HOUSTONJOURNAL OF MATHEMATICS
@ 199s University of Houston
Volume 24, No. 3, 1998
INVARIANT
DIFFERENTIAL
PROLONGATION
AUREL
BEJANCU,
FORMS
OF THE
L. HERNANDEZ
ENCINAS
Communicated
ON
THE
COTANGENT
FIRST
JET
BUNDLE
AND J. MUfiOZ
MASQUti
by the Editors
ABSTRACT. The structure of the differential forms on J1 (T*M)
invariant under the natural representation
which are
of the gauge algebra of the trivial
principal bundle r : A4 x U( 1) + M and the structure of the horizontal forms
on J’ (T*M)
which are invariant under the Lie algebra of all infinitesimal
automorphisms
of K:
A4 x U(1) + M are determined.
1. INTRODUOTI~N
The main goal of this paper is to determine
J1 (T’M);
tion of the gauge algebra
J1 (T*M).
the structure
that is, the forms which are invariant
The variational
the structure
of the trivial principal
problems
of the horizontal
of gauge forms on
under the natural
bundle
as well.
forms do not depend
tional problems.
mapping
Unlike gauge forms,
on arbitrary
K (j$)
2-form
and
forms which are not only gauge invariant but also
functions
aut(M
on A2T*(A4),
automorphisms
x U(l))-invariant
horizontal
and do not provide interesting
In fact, such forms are isomorphic
for the canonical
-+ iti on
defined by such forms are also studied
invariant under the Lie algebra aut (A4 x U( 1)) of all infinitesimal
is determined
representa-
7r: A4 x U(1)
to lR[ K* &],
and K: J1 (T*M)
where 0,
-+ /j’T*(IM)
variastands
is the
= d,w.
1991 Mathematics
Subject
Classijkation.
Primary:
53A55, 81RlO;
Secondary: 17B65, 17B66, 22365, 53CO5, 58A10, 58A20, 58315, 58330, 58F05.
Key words and phrases. Contact form, cotangent bundle, gauge algebra, gauge invariance,
jet bundle, Lagrangian density, Lie algebra representation, variational calculus.
A. Bejancu supported by the European Community under contract No. CIPA-CT92-2103
(841).
L. Hernkdez
Encinas and J. Muiioz Masque supported by DGES (Spain) under grant PB95-
0124.
421
422
BEJANCU,
ENCINAS
AND
MASQUti
The first motivation
for these results is the geometric formulation
of Utiyama’s
theorem ([22]) c h aracterizing
gauge invariant Lagrangians,
which is nowadays formulated
as follows.
Let 7~: P + M be an arbitrary
principal
G-bundle.
If we con-
sider the induced action of G on TP, the quotient vector bundle Q = T(P)/G -+
M exists, I’(M, Q) is identified to G-invariant
vector fields on P, and we have
an exact sequence ([l, Theorem
11, [5, §4]), 0 + L(P) + Q + T(M) + 0,
L(P) being the adjoint bundle, whose splittings are the connections
on P so that
connections
can be identified
with
the sections
of an affine bundle
p: C(P) +
M modelled over T*(M) 8 L(P) ([5, Definition
4.51). A Lagrangian
density
Cdqi A . . . A dq,, C: Jl(C(P))
+ Et, is gauge invariant
if and only if C factors
IE: J1 (C(P)) --+ A” T*(M) @IL(P) through
by the curvature
mapping
C: /n\2T*(M)~L(P)+R,
tation
of the gauge
algebra
result was the starting
objects in differential
geometric formulations
a function
which must be invariant under the natural represenon the curvature
bundle (cf. [2], [3], [5], [S]). This
point
for the study
of gauge invariance
and gauge-natural
geometry ([5], [6], [7], [21]) and it lies on the basis of the
of gauge theories (e.g., see [5], [7], [13], [17]). Because of
the importance
of Utiyama’s result it seems natural to try classifying differential
forms of arbitrary
degree and not necessarily horizontal (as Lagrangian
densities)
under
bundle
the representation
of connections
corresponds
to the classical
tion on the geometry
the bundle
of the gauge algebra
of a principal
bundle.
electromagnetism,
of differentiable
of connections
of the bundle
on the first jet prolongation
In the abelian
this problem
case, G = U(l),
poses a general
of the
which
ques-
manifolds, due to the elementary
fact that
idenrr: M x U(1) -+ M can be naturally
tified with the cotangent bundle of M, thus inducing a Lie algebra representation
of aut (M x U(1)) into the vector fields of J1 (T* M). In [ll] we solved what
could be called the ‘geometric part’ of this problem; i.e., we classified gauge invariant differential forms on Jo (T* M) = T* M, proving that they are determined
by the symplectic
form of the cotangent
the corresponding
problem
answering
to the original
bundle.
In the present
for the forms on the first prolongation
motivation
of the problem.
work we solve
bundle
thus
As could be expected
from
their physical meaning, the algebra of gauge invariant forms on J1 (T* M) is larger
than that of T*M, and, in fact, there are plenty of such forms which are related
to the canonical
contact
structure
is determined
the present case involve
differential
case, which have been strongly
gauge invariance.
system
on the jet bundle
and whose module
by the curvature
mapping.
Accordingly,
the proofs in
different ideas and techniques to those of the geometric
influenced
by the approach
of [5] in what concerns
INVARIANT
DIFFERENTIAL
FORMS
ON J’(T* M)
423
2. THE FUNDAMENTAL REPRESENTATION
In this section we define
infinitesimal
automorphisms
structure
t stands
u(1)
then
group
U(1)
(i.e., A E u(1)
each connection
in U(1),
and A is the standard
is the element determined
form WY can be identified
fields of J’ (T’M).
basis
on M x U(1) can be identified
If
of the Lie algebra
by R -+ U(l), t e exp(it)),
with the ordinary
one-form w
wr = (dt + r*w) 8 A; in other
by the formula
of connections
of the Lie algebra of all
7r: M x U(1) -+ M with
= {z E @: ]z] = 1) into the vector
for the angle
on M defined
the natural representation
of the principal
bundle
words,
with the cotangent
the bundle
bundle,
i.e.,
c (M x U(1)) ” T*(M).
2.1. Infinitesimal automorphisms.
Gau(P))
the group
(respectively
of all automorphisms
the group
is G-invariant
the Lie algebra
of Aut(P)
(cf. [lo, III.
= F (M, Q).
gau(P)
as being
vector
r-vertical
vector
the angle
in U(1) defines
Vt E R. Because
Accordingly
vector
a global
with the standard
field on M x U(1) such that
M
field X on P
fields on P as the “Lie
§35]), [3, 3.2.9-3.2.171).
field A’ associated
by
n: P +
G-bundle
@‘t E Aut(P),
of G-invariant
(respectively
A vector
Similarly,
On P = M x U(l),
unique
of P).
we set gau(P)
= I’(M,L(P))
the “Lie algebra” of the gauge group.
aut(P)
fundamental
by Aut(P)
of a principal
transformations
if and only if its flow at satisfies
of this we consider
algebra”
of gauge
Let us denote
one-form
basis
dt(A*)
we write
and we think
of
dt and the
A E u(l)
is the
= 1. Note that
A*
is U(l)-invariant
(and hence A* E gau (M x U(1))) since U(1) is abelian.
Let
(N; 41, . . . . qm), m = dim M, be an open coordinate domain in M. Then, a vector
field X E X (N x U(1)) is U(l)- invariant
see [ll, III.B, Proposition
l-(4)]),
In particular,
X belongs
Remark. Decomposition
X’ E X(M)
defines
9
E
(2.1) has a global meaning.
a vector
C”(N).
As the bundle
is trivial,
every
field on M x U( 1) and every X E aut (M x U( 1)) is
as X = X’ + gA*, X’ E X(M) being the 7r-projection of X
Hence aut(M x U(1)) can be identified with the sections of the
decomposed
and g E Cm(M).
as (e.g.,
to gau (N x U(1)) if and only if,
x = 9 (41, . .. . qm)A’,
(2.2)
uniquely
if and only if it can be written
424
BEJANCU,
vector
that
bundle
aut(M
T(M)
ENCINAS
AND MASQUti
$ (M x R). Recalling
gau (M x U( 1)) is abelian,
is th e semidirect product of X(M)
representation;
i.e., p: X(M) + Der(gau(M
the adjoint
it follows
and gau (M x U( 1)) via
x U(l))),
p(X’)(gA*)
=
x U(1))
[X’, gA*] = (X’g)A*.
2.2. The
action
principal
bundle
on T*(M).
of Aut (M x U(1))
r:
Each
P --t M, acts in a natural
(cf. [12, II $61) by pulling-back
connection
automorphism
Q of a
of P
way on the connections
forms, i.e., (W1)*~r
= u+.r,
and this
action induces an action of Aut(P)
on the bundle of connections
C(P), which
we denote by & : C(P) --+ C(P), in such a way that if p: C(P) -+ M stands for
the bundle projection,
then we have p o 6 = cp o p, where cp: M -+ M is the
diffeomorphism
induced by Q on the ground manifold.
If X E aut (M x U(l)),
then its flow @t is a family of automorphisms
of M x U(1) and &t induces a
flow on C (M x U(1))
field generated
mapping
2 T*(M).
aut(M
x
U(1))
by _% E X(T* M)
Let us denote
by _%. The local expression
+ x(T*M),
the vector
of _% and the basic properties
of the
X ti r?, are as follows (for the details
see
[ill):
1. If X is given by the formula
(2.1), then
(2.3)
where (qi,p,),
1 5 i < m, is the coordinate
system
induced
by (N; ql, . . . . qm)
in p-l(N)
C T*(M); i.e., w = zzlpi(W)dzqi,
VW E T,‘
(M), 5 E N.
2. For every X E aut (M x U( 1))) X is p-projcctable
and its projection
is X’,
the projection
of X onto M.
3. X e
x is a Lie algebra
2.3. Infinitesimal
homomorphism.
contact transformations.
ifold p: E + M, let us denote
with bundle
projections
Given
by J1(E) the l-jet
pl : J1(E) -+ M, pl (jis)
an arbitrary
bundle
fibred man-
of local sections
of p,
= s; ~10: J1(E) + Jo(E) = E,
PIO (.$) = 4%). Th e manifold J1(E) is endowed with a canonical Pfaffian differential system S, locally generated by the contact one-forms Bj = dyj--Czl
yidx,,
1 2 j < n, where
m = dimE
are the coordinates
of p: E +
M;
X E X(E),
there
is plo-projectable
induced
i.e., yi(jJs)
- dim M, and (xi, y3; y!), 1 5 i 5 m, 1 < j 5 72,
on .J1 (E) by a fibred
coordinate
= (a(yj o s)/dxi)(z)
([4], [9], [14], [15]).
system
(xi, yj)
Given
vector field X(1) E X(J’(E))
such that 1) X(1)
onto X, and 2) X(1) leaves invariant the contact system S; i.e.,
exists a unique
INVARIANT
Lx(,,S
C S.
mation
attached
Lie algebra
generator
The vector
DIFFERENTIAL
field Xcl) is called
to X, and the mapping
injection.
If X is pprojectable,
of the usual
FORMS
l-jet
prolongation
ON J’(T’M)
425
the infinitesimal
X(E)
+
contact
ZE(J’(E)),
transfor-
X ++ Xcl),
is a
Xcl) coincides with the infinitesimal
of the flow of X (cf. [15], [18], [19]).
2.4. Representing
Aut (A4 x U(1)) into X (J1 (T*M)).
By composing the representation
aut (M x V( 1)) + ZE(T*A4) given in $2.2, with the l-jet prolongation described
in 52.3, we obtain
X(T*(M))
--f X(Jl(T*M)),
formulas for jet prolongation
the natural representation:
In particular,
the natural
representation:
for fi = 0 we find the l-prolongation
(gA*h = -g g
(2.5)
aut(A4
x U(1))
+
X ++ _% H _%(I). Formula (2.3) and the general
(e.g., see [19]) yield the following local expression of
of the gauge representation:
& - $ & $.
3
3.
In this section
gT*M-INVARIANCE
to each manifold
M we attach
an abelian
Lie subalgebra
ATOM c
X(T* M) which generalizes
ance with respect
the gauge algebra representation
and we study invarito such a subalgebra as a first step in studying gauge invariance.
3.1. The gauge algebra of T’M.
We denote by gT*M and we call it the gauge
algebra of T’M, the abelian Lie algebra of all p-vertical vector fields on T’M which
are invariant
under translations
of the fibres. More precisely: every w E T,(M)
gives rise to a translation
T,,, : T,*(M) + T,‘(M), am
= w + w’. A p-vertical
vector
field X E X(T* M) belongs
fibre is invariant
under
translations;
to gT.M if and only if its restriction
i.e., ~~
(XIP-~(z))
= XI~-I(~),
to each
Vx E M,
VW E T,‘M.
domain of M and let (qi , pi), 1 5 i 5
Let (N; 41 , . . . , qm) be an open coordinate
m, be the coordinate
system induced in p-l(N)
(see 52.2-l.).
Then, it is not
difficult
to prove that
a vector
field X E X(T*M)
belongs
to gT*M if and only if,
426
BEJANCU,
ENCINAS
AND
MASQUli
locally, it can be written as
x = 29i(Ql,...,4
(3.1)
&
i=l
9i E
c-(N).
Hence the mapping which associates ~~-l(~) to each open subset U s M is a
locally free sheaf of CE-modules of rank m. A vector field X E STEM belongs,
locally, to the image of the gauge representation (see (2.3) for fi = 0) if and only
1 5 i 1. m, thus explaining
if there exists g E Cm(N) such that g2 = -ag/dqi,
our terminology. In fact, we have
Proposition
3.1. Let 6’1 = CEipidqi
bundle p: T*M -+ M. Then
1. A p-vertical
be the Liouville
vector field X E X (T’M)
form
belongs to gT’M
on the cotangent
if and only if Lxfi,
is p-projectable.
2. The mapping L: gT*M +
R(M),
L(X)
=
ixdQi,
is an isomorphism of
C” (M)-modules.
Denoting
by Z1(M),
have gau(M
L-~(Z~(M)),
B1(M)
the closed and exact l-forms
x U(1)) = L-‘(B~(M)).
and contact
f12,, is said to be gT*M-invariant
by W(T*M)
that O(T*M)
ifwe
on M, respectively,
forms.
we
define gauloc (M x U(1)) =
then gau l0c (M x U(1)) /gau (M x U(1)) % H1(M;
3.2. &PM-invariance
0, where X(i)
Moreover,
R).
A differential n-form on Jl(T*M),
if for every X E gT*M we have Lx(,, fl,
=
stands for the l-jet prolongation of X (cf. $2.3). Let us denote
the set of all gT*M-invariant n-forms on J1 (T*M).
It is clear
= @,“==, W(T*M)
is a graded C”(M)-subalgebra
of fi.(T* M) =
is said to be of type
W(T* M). A gT*M-invariant form 0, E W(T*M)
if either R,(U) = 0 or R,(w) # 0 and
(s, t, n - s - t) at a point ti E Jl(T*M)
then n - s - t should be even, say n - s - t = 214 and the two conditions below
@:=a
hold true:
1. There exists a linearly independent system of tangent vectors Xf, .. . . Xf+,
T,(M)
such that for every local section w of p with jiw
system of tangent vectors Xo, X1, ...,XS+U E T,(M)
i(jlw),x,O...i(jl,),xo
s+u% # 0,
E
= ti and every
we have
i(jlw),xoi(j~,),x,...i(j~w).x,+,R,
= 0.
3 There exists a linearly independent system of pie-vertical tangent vectors
such that for every system of pm-vertical tangent
Yic, . .. . Y, E T&lT*M)
vectors Yo, Yi, . ...YU E T,(Jl(T*M))
iY:,...iytf12n # 0,
we have
iyoiyI...iyU512, = 0.
INVARIANT
DIFFERENTIAL
FORMS
ON J’(T’ M)
Note that the above two conditions
completely
determine
u. We denote by 0S~t~2U(V) C W+t+2”(V) the subspace
427
the integers
s, t and
of the forms of type
(s, t, 221).
Theorem 3.2.
1 5 j 5 m,
1 5 i 5 m be as in 32.2-l.,
Let (N; qi,pi),
be the coordinate
system
and let pi, 1 < i 5 m,
(cf. $2.3).
on J’(p-lN)
induced
We set
1 5 i 5 m. Then, the forms $ohij = dqhl A . . . A dqh, A
oil A . . . A Bi, A dej, A . . . A de,,, with h = (hl, . . . . h,), 1 < hl < . . . < h, < m;
Bi = dpi - ~,“=,p~dqj,
i = (il , . . .. it),
1 5 il < . . . < it 5 m; j = (jl, . . . . ju),
5 . . . 5 j, < m;
as a Cm(N)-module.
Furthermore,
s + u 5 m, are a basis for W~t~2”(p-1N)
on(T*M)
= @s+t+2u=n @SJJ”(TflM).
If the local expression
PROOF.
1 5 jl
of a vector field X E gT*M is as in (3.1), then
(3.2)
Hence Lxcl,dqh
= 0, Lx(,,Bi
= 0, Lx,,,dBi
above
forms
are linearly
(h, i, j) satisfying
= 0, and the C”(N)-module
spanned
is contained
in 0 (p_lN).
Moreover, it follows
vhzj E @S~t~2U(p-1N). We shall prove that the
by all forms in the statement
from the very definitions
that
independent.
the conditions
by I the set of indices
Let us fix an index (a, b, c) E I.
Let us denote
of the statement.
Let ai < . . . < ak_, be the complementary
set of {al, . . . . a,} in (1, . . . . m}. As
s + u 5 m we have u 2 m - s. We set a’ = (al,, . . . . aL_,),
a” = (a:, . . . . a:), so
that {a”} 5 {a’}. (F or any ordered system X = (Xl, . . . . Xk) we denote by {X} the
underlying set; i.e., {A} = {Xl, .. . , A,}.) Assume for some functions fhij E C”(N)
we have C (h,i,j)EI fhG(PW = 0. By taking interior products successively
with
alap;:,
) . . , a/a@
Again taking
. . . A dqh8 A dq,;
f&c
particular,
in the above
interior
products
A . . . A dq,:
Accordingly,
= 0.
s + t + 2u = n, span
the local expression
. ..Adqh.,
equation,
xch,i,cjEJ
fhic(Phia”
we only need to prove that
W(p-‘N).
of an n-form
h = (hl < . . . < h,);
we obtain
= 0.
with a/apl, . . . . a/apt, we have z(h,b,c)E1fhbC dqh, A
= 0. Hence, fhbc = 0 whenever {h} 17 {a”} = 8. In
Let 0,,
in the basis
8, = &A.../&,
the forms
phij,
fiirc dqh A Bi A dp3, be
= xa+t+U=n
(dq,,f?a,dp$,
with dqh = dqh,A
i = (il < . . . < it); dpjk = dpi:
A...A
By taking X in (3.1)
dp& (.%k) = ((_Gl)
< . . . < (jU, kU)), lexicographically.
from (3.2) we deduce that
successively equal to a/Bpl, qa(d/8pb) and q,qb(a/ap,),
$1)
is equal
respectively.
= 0 (hence
to
a/aPl,
qa(d/aPb)+a/aP:
The invariance
fiik
E C-(N))
condition
and %??b(a/aPc)
Lxcl,fi2,
+qa(d/dp~)+qb(d/dP~),
= 0 yields afi,,/apl
and fiik = 0 for a $2 {h},
a E {lc}.
= aj&/ap”,
Hence if filik # 0,
428
BEJANCU. ENCINAS AND MASQUE
we can write
zt &dql,
and then, Q, = Cs+t+2u=n
(ck,
dqku A
Remark.
n {I} = 0, 1 = (Z1 < . < Zs), s = c7- u,
{h} = {Ic} U {Z}, {Ic}
A . A dql* A 0%A @,,
dqk, A dp3,:) A . ..A
d$,)
0
p: E +
A form R, on a fibred manifold
M is p-horizontal
if ixR,
X E Z(E).
Theorem 3.2 implies that gT*M-invariant
by contact forms, their exterior differentials and pi-horizontal
for every p-vertical
are spanned
= 0
forms
forms.
4. gau (M x U(l))-INVARIANCE
Let p, : A” T*(M)
n-form
+ M be the bundle
on A” T*(M);
i.e.,
&(X1
E G,n(AnT*M),
Liouville’s
form on the cotangent
abelian
case the geometric
PI,
form (~5
formulation
4.1.
given by 4(jzf)
= ji(df),
d,
I
(J2(M,Wo)
2. A function
= {&I,
f E C”
(T*M)
i.e., for w2 E A”Tj(M),
expression
$*(ph)
= ya+(i),
= 8/8pj
IQ/ 5
+ a/ad,
part 2. follows from 1. Finally,
AT*(M)
T” M)
takes the following
proving
x U(l))}.
E gau(M
x U(1))
exists such that f = g o K.
system
induced
by (41, . . . . qm) on
pij(wz)d,qi
A d,qj.
induced
on J2(M,IW)
by
It follows from the definition
Hence
l., taking
&(d/dy,)
= a/ap,,
into account
the local
in (2.5). As the fibres of K are connected,
3. follows from the local expression
of the exterior
0
R, on J1(T* M) is said to be gau(M x U( 1)) -invariant
As gau(M
if Lgc1,f12, = 0 for all X E gau(M x U(1)).
An n-form
invariant)
Then,
= pi - pg.
system
(x).
1, 1 5 i 5 m.
thus
+ 0,
= 0,VX
w2 = C,.,j
= (&lf/dqa)
of the gauge representation
differential.
than
3.1. In the
of vector bundles over M,
as above we have n’(pij)
of M; i.e., y,(jzf)
+*(d/dy(ij,)
theorem
: X E gau(M
Let (yol), IcrI < 2, be the coordinate
of 4 that
in Proposition
satisfies Xcljf
g E C-(/j”
with the same notations
. . ..qm)
where
fir is none other
and we have
(ji(df))
(J’(T*M))
-%
1 5 i < j < m, be the coordinate
A2T*M;
(41 ,
P
= d,w,
if and only if a function
PROOF.
of Utiyama’s
There exists an exact sequence
0 -+ 52(kf,Iw)0
3. Let pij,
. . . . (P,).-GL
Note that
as introduced
be the canonical
[31, [51, PI):
Proposition
1. &Tj;f
bundle
and let 0,
= v~, ((P,)JI,
E A”TG(M).
w,
Xlr...,Xn
projection
,..., Xn)
x
(or gauge
U(1)) c
INVARIANT DIFFERENTIAL FORMS ON J’(T* M)
gT*M,
it follows from the very definitions
invariant.
Moreover,
differential
form on A” T*(M)
results
gauge forms.
gT*M-invariant
it follows from 2. of the previous
Also note that
are equivalent
is gauge invariant.
gau(M
forms are gauge
proposition
that
In fact, as a consequence
we shall see that @(T*M)
in the next section,
invariance
that
429
and K*R’(A’
x U(l))-invariance
every
of the
T*M)
and gauloc(M
span
x U(l))-
notions.
5. LOCAL STRUCTURE
OF GAUGE FORMS
Let us denote by G& the sheaf of gauge n-forms on M. We set GM = @,“==, 6b.
Theorem
With the above notations
5.1.
1. For 1 < n 5 m = dim M,
following
we have
G’& is locally generated
forms:
dqhl A ... A dqh, A oil A ... A 8t3 A dBj, A .
(5.1)
over KICKS T*M by the
with 1 5 hl <
A dBj, A dpk,l, A . . . A dpk,l,
. . < h, 5 m, 1 < iI < . . . < i, < m, 1 5 jI
<
. < j, 5 m,
l~Ic,<l,<m,l~cuLu,r+s+2t+u.=n.
2. For n > m, G& is locally generated
together
(5.1)
(5.2)
dql A .
with the following
A dq,
ower IG*C~~ T*M by the forms in formula
forms:
A dp,, A . . A dpi, A dPj,: A . . . A dpt , s + t = n - m.
For every n > 0, GG is a locally free sheaf of ~*Cr~~~~-rnodules
PROOF.
Let us consider
(5.3)
0,
= Afil...i,,dqi’
n!
the local expression
of an n-form,
A... Adqi” + ~fi’~~~i’.~j,
of finite
1 5 n 5 m,
A... Aoj,+
rank.
430
BEJANCU.
where the coefficients
respect
By imposing
direct
are functions
to the indices
stein’s convention
AND
on Jl(T*M),
MASQUJ?
skew-symmetric
separately
with
ir, . . . . i,.; jr, . . . . j, and (hr, ICI), . . . . (ht, IQ), and we use Ein-
for repeated
indices.
= 0, VX = g(ql,...,qm)Al
Lgi,,,R,
calculation
ENCINAS
shows that
R, is gauge invariant
(cf.
(2.2))
and using
(2.5), a
if and only if all coefficients
on A” T* (M) and satisfy
f’s in (5.3) are defined
(5.4)
(5.5)
E(O) being the signature
fil...Zndqil
of a permutation
0 of (ir , .. . . ir).
Thus in (5.3) the terms
A... Adqin, f31...jn0j1 A... Ae3,, and f/;,;:;j,“dqil
A... Adqir ABj, A... Aej,V
are as in the statement.
We proceed to calculate the other terms in (5.3) showing
that they are expressed by means of the forms in (5.1). To do this we first note
that from (5.4) we obtain f~I1,::~; + fi::::L; = . = f~~,::;c”,- + f:I1,:;f; = 0.
Thus,
we have f,h:.::t;dpk:
where we set
p,’ -p;
f;:$.:; = f
(cf. Proposition
. ../\$.r\dp;;
r\...r\dpk,:
Next, we examine
A . . . A dpk”, = 2-nf(h1k1)...(h-kn)dph~k~
4.1-3.). Similarly,
= 2-tfj,...i,,(h,k,)...
(5.6).
As g is an arbitrary
First,
function,
+ f!hhl)...(hn-lkn-l)
# il,
f!ilh)...(h,-lkn-l)
$1
for ICI # il.
by using
(htkt)ejlA...Aej,Adphlkl
21
# il,h
f~::::~t’hl...htOJlr\
A...Adph&.
we obtain
21
Then,
using (5.5) we obtain
of K* (pz3) =
for r = 2 it becomes
f~h,k,)...(hn-,kn-~)
forhl
A . . . A dph,k,,
...(hk)...,and we write p,j = pi - pi, instead
(5.8) we obtain
+ f(hil)...(h,-lk,-1)
a1
_
-
o
7
= f(klh)...(h-lkn-l)
kl
I
INVARIANT
As f~~&&.(hn--lkn--l)
DIFFERENTIAL
is skew-symmetric
FORMS
ON J’(T*
431
M)
with respect to (hrlcr), ,.., (&-r&r)
we d:duce that (5.8) holds Vl(htI~~),2 < t < n - 1. Thus, by using (5.9) we finally
’ expressed by means of
conclude that f!hlkl)..,(hn--lkn--l)dqll
/\dp;: /, ... ,‘,dpkn-l
h,_l 1s
dqi, Adphlkl * .f:Adph,_,k,_I
and de,, A... Adej, Adphlkl //. . . r\dphlkt, 2s+t
For r > 2 the sum in (5.6) becomes
= n.
which implies
(5.16)
f(ht,kl)...(htkt)
+ f!klhl)‘
.‘(htkt)= 0>
%I...%,_1
21 ZP 1
for {hr, /cl} fl {ir, . . . . i,_r}
= 0,
(klzl)(hzkz)...(htkt)
= f(klkl)(hzkz)...(htht)
1
kl
(5.11)
for ICI $! {ir, . . . . G-r},
(5
12)
f
!Zlif)(h?k2).--(htrC,) + f !i?il)(hzkz)...(htkt) + f ii?ir!(h2rc,)...(h,k1)+
212223...ir_-1
Z2ZrZ3
1,
1
2122~32P 1
+ f!itir)(h2k2)...(htk’)
+ f!irit)(hz”z)...(h:“t)
f !i,l2)(h?k2)...(htkt)
22b13
a?
2,2123...ir__1
1
for 3 5 r 5
71 -
ZrZ123...ir_1
1.
Further, we decompose the term we are interested in as follows:
=
o
1
432
BEJANCU,
(5 13)
f@+...@trct)dqi,
ENCINAS
A ,, A
&j-l
A
t,...zy_l
AND
dp”h: A . . . A dpl”h’=
,
(hlkl) ;’ (htkt)dqil
f,,...i,_‘
c
MASQUJ?
A . A dqiT-’ A dp;;
A . A dp;: +
{hl,kl}n{zl,...,i,-l}=0
Adp;;
A dpk
A . . . A dpk +
Adp;I
A dp;2, A . . . A dp;“, +
(r _ l)f!if,‘t)(hz”z)...(htkt)
x1 ZT 1
f!i211)(hzkz)...(htIct)
21
ZF
dqil A . . . A dqir-l
dqZ1A . A dqir-'
1
A
A
dp;;
A
dp;“,
dp;; A dp;", A
A
A ...
Adpi’, +
dpkl)
By using (5.10) we obtain
(5.14)
chllcl) (htkt)dqil
f&..Zr--.;’
c
{hl,kl}n{zl....,i,-l}=s
;f~~~~~~$W
c
A ,., A dqiv-’ A dp;; A .,. /‘, dp;: =
dqil
A . .
Adqi’-’
{hl,kl}n{il,...,z,-1)=B
Adphlkl A dpk’, A . A dp;“,
Let us denote
by F the sum in the first brackets
(5 15) F = f!klZl)(hzk2)...(h,kt)dq21
Zl...Z,_l
A . . ,, dqk’
(klkl)(hzkz)...(h,kt)dq”2
(-1)Tfkli2...i,_l
Similarly,
(5
16)
let us denote
G
=
of (5.13).
A dpklz, A dp”hZ
z A
A ... ,, dqir-’
2*22i3...ir--1
dqil A
A dqir-l
2P
1
~1~2~3
AdpI; A dp’i; A . .. A dp;‘,
27
1
,, dp;“,
of (5.13).
A dpili2 A dp;“, A
f(iti2)(h?‘C2)...(htkt) + f!iZil)(h?lCZ)...(htkt)
ZlZZ~3
/, dekl
by G the sum in the last brackets
f(iliz)(hzkz)...(htkt)
From (5.11) we obtain
A dpf”t+
/, ,,. ,, ,jpkt,
Thus,
A dp;:$
dqZ1 A .. . A dqiT-l
INVARIANT
on m the following
By induction
Lemma 5.2.
Let (Aij)
tions satisfying
lemma
FORMS
ON
J1(T* M)
433
is easily checked:
be a m x m skew-symmetric
Aij + Ajk + &
dq, A dp;, mL2.
S,
DIFFERENTIAL
matrix
= 0, i, j, k E [l, m].
of
differentiable
Set S,
= Cicj
func-
Aijdq,
A
Then,
= Al,dql
A de,
c
dqi A dp; + de,
a=1
(hakz)...(htkt) _
(~lzz)(hzkz)
(htkt) +f!izil)(hzkz)...(htlc,)
Let us apply thelemmatoAilZZi3...ir_-l- fili2i3...iF_-1
“’
taking
into account
that for any set of indices {is, . . ..&I}
212223...2y_1
and {(h&2),
. . . . (htk,)}
the m x m matrix
A!h?!Z)‘~.(h”k”) (whose entries are il, i2) is skew-symmetric
( Z12223...Zr-_1 >
and it satisfies the relations in the lemma, as follows from (5.12). Hence each
sum S(h2”z)...W”)
_ A@+)..:(%)
212223,,,2,_1 dqil A dqt2 A dpi:, which appears in (5.16) is
mzs...i,_1
expressed as in the lemma. Thus, (5.16) becomes
(5.17)
C
=
f!itiz)!hzkz)...(htkct) dqi’ A . A dqiT-’ A dpiIiz A dpki A . . . A dpk +
z1z2...2,_1
(-1)~-‘ni”,21c,z.):1~~~“t’ dql A de1 A dqi3 A . A dqir-’
Adp;;
A . . . Ad&
+
(-l)‘-lF
Aj;zi;;r(‘:‘kt)dqj
j=2
A
‘2
dq, A dpji + dej
i=l
Adqa3 A . . . A dqzr-l
))
A dp;2, A . . . A dp&
As j~h~“~’ ..(htkt) is skew-symmetric
with respect to the upper indices, the calcula%I...?,_1
tions performed with respect to (h 1 k 1) in order to obtain (5.14), (5.15) and (5.17)
can be repeated for any other of the indices (h2k2), . . . . (h,kt).
Thus, the right
hand side of (5.13) is finally expressed as a linear combination
with coefficients
in CrzTeM
of the differential
forms in (5.1).
The above proof still works for the
term fj~,:;~~s,~I~::2t dqil A... Adqir Ad,, A . Aej, Adpk: A
Ad&,
since conditions
(5.6) and (5.7) are the same for both types of coefficients.
Now, let us assume
R,
=
n > m. Thus,
f f;;,::;;dql
c
a+b+m=n
R, is locally expressed
A . . . A dq,
&
A dp;:
f;;:::k:hl-hbdql
’
Adp;:A...Adp;;+fl:,,
A . . . A d&
A . . . A dq,
as follows:
+
A dpj, A . . . A dpj,
434
BEJANCU,ENCINAS
i-2; =
$ f,“:,::,“,-dp;;
AND
A . . . A dpk,“n+
MASQUti
c
r+s=n,r<m
-& f,j,l,::*dqi’
.
A . . . A dq2’
A ej, A . . . A ejs +
A .. . h
ej, h dp';,: h
. . . h dpk +
A ... A dqi-A dp:;A ... Ad& +
Adp;:
A... Adpt.
Note that the above proof also works for R’,. Hence we conclude that fl2:, is gauge
invariant if and only if 0; is a linear combination
with coefficients in A” T*(M)
of the forms in (5.1) for r < m. Furthermore,
it is easy to see that R, is gauge
invariant if and only if OR:,is gauge invariant and all f~~,;:~, fi::::g’hl”.hb
belong
to C’rz T_M. Moreover,
although
and (5.2) are not linearly
following
relationship:
in the general
independent
case the differential
over &*Cw(A2
T*M),
CL”=, dqi Ao, +p;, (dG’1) = tic&
forms in (5.1)
as they satisfy
= cicj
K*(pij)dqi
the
A dqj,
L&, stands for the canonical form on r\“T*(M) (cf. §4), their matrix in
the basis induced by (dqh, & , dp,k) h as constant entries, thus showing that the
This proves the
rank of the system of forms (5.1) and (5.2) is locally constant.
0
last part of the statement
and completes the proof.
where
Remark. The genera! formula for the rank of $$ seems to be too complicated
to
be written down explicitly; for example, for m 2 2, we have rkGa = m(m+ 3)/2,
rk$,_, = m(m+l)(m2+5m+2)/8,form
> 3,rkQ& = m(m+l)(m+2)(m3+4m25m + 12)/48. The calculation
of the rank for n > m is much more difficult; for
example the ranks of $$, n > 3, for a surface (dim M = m = 2) are rkG$ = 17,
rkG& = 17, rkGk
= 22, rkG& = 15, rkG;1, = 6, rk@,_, = 1, rkG’J$, = 0, Vn 2 9.
6. Aut (M x U(l))-INVARIANT
A differential
if Lx-(~,~,
n-form
R, on J1 (T’M)
HORIZONTAL
FORMS
is said to be aut (M x U(l))-invariant
= 0 for every X E aut (M x U(1)).
Theorem 6.1. The R-algebra of aut (M x U(l))-'znvariant horizontal forms on
J’ (T*M)
is !LFL[K*
C?J], where K : J’ (T*M)
and L?s stands for the canonical 2-form
t
r\” T*(M)
on A” T*(M)
is as in Proposition
(cf. $4).
4.1
INVARIANT
PROOF.
We have
local expression
Rz =
DIFFERENTIAL
xi.+
Pijdqi
FORMS
A dqj
=
ON Jl(T’A4)
f Ci,jPijdqi
435
A dqj.
From
and
(2.4) it is not difficult to see that every differential
in W[K* Rs] is aut (M x U(l))- invariant.
Conversely,
let us assume that
this
form
R,
=
$ fil...i,dqi,A...Adqi,
= 5 fil...i,dqi’A...Adqin
is an aut (M X U(l))horizontal
n-form on Jl(T*M),
where fil,,,i, is skew-symmetric
in the
C~,,,.,i,=,
invariant
indices ii, . . . . i,. In order to prove that f12, E W[K*&] , we obtain some formulas
for calculation
of determinants
of square matrices whose entries are pi. First, we
denote
by wil..,i,
of dqil A . . . A dqin in L’s, where n = 2r.
the coefficient
Note
that we have wil,..i,, = eiZPiliaWia,,,in + . . . + c~,,PilinWi~,,,i,_l, where the W’S in the
right hand side are some coefficients in 0i-l
and eik, 2 < lc 5 n, is the sign of the
. . . . in). Then by induction
on n we obtain
permutation
(ii, ik, iz, . . . , ik-l,ik+l,
Lemma 6.2. Let n > 2 be a natural
(6.1)
&l...i,
=
Pa122
Pilia
Pili4
PiSia
0
Pi324
.
.
PiniS
I Pini2
1 WZi3i4...in
is aut (M x U(l))-’ invariant,
Theorem
5.1 the functions
invariance
condition
..
...
..
Pini4
we have
Pili,
Pisin
...
:
...
0
WiliZ...in ’Wi324...in7 if n = 2r,
=
As R,
With the above notations
number.
WiliSi4...in, if 72 = 2r + 1.
it is also gauge
fil,..i,,
are defined
when applied
to a vector
invariant
and according
to
on A” T*(M).
In addition,
the
field X E X(M), X = f”(a/aq2),
yields
(6.2)
{
..
ft%$’ _
fil,,.i,
{dfi’
A dq'" t
A
dqi2 A . . . A dqin + ... + dqi’ A ... A dfin}
Take fi = ai E R, 1 5 i < m, to conclude
(6.2) b ecomes
(6.3)
gftx*...i.
- gfti,i,...i,
a2
ftil...i,_1
fil..,i,
that
+
only depends
. ..t
-Ptk-p
aft
dqh
afi,...i,
8P”,
=
o,
= 0.
on Pij. Thus
436
BEJANCU,
Without
loss of generality
g
{
AND
we may consider
(1, . . . . n) for (ii, . . . . in). Then
(6.4)
ENCINAS
from now onwards
(6.3) is arranged
fta...n-d+}
MASQUI?
the permutation
as follows:
+ ...+
1
For h,lc E [l,m], h # Ic, set Xh,” =
From (6.4) we obtain the homogeneous
afl...n/dpt.
linear system
Let us fix h E {n + 1, . . . . m}.
with m equations and m - 1
unknowns,
Suppose
l<t<m.
c ptk Xh’” = 0,
k#h
(6.5)
m is odd.
6
=
Thus we can consider
0
p12
ph-1,l
ph-1,2
...
ph+l,l
Ph+1,2
...
Pm2
‘.’
...
the non-null
determinant
pl.h-1
Pl,h+l
.
0
ph-l,h+l
...
=
(w12...(h-l)(h+l)...m
In case m is even we consider
?‘hl
Ph2
Pa1
0
Ph-1.m
0
ph+l,h-1
Ph+l,m
..
Pm1
Plm
..
Pm,h-1
Pm,h+l
...
0
Ph,hbl
Ph,h+l
...
Phm
p2,h-1
p2,h+l
...
2’2m
ph-l,h+l
...
1”.
the determinant
.
..
8
=
ph-1,l
ph-1,2
ph+l,l
ph+1,2
.
. .
0
According
to Lemma
6.2,
Pm2
we have
“.
6’ =
pm,h-1
for only
depend
on phk with
pm,h+l
.
0
mhl2...(h-l)(h+l)...m’W23,,.(h-l)(h+l)..,m
Hence in both cases (6.5) only has the trivial
looking
Ph+l,m
..
..
Pm1
Ph-1,m
.
0
ph+l,h-1
solution.
1 5 h, Ic < n.
Thus the functions
From
#
0.
we are
(6.4) we obtain
the
INVARIANT
DIFFERENTIAL
FORMS
ON J’(T*M)
437
system
f12...n
(6.6)
-
1.k
plkx
=
0, P2kXl>k
=
0, . ..) P,kXl,k
=
0,
2<ksn.
Two cases have to be examined:
Case
n =
1:
2r + 1.
are the last n first equation
invariant
Then
in (6.6)
Case 2: n = 2r. There
XL2
~
XL”
whose
equations
Hence
from the
there exist no aut (A4
solutions
{X1,2,
12...n (P121 . . ..Paj.
=------=K
..’
system
solution.
x
U(l))-
of odd degree in J1 (T*M).
exist non-trivial
=
A12
linear
only has the trivial
it follows f~,,.~ = 0. Thus,
forms
horizontal
(6.7)
the homogeneous
1 ones in (6.6)
Al,
. . . . X1,n},
given by
..., PTL-l,n),
where
A12
IO
PYn
P34
:
=
n”.n
I-,&l
---
p34
p24
0
,A13 = / .
:I
pn3
p23
=
and Krs...,
systems
Pn3
...
p23
0
.
equation
tem
{1,3,4
Pn4
“’
0
,
‘..,
0
which will be determined
by using the other
We first replace X1,2, . . . . X1>” from (6.7) in the first
of (6.6) and obtain
fl...n =
we consider
fl...n
-
p2kx2’”
A23
the coefficients
=
= Kl...,,
0,
. . ..P.~,
a_ftfaq 2
of
plkx2’”
=
P31
p34
P41
0
Pnl
Yi)j
do not depend
. . . . pn--l,n)A12...n.
=
in (6.4)
0, p3kx2’”
=
and we obtain
o ,...,
it follows:
p,kx2’”
the sys=
0,
k
X1>2/A12 = X2s/Ass
E
ZZ
where we have set
.
denote
K12...n(p12,
F rom the last m - 1 equations
,..., n}.
. . . = X2>“/A2”
Next,
.
:
P3,n-1
function
from (6.4).
(6.8)
Further
:
l&$,-l
.
h-1,3
is a differentiable
obtained
P4n
-,
Pzn
P2,n-1
p3n
.
;’
..
1:
lP2n
Aln
...
pn4
..
.
..
. .’
= dKl,,,,/dpi.
on {pr2,...,pr,}
p3n
0
P34
‘. .
p3,71-1
p31
p4n
P43
0
. ”
P4,n-1
p41
.
..
Pn4
.‘.
Pn,n-1
PTLl
,..., A2n =
:
0
Pn3
Then
and
taking
.
into account
{p21,...,~2~},
that
respectively,
A”’ and Aarc
and by using
BEJANCU.
438
dXh>” /ad.
y1,2
MASQUfi
the system
=
y2,3
yl,n
yL3
-A12
consider
AND
= aXitj Jap”,, we obtain
(6.9)
From
ENCINAS
a13
=
obtained
.‘.
y2,n
=-=a23=...=Aln
by vanishing
A2n
the coefficient
.
of d_ft/dq3
in (6.4) we
the equations
(6.10)
3,k =
p4kX
Then by using
0
p,kX3’k
, .‘.I
(6.8) and Lemma
0,
x3’l
=
-Kl...nw3456...n
=
-K
x3,4
=
Y3,4Wl2...*
. . . . n}.
we obtain
’w4256...n,
W1456...n,
l...nW3456...n
’w34...*+
( W1256...n . W3456...n -t WlZ...n . W56...n ) i
.. . .. . .. . .. .. . .. .. . .. .. . .. ... .. ..*............
K l...n
. . . . . .. .
x3-
=
Y3,nW12...n
Kl...,
. w34...n+
(W1245...n-1
W3456...n + W12...n
Replace X3)‘” from (6.11) in (6.10) obtaining
0,4 < k 5 n, with non-trivial
solutions
(6.12)
{1,2,4,5,
k E
6.2, by direct calculations
~3~2
(6.11)
=
y3,4
W56...n
=
y3,n
W456...n--1
The equation
obtained
, ..‘I
by vanishing
prkX3’k
(6.13)
W456...n-1
the systemp4kY3’”
y3,n-1
W45...(+2)n
--
1.
= 0, . . ..pnkY3>IC =
y3,n
W(n-1)45...n-2
the coefficient
= 0, k E {1,2,4,5,
of 8.f’ Jag3 in (6.4) is
. . . . n}.
By using (6.11) and (6.12) in (6.13) we obtain
y3,4
Kl ..n
------sign
w345...n.
-
. . . . .. .
(6.14)
y3,n
=
I
By induction
(3, 4, !i,6,
. . . . n)
w56...n,
. .. . .. .. .. . .. .. . .. . .. .. .. .. .. .. .. .. .. .. ... . ...
K1v.n
------sign
W345...n
(3, n, 4,5, . . . . n - 1) ~45...~-1.
on n, and by using the next homogeneous
from (6.4) by vanishing
ykhfl
=
the coefficients
K
_ *sign(h,h+1,3,4
of aftJaqh,
linear
systems
4 5 h 5 n, we get in general
,..., L,h-
,..., n).
W34...n
(6.15)
W34...XGi...n~
. . .. . . .. .
yh,n =
. .. . .. . .. .. . .. .. . .. .. . .. .. .. .. .. .. .. .. .. .. .. ..
KL..,
--sign
(h, n, 3,4, . . . . K, . . . . ^n)w,,,,,-,,,,,,
W34...n
obtained
INVARIANT
DIFFERENTIAL
FORMS
ON
J'(T*M)
439
where the circumflex over a term means that it is to be omitted.
Moreover, by
using (6.8) and the first equation in (6.11) we obtain X3,1 = Y3)1wl...n . w~~...~ +
Kl...nW245...n ' W34...n = -
KI...~w~~...~ . w425...n. Hence Y3,1 = 0. Then from (6.9)
it fohows
Yl,k = Y2,k = 0, 1 5 Ic 5 n.
(6.16)
Finally, taking into account that K I...~ only depends on pij, 1 5 i,j < n, and
by using (6.14)-(6.16) we obtain dKr..., = - (KI...~/w~~...~) d(ws4...n); that is,
KI..., = oIw~...n, cr E B. Replace KI,.,, in (6.8) and obtain fr..., = a~~...~,
which finishes the proof of the theorem.
cl
Remark. For even dimensions, m = 2r, we have G’; = Pfaffian(pij) dql A . .. A dq,.
7. VARIATIONAL
PROBLEMS
DEFINED
Let p: E -+ M be a fibred manifold.
form R, on J’(E)
BY
INVARIANT
FORMS
The horizontal part of a differential n-
is the n-form h(n2,) on J?+‘(E)
= Q,((Y~)*&-+I)+XI,
. . . . (jr~),(pr+l)*Xn),
and every local section s of p ([14]).
such that h(R,)(X1,
with Xl, . . ..X.
Assume M is oriented by a volume form II,.
. . . . X,)
E Tjj+l,(JT+lE)
Each form R,
on J’(T*M)
of degree m = dim M gives rise to a functional on the space of l-forms on M
with compact support, C: I’,(M,T*M)
-+ Iw, C(w) = JM(jrw)*R,.
The first
variation of C at a point w E P (M, T* M) is the linear functional S,C: ggeM + Iw,
&3(X)
= J&w)*
(Lx,,, R,), g&M being the subspace of vector fields X E
gT*M whose support has compact image on M, and Xc,) is the r-jet prolongation
of X. A linear form w is said to be an eztremal of C if 6,C = 0. Two mforms R,,
nk on J’(T*M)
are equivalent (cf, [18]) if for every l-form w on M,
&,C = 13~2’. Obviously, if w, and wh are equivalent, C and 6’ have the same
extremals. The form R, is said to be variationally trivial if it is equivalent to
zero; or equivalently, if every l-form on M is an extremal of its functional. Two
forms of different orders, say f12, on J’(T* M), Rk on J” (T’M), and r’ > r,
are said to be equivalent if pr, ,.C&,,and nk are equivalent. We are interested in
the variational problems defined by gauge invariant m-forms. Such forms admit
a large subspace of symmetries (precisely gau (M x U(1)) C gT*M) but not so
large to be variationally trivial.
Theorem
7.1.
With the above hypotheses and notations we have
1. Every m-form is equivalent to its horizontal part.
2. All gT.M-invariant
m-forms
are variationally trivial.
440
BEJANCU.
3. Given
invariant
a gauge
such that R,
ENCINAS
m-form
n,,
is an m-form on A” T*(M),
and only if fir,, is horizontal.
5. Let f12, = L u,
2 defined by R,,
w + n is also an extremal
cz”=,
1. Let f,
a,,
Q,,
exists on A” T*(M)
(af lap:)
If w is an extremal
m-form.
then for every
if
closed l-form
77 on M,
of C.
be arbitrary
differentiable
functions and forms, reWe have h(R, A Cl;,) = h(R,) A h(nk,), h(df) = df -
eh, where cr = (cyi, . . . . cu,) E KY, ICY= CY~+...+CY,, 0; =
dp; - C”3_-1 pi,+(jjdqj,
are the coordinates
horizontal
onto Jl(T*M)
projects
n;,
on Jr (T*M).
cy,,=,
an m-form
then h(K*fi,)
be a gauge invariant
of the functional
PROOF.
MASQUh
and &*fi2, are equivalent.
4. If %,,
spectively
AND
(j) being
the multi-index
(j)% = bij, 1 5 i 5 m, and (pb)
on J’(T* M); i.e., (pL)(jgw)
induced
= (dla’(pi o w)/aqC1) (x).
X (r+i) = Cz”=, Cafe,
(al”lqi/aqa)
(d/dpi).
Hence
=
Lx ,T+,,e~ = 0, and for every l-form w we thus have (jr+‘w)*(Lxc,+,,h(df))
(j’w)* (Lx(,,(df)).
Part 2. f o 11ows from the definitions and 3. follows from 2. tak-
If X is given by (2.2), then
ing into account (5.1). In order to prove 4., locally we set fi2, = CF==, Ch,% 3 fbz3
dqh A dpij, where h = (hl, . . . . h,_,),
1 I hl < . . . < h,_,
5 m, dqh = dqhl i
A
dqh,_,,
and i = (il , . . ..iT). j = (jl, . . . . jr), 1 I ik < .ik 5 m, 1 5 k I r, (i~,j~)
4
for lexicographic order), dp, = dpi,j,A...Adpi.j,.
Let k1 <
of h; i.e., {kl, . . . . kr} = { 1, . . . . m} - {hl, . . . . h,_,},
let ch be the signature
of the permutation
(1, . . . . m) H (hl, . . . . h,_,, kl, . . . . kr),
.
+
(iT,jT)
stands
(4
. . . < k, be the complement
of the set {kl, . . . . kr}.
and let IIf, be the group of permutations
Ldql r\...r\dqm,
.
only
invariant
C,<j
C =
we have
(dgj/dqi
- dgi/dqj)
Let us consider
Ch,i,j
group O,(g)
= f12,(A(Xl),
Hence
As the Lagrangian
c o K, ,!? E Cm(A2 T*M).
(x) (ac/aPij)
($lc,(kl))
-
&,[kl)))
C E Cm(J’(T*M))
density
Hence
in 5. is gauge
(X(llC)
(j,&)
=
0
(dzw).
metric g on M with volume
form ug. For
(on the right) on A”Tz(M)
by setting (0, . A)(Xl, . . ..X.)
A gauge invariant horizontal m-form 0, = Cv, is said
. . . . A(X,)).
to be g-invariant
R is invariant
’ K,
for the orthogonal group of the scalar product induced
space at x; i.e., O,(g) is the group of R-linear mappings
such that g (A(X), A(Y)) = g(X, Y), VX, Y E T,(M).
The
T,(M)
acts
= (i) + (j).
(ij)
0.
ehe(o)(fh23
=
stands
by g on the tangent
+
CDEII,,
a pseudo-Riemannian
every 2 E M, O,(g)
A: T,(M)
x:0
- P:;~~~~~,, , with
>
if Vr > 0, f&j =
Ir;;vBckplj
(
if and
’ =
h(,*!?,)
Then
if in the decomposition
under
the action
C = Lo K, the function
of the groups
O,(g);
c:
A2 T*(M)
i.e., if for every
-+
z E M,
INVARIANT
A E O,(g),
DIFFERENTIAL
w2 E /j2TG(M),
FORMS
_
.E(w, . A)
1)/2
Lk:
Iw, 1 5 Ic < m’, given by &(wz)
+
is the pseudo-Riemannian
g($-‘w,4-‘w’),
4(X)(Y)
Vw,w’
= g(X,Y),
‘U& E T,‘(M),
Theorem
E T,“(M),
X,Y
g(“)(wl
PROOF.
The action
of O,(g)
where
ring of polynomial
where S’ stands
1s isomorphic
on A’T;(&f)
of g.
From
invariants;
i.e., S’ (A” T,(M))
for the graded
symmetric
and
differentiable
Fz, which
function
x E M, thus finishing
) . . .. (,&J,]
equations
Remark.
For m = 2r the unique
correspond
on Jl(T*M)
= rd(ixp;,dQ,)
)
over V* can be identified
as ,& = F, o ((cl),
funcfor
, . . . . (&t),),
can be taken to depend
smoothly
on
0
Maxwell’s
* (ixp&,dQl)
theory
the proof.
Remark.
m-form
invariant
is a basis for the
that every invariant differentiable
Iw can be written
+
. . ..&f
OzCg) = IF![(Q,
From [16] we thus conclude
tion & : A” T,*(M)
to that of O(s, m - s)
algebra and we use that for a finite-
real vector space V, the ring of polynomials
rd( (j’w)
if
Iw such that for every
classical
13, $8, 381) we know that cl,
dimensional
Lx(,,R,
=
is the polarity
wi)).
F: M x JIP’ 3
with S’(V).
horizontal
T,*(M)
--t Iw is g-invariant
A” T*(M)
2:
function
s is the signature
see [20, Chapter
certain
where gck)
i.e., g(l)(w,w’)
sf(wz) = F (2,~:l(wa),...,c,/(w:!)).
on A” (iIJ?)*,
(e.g.,
+
functions
and for Ic > 1 and every WI ,..., wk,wi, . . . .
function
only 2f there exists a differentiable
wz E A2Tj(M),
= g(2k)(w$,wg),
by g on AkT*(AI);
where 4: T,(M)
E T,(M),
m/2 if m is
g-invariant
A . . . A wk, w{ A . . . A wk) = det(g(‘)(wi,
A differentiable
7.2.
We have m’ natural
metric induced
441
We set m’ =
= C(w2).
even, m’ = (m A2 T*(M)
‘f m is odd.
1
J’(T*M)
ON
to the Lagrangian
(up to scalar factors)
is R,
aut (M x U(l))-invariant
= &*Q$, which is variationally
A K*(Q~-‘),
A (dw)r-l),
C = I?~OK, for m = 4.
VX
E ggeM.
Hence
trivial as
(jlw)*(Lx(,,Rm)
an d we can apply Stokes’ theorem.
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Received
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(Bejancu)
6600-1~~1,
version
received
DEPARTMENT OF MATHEMATICS, TECHNICAL UNIVERSITY “G.H.
10, 1998
ASACHI” OF IASI,
ROMANIA
E-mail address: reluQmath. tuiasi.
(Encinas)
ro
DEPARTAMENTO DE DIDACTICA DE LAS MATEMATICAS Y DE LAS CC.EE.,
SIDAD DE SALAMANCA, PASEO CANALEJAS 169, X008-SALAMANCA.
E-mail address: encinasQgugu.
(MasquB) CSIC-IFA,
E-mail
July 15, 1997
February
address:
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C/SERRANO
jaimeQiec.
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