation are those that satisfy 0 s X @ ⌫ s y␣ Ž f . X @ df  ␥ ; that is, X @ df  s 0. The results obtained in Section 2 show that this is the case if and only if f  g hoŽ0, 0. ¬ 1 F  F r 4 . Now, f  g hoŽ0, 0. can always be chosen so that the f ’s have their range in the group space of Gr Ž m, ⺢. because the composition of any smooth function with an element of hoŽ0, 0. is an element of hoŽ0, 0. Ži.e., simply compose hoŽ0, 0. with a function from ⺢ to ⺢ with bounded range, such as arctanŽ⭈. or tanhŽ⭈. with an appropriately chosen constant.. This ˜ s y␣ Ž h. dh  ␥␣ , with h ␣ g hoŽ0, 0. ¬ 1 argument gives the evaluation ⌫ F  F r 4 , where at least one of the h’s is not constant. Now, dh  g hoŽy1, 1. and the fact that hoŽy1, 1. is a module over hoŽ0, 0. shows that ˜ g hoŽy1, 1.. It follows that ⌰ s 0 at all regular points of ⌫, ˜ by noting ⌫ the validity of the Maurer᎐Cartan equations, and hence ⌰ , 0 over S.
˜ s 1␣ Ž h. dh1 ␥␣ with h1 s An interesting example is given by ⌫ 2 2 arctan ln < R q ct < y ln < R y ct <4 , R s x q y 2 q z 2 , on ⺢ 3 = ⺢ with coordinate functions x, y, z, t 4 , and the remaining h’s all constant. This gives super conductor singular solutions that are singular on the entire null cone of Minkowski 4-space with vertex at the origin! More elementary examples are obtained with h1 s arctan Ž yrx . or h1 s tanhŽ yrx ., while outrageous properties are associated with h1 s sinŽ yrx .. A classification of all super conductor singular solutions can be obtained in terms of loop ˜ integrals of ⌫. Let ⍀ be a column matrix of k-forms that forms a representation space for the action of Gr 0 Ž m, ⌳0 ., and let ⍀ be constrained by the requirement that its entries satisfy the differential system d⍀ q ⌫ n ⍀ s ⌺, where ⌫ is a matrix of gauge compensating 1-forms for the local action of Gr 0 Ž m, ⌳0 .. Accordingly, ⌫ has the evaluation ⌫ s Ž Aw ␣␥␣ y dA. Ay1 , w ␣ g A 1 ¬ 1 F ␣ F r 4 , and Theorem 3.1 shows that all such solvable systems have the representation ⍀ s A d q y H Ž w ␣␥␣ n d . 4 ,
Ž 5.19.
⌺ s A d q w ␣␥␣ n q Hd Ž w ␣␥␣ n d . y w ␥ n H Ž w ␣␥␣ n d . 4
Ž 5.20. for any choice of A g Gr 0 Ž m, ⌳0 . and any choice of the antiexact quantities , 4 . If the torsion forms, ⌺, all vanish, then the entries of ⍀ are said to be gauge co¨ ariant constant Ži.e., d⍀ q ⌫ n ⍀ s 0.. In this event s H ⌺ s 0, and the available choices of d are limited by the con-
EXTERIOR DIFFERENTIAL EQUATIONS
613
straints Hd Ž w ␣␥␣ n d . s w ␥ l H Ž w ␣␥␣ n d . .
Ž 5.21.
REFERENCES 1. D. G. B. Edelen, ‘‘Lagrangian Mechanics of Nonconservative Nonlolonomic Systems,’’ Noordhoff, Leyden, 1977. 2. D. G. B. Edelen, ‘‘Isovector Methods for Equations of Balance,’’ Sijthoff & Noordhoff, Alphen aan den Rijn, The Netherlands, 1980. 3. D. G. B. Edelen, ‘‘Applied Exterior Calculus,’’ Wiley, New York, 1985. 4. D. G. B. Edelen, A metric free electrodynamics with electric and magnetic charges, Ann. Phys. Ž N.Y.. 122 Ž1978., 366᎐400. 5. D. G. B. Edelen and D. C. Lagoudas, ‘‘Gauge Theory and Defects in Solids,’’ North-Holland, Amsterdam, 1988. 6. E. Cartan, ‘‘Les systemes differentiels exterieurs et leurs applications geometriques,’’ Hermann, Paris, 1945. 7. R. L. Bryant et al., ‘‘Exterior Differential Systems,’’ Springer-Verlag, Berlin, 1991. 8. P. A. Griffiths, ‘‘Exterior Differential Systems and the Calculus of Variations,’’ Birkhauser, ¨ Boston, 1983. 9. D. G. B. Edelen, Explicit solutions of a class of exterior differential systems, Quaestiones Math. 20 Ž1997., 1᎐15. 10. D. G. B. Edelen and J. Wang, ‘‘Transformation Methods for Nonlinear Partial Differential Equations,’’ World Scientific, Singapore, 1992. 11. E. Cartan, Les systems ` de Pfaff `a cinq variables et les ´equations aux derivees ´ ´ partielles du ´ second ordre, Ann. Sci. Ecole Norm. Sup. Ž 3 . 27 Ž1910., 109᎐192. 12. F. W. Warner, ‘‘Foundations of Differentiable Manifolds and Lie Groups,’’ SpringerVerlag, Berlin, 1983.