A gauge-theoretic description of $μ$-prolongations, and $μ$-symmetries of differential equations

0.31 MB·English

by  G. Gaeta

A gauge-theoretic description of µ-prolongations,
and µ-symmetries of differential equations
9
0
0
Giuseppe Gaeta∗
2
Dipartimento di Matematica, Universita’ di Milano
n
a via Saldini 50, 20133 Milano (Italy)
J

0 To be published in J. Geom. Phys. Version of January 2009
2

]
h Summary. We consider generalized (possibly depending on fields as well as
p on space-time variables) gauge transformations and gauge symmetries in the
-
h contextof general– that is, possibly nonvariationalnorcovariant– differential
t equations. In this case the relevant principal bundle admits the first jet bundle
a
(of the phase manifold) as an associated bundle, at difference with standard
m
Yang-Mills theories. We also show how in this context the recently introduced
[
operationofµ-prolongationofvectorfields(whichgeneralizestheλ-prolongation

1 of Muriel and Romero), and hence µ-symmetries of differential equations, arise
v naturally. This is turn suggests several directions for further development.
6
MSC: 58J70; 35A30; 58D19; 76M60
9
0
3
1 Introduction
.
1
0
The analysis and use of symmetry properties of differential equations [10, 15,
9
24, 26] is by now recognized as the most powerful general method to attack
0
: nonlinearproblems,andwidelyusednotonlyinPhysics(wherethis theorywas
v
first extensively applied) but also in Applied Mathematics and Engineering.
i
X The original theory of Lie-point symmetries was over the years generalized
r in several directions [5, 10, 15, 24, 26]. All these make use of the fact that
a
once we know how a vector field acts on independent as well as on dependent
variables (i.e. fields), we also know how it acts on field derivatives; the lift of
a transformation from the extended phase manifold (space-time variables and
fields;withdueaccountoftherelevantside–e.g. boundary–conditions,thisis
thephasebundle)toitsactiononfieldderivativesisknowninthemathematical
literature as the prolongation operation.
Despite its success, the symmetry theory of differential equations was until
recently not able to cope with certain very simple problems which could be ex-
plicitlyintegrated,yetseemedtohavenosymmetryunderlyingthisintegrability
(see e.g. [17, 24] and references therein).
∗[email protected]
1
Muriel and Romero [17] were able to solve this puzzle in analytical terms
by considering a modified prolongationoperation, and thus a new kind of sym-
metries, for scalar ODEs and then also for systems of these [18, 19] (see also
∞
[6] in this respect). These depend on the choice of a function, denoted λ
C
in their papers; referring to this fact the new kind of symmetries are known as
∞
-symmetries, or λ-symmetries.
C
The geometrical meaning of λ-prolongations was then clarified in [25], by
means of the classical theory of characteristics of vector fields. A different geo-
metrical characterization, in the language of Cartan exterior differential ideals
(ideals of differential forms), was proposed in [12].
This opened the way for generalizing λ-prolongations and λ-symmetries to
the framework of (single, or systems of) PDEs [12]; in this case the central
object is a matrix-valued differential one-form µ = Λ dxi (the Λ being ∞
i i
C
matrix functions satisfying the horizontal Maurer-Cartan equation), and these
are therefore called µ-prolongations and µ-symmetries.
It was then realizedthat for eachvectorfield Y obtainedas the µ-prolonga-
tionofsomevectorfieldX,thereisavectorfieldY,locally(andgloballyunder
certain conditions) gauge-equivalent to Y, obtained as the standard prolonga-
e
tion of a vector field X, locally (and globally under certain conditions) gauge
equivalent to X; see [8] for details.1
e
Thisresultcallsforamorecompletegeometricalunderstandingofitsorigin,
and shows that gauge transformations play a role also out of the well-known
framework of Yang-Mills theories [1, 4, 9, 14, 22, 23], and actually also for
non-variational problems and for non-invariant equations (or non-covariant
equations in physical language). The first task is to extend the formalism so to
fully include the gauge variables, not just leaving them to the role of external
parameters.2
In this note we provide a formalism including gauge variables, i.e. set the
problem in an augmented bundle; and give a precise formulation of the µ-
prolongation operation in terms of such an augmented bundle and correspond-
ingly an enlarged set of variables.
The reader should be warned that this geometrical understanding does not
– at the present stage – correspond to a substantial computational advantage;
thus the geometric construction presented here has – at the present stage –
interest only per se, i.e. for the understanding of the Geometry behind twisted
prolongations. On the other hand, our work also suggests how to extend the
applicationsofµ-symmetries,andhowtofurthergeneralizethem;thesematters
will however only be shortly mentioned in our final discussion, see section 9,
deferring the implementation of such suggestions to a later time.
We will assume the reader to be familiar with basic jet-theoretic material
1This also has some interesting consequences in the frame of variational problems: it is
possibletoextend Noethertheorytoλandµsymmetries;see[7,21].
2In order to avoid any confusion, we stress that here we consider gauge transformations
moregeneral thanthoseconsideredinstandardYang-Millstheory: (1)thesemaydepend on
the field themselves and not only on space-time variables; (2) we allow nonlinear actions on
thefields. Seethediscussionlateroninthispaper.
2
and with the standard theory of symmetry of differential equations, as given
e.g. in [10, 15, 24, 26]. We will freely use standard multi-index notation; thus
a multi-index J = (j ,...,j ), where j N, will have order J = j +...+j ;
1 k i 1 k
and for the same multi-index J we will∈have D =Dj1...Djk|. |
J 1 k
We use the formalism of evolutionary representatives of vector fields [10,
15, 24, 26], which provide an action on sections of a bundle via (generalized)
verticalvectorfieldsdescribingtheactionofgeneral(proper)vectorfieldsinthe
bundle. Adiscussionwithoutresortingtoevolutionaryrepresentativeswouldbe
equivalent, but would require more involved computations; on the other hand,
our discussionimmediately extends (with the standardcautions [10, 15, 24, 26]
and obvious modifications) to the full class of generalized vertical vector fields.
Acknowledgements.
IamindebtedtoGiampaoloCicognaandPaolaMorandoforseveralconstructive
discussions, and to Giuseppe Marmo for ongoing encouragement. I would also
liketothankDiegoCatalano-Ferraiolifordiscussionsabouthisviewofauxiliary
variables in the λ-symmetries formalism.
2 Underlying geometry
In this section we will first set some notationregardingstandardconstructions,
and then introduce the gauge bundles, which are the essential part of our
construction.
Given a fiber bundle , the space of sections in it will be denoted as Γ( ).
P P
The algebra of vector fields in will be denoted as ( ), and that of vertical
P X P
(with respect to the bundle projection) vector fields as ( ).
v
X P
2.1 Bundles, differential equations, and symmetry
When dealing with differential equations, the independent variables will be de-
noted as x B; dependent variables as u U. Here B and U are smooth
∈ ∈
manifolds. We will use local coordinates x1,...,xm in B and u= u1,...,un
{ } { }
in U. We stress that all of our considerations will be local.
We will then consider the bundle (M,π,B) with fiber π−1(x)=U; thus its
total space is M B U. The bundle M will be our phase bundle.3.
≃ ×
Differential equations ∆ of orderk identify a submanifold in the total space
of the Jet bundle JkM, the solution manifold S JkM.
∆
⊂
Sections of M are naturally prolongedto sections of JkM; the function u=
f(x)isasolutionto∆ifandonlyiftheprolongationofthecorrespondingsection
σ =(x,f(x)) in Γ(M) to a section in Γ(JkM), call it σ(k), is a submanifold of
f f
S .
∆
3Physically,B shouldbethoughtasaregionofspace-time,andU asamanifold(possibly
a space) in which the field ϕ takes values; the global structure of the bundle also carries
informationontheboundaryconditionsthefieldsaresubjecttoattheboundary(ifany)∂B
ofB.
3
Similarly,givenavectorfieldX actinginM,thereisanaturalprolongation
of X to a vector field X(k) in JkM. The vector field X is a symmetry of ∆ if
and only if it maps solutions into solutions; equivalently, if X(k) :S TS .
∆ ∆
→
Westressthatadifferentfibredstructureisalsopossibleforthejetmanifolds
M(k) = JkM. These can also be seen as bundles over M; we will denote
the bundle maps for these structures as σ , so we have (JkM,σ ,M). The
k k
compatibility between these two structures is given by π =π σ .
k k
◦
Remark 1. As already mentioned, one can consider – beside the natural pro-
longationoperationforvectorfields mentionedabove–some“twisted”(or“de-
formed”) prolongation operations, known in the literature as “λ-prolongation”
(whenthedeformationisrelatedtoapoint-dependentscalefactor)or“µ-prolon-
gation” (when the deformation is related to a general point-dependent linear
map). These were first introduced by Muriel and Romero [17], and quite sur-
prisingly turn out to be “as useful as the naturalones” in analyzingdifferential
equations. Thepurposeofthispaperistoinvestigatethegeometricalstructures
behindthisseemingly“unreasonableeffectivenessoftwistedprolongations”.
⊙
2.2 Prolongation of vector fields in JkM
The prolongationof vector fields fromM to JkM goes essentially throughcon-
sideration of partial derivatives of functions corresponding to a general sec-
tion in M and its transformed under the action of the vector field X. As
well known, if σ = (x,u) : u = f(x) , then the one-parameter group
f
{ }
generated by X = ϕa(x,u)(∂/∂ua) + ξi(x,u)(∂/∂xi) maps σ into σ with
f f
fa(x)=fa(x)+ε[ϕa(x,f(x)) ξi(x,f(x))ua]. Propertiesoftransformationfor
− i e
partial derivatives are readily derived from this expression.
e
In the case of a vertical vector field (including evolutionary representatives
of general vector fields) the prolongation formula is specially simple: in multi-
indexnotation,thevectorfieldX =ηa(∂/∂ua)isprolongedtoY =ηa(∂/∂ua),
J J
with ηa =D ηa. In particular we have in recursive form
J J
ηa = D ηa . (1)
J,i i J
2.3 µ-prolongations in JkM
Letusbrieflyrecallhowµ-prolongationsaredefined,restrictingagaintovertical
vectorfieldsforthesakeofsimplicity(see[8,12]forthegeneralcase). Westress
thathereweworkinM andJkM, andnot inthe augmented(jet) bundle to be
defined below.
VerticalvectorfieldsinM areprolongedtovectorfieldsinJkM viaamodi-
fiedprocedure basedona horizontalone formµ with values in a representation
of a Lie algebra [1, 9, 13, 27]. The form µ should satisfy the horizontal
G
Maurer-Cartanequation
1
Dµ + [µ,µ] = 0 . (2)
2
4
In terms of the local coordinates introduced above, we have
µ = Λ (x,u,u ) dxi (3)
i x
with Λ some n n matrices (these are related to the Lie algebra of a Lie
i
× G
groupGactinginU,see[8,12]fordetailsonthisrelation). Withthis notation,
the horizontal Maurer-Cartanequation (2) reads
D Λ D Λ + [Λ ,Λ ] = 0 . (4)
i j j i i j
−
WewriteverticalvectorfieldsinM asX =ηa(x,u,u )(∂/∂ua),andthecor-
0 x
respondingverticalvectorfieldsinJkM asY =ηa(∂/∂ua). Theµ-prolongation
0 J J
formula for vertical vector fields is then, in recursive form (cf. (1) above),
ηa = D ηa + (Λ )a ηb . (5)
J,i i J i b J
The condition (4) guarantees the ηa are well defined, see [8, 12].
J
It was shownin [8] that (locally,and possibly globallyas well) µ canalways
be written as µ = g−1Dg, and correspondingly µ-prolonged vector fields are
related to standardly prolonged ones via a gauge transformation. We refer to
[8] (in particular Theorem 1 in there) for details.
3 The gauge bundles
Let G be a Lie group and ǫ:G e the operator mapping the whole Lie group
→
G into its identity element. We assume G acts on U via a (possibly nonlinear)
representationT :G U U;atapointp U theactiononV :=T U (thisis
p
× → ∈
the relevantactionwhendiscussing howG actsonvectorfields)is describedby
the linearization Ψ = DT of T. This also induces an action of the Lie algebra
of G in V =T U via the linear representation ψ =DΨ.
p
G
Let (ℓ ,...,ℓ ) be a basis of left-invariant vector fields in ; we write L =
1 r i
G
ψ(ℓ ) for their representation. Any element ξ can be written4 as ξ =αmℓ
i m
∈G
and acts in V via the vector field ψ(ξ)=αmL .
m
3.1 The basic gauge bundles
We introduce a principal bundle (P ,ǫ,M) over M with bundle map ǫ, fiber
G
ǫ−1(p) = G, and total space P M G = M. This will be called the
G
≃ ×
global gauge bundle. Sections γ Γ(P ) are described in local coordinates
G
∈ f
by g =g(x,u).
ThetotalspaceP =M Gcanalsobegiventhestructureofafiberbundle
G
×
(M,π,B) over B with projection π = π ǫ. The compatibility between these
×
two structures is given by π =π ε.
fLeet us consider a reference se◦cetion ̟ Γ(P ); in a tubular neighborhood
G
e ∈
G M G M of̟ (with G aneighborhoodofzeroinG, and
0 0 0 0
≃ × ≃ ×G ≃G G
b 4Withαm=hℓm,ξi,whereh.,.iisthescalarproductinG. Theαmarenaturalcoordinates
inG,andusingtheseℓm isgivenbyℓm=∂/∂αm.
5
aneighborhoodofzeroin )wecanuselocalcoordinates(x,u,α), whereα=0
G
identifies the section ̟. We will, for ease of notation and discussion, work in
M ; it should be kept in mind that our results will be local in (and hence
×G G
a fortiori in G). The projection from to 0 will be denoted as ρ.
G { }∈G
The manifold M can alsobe giventwo different fiber bundle structures:
×G
we can consider it as a bundle (M,π,B) over B with total space M =M ,
projection π =π ρ and fiber π−1(x)=U ; or as a bundle ( ,ρ,M)×ovGer
M with total spa×ce =M =Mc b , proj×ecGtion ρ and fiber ρ−G1(Bcp)= . The
b GB b ×G G
compatibility between these two structures is given by π =π ρ.
c ◦
We will denote M =(M ,π,B) as the augmented phase bundle, and
=(M ,ρ,M) as the l×ocGal gauge bundle. In thbe following we will also
GcaBll these ×theGaugmcented bundble and the gauge bundle for short.
Remark 2. We stress that M is not an associated bundle for this principal
fiber bundle, contrary to what happens in Yang-Mills theories (where g =g(x)
and does not depend on u); this is the reason for some features which could
appear odd to readers familiar with standard Yang-Mills theory. On the other
hand,(J1M,σ ,M),is anassociatedbundlefor(P ,ǫ,M),whichactsonfibers
1 G
ǫ−1(m) via the representation ψ =(DΨ).
⊙
3.2 Higher order gauge bundles
WewillalsointroducegaugebundlesassociatedtohigherjetspacesJkM. Thus
we consider the order k augmented jet bundle M(k) = (JkM ,π ,B) with
k
fiber π−1(x) = U(k) ; and the order k jet gauge bundle J×k G = (JkM
,ρ ,JkkM) with fibe×rGρ−1 = . (Similarly, hcigher order bunGdlBesbwith tot×al
G k b k G
space M(k) JkM G could be defined. We will not enter into such details.)
≃ ×
Moreover,recall that JkM can be seen as a bundle over M with projection
f
σ , and correspondingly JkM can be seen as a bundle over M with projection
k
σ .
k
c c
The situation is summarized in the following diagram, which also embodies
tbhe different double fibrations considered above.
JkM ρk JkM
−→
c πk πk
ց ւ (6)
σk bπ B π σk
 
 
yb րb ρ տ y
M M
−→
c
3.3 Total derivative operators in gauge jet bundles
As well known, the prolongation operation is usually performed by applying
the total derivative operators in JkM; see section 2.2 above. As the gauge jet
bundles we are considering have a peculiar structure (that is, they are order k
jetbundlesforwhatconcernstheuvariables,notforwhatconcernstheαones),
6
we should discuss what are the total derivative operators to be considered in
this case.
Jet spaces are equipped with a contact structure , defined by the contact
C
forms χa := dua ua dxi; note that there are no contact forms associated to
J J − J,i
gauge variables, as we are not considering jets of these. The total derivative
operators D can then be defined in geometric terms as the vector fields (with
i
a component ∂/∂xi) annihilating all the contact forms in ; it is immediate
C
to check that this requirement yields just the usual total derivative operators
(associated to the u variables alone) D = (∂/∂xi)+ua (∂/∂ua). We stress
i J,i J
that one should not add also components “along the gauge variables”, i.e. of
the form αm(∂/∂αm).
i
Thus,the prolongation operation leading from M to JkM should bebased on
the usual total derivative operators D , and hence does not involve derivation
i
c c
with respect to the gauge variables.5
Thisfactshowsthatasubstantialdifferenceexistsbetweenthegaugebundle
and the bundle obtained by simply adding new dependent variables αm.
4 Prolongation of vector fields in M and in M
g d
GivenavectorfieldinM,thisisnaturallyprolonged (orlifted)toavectorfield
in JkM. The same applies for vector fields in M and M, which are naturally
prolonged to vector fields respectively in JkM and JkM.
f c
We will denote by Pr(k)[ ] the operator of prolongationof vector fields in a
P f c
bundle to vector fields in the jet bundle Jk , and omit the indication of the
P P
bundle (i.e. just write Pr(k)) when there is no risk of misunderstanding.
P
4.1 Prolongation of general vector fields
We willgive some explicitformulas inthe localcoordinates(x,u,α) introduced
above. With the (x,u) coordinates in M, any vector field in M is written as
X =ξi(∂/∂xi)+ϕa(∂/∂ua)+Bmℓ , with ξi,ϕa,Bm depending on (x,u,g).
m { } f
Passing to the restriction X of X to , i.e. its expression in M, and in-
e G
troducing also the local coordinates α in (recall ℓ = ∂/∂αm), we have
m
b e b G c
X =ξi(x,u,α)(∂/∂xi)+ϕa(x,u,α)(∂/∂ua)+ Bm(x,u,α)(∂/∂αm).
As usual in considerations involving vector fields on jet bundles, it will be
b
convenient to work with evolutionary representatives [10, 15, 24, 26]; we will
consistently use these. The evolutionary representative of X is
∂ ∂ b
X X = Qa + Pm , (7)
≡ v ∂ua ∂αm
b
5It may be worth stressing, just to avoid any possible misunderstanding, that albeit a
vector field in M (respectively, in M) will have components both in the M and in the G
(respectively, G) directions, the prolongation operation should be applied only to the M
components, aseobviousfromthedefibnitionofJkM andJkM above.
e b
7
where Qa :=ϕa uaξi, Pm :=Bm.
− i
ThecoordinateexpressionoftheprolongationX(k) (JkM)ofX isgiven
∈X
by the (standard) prolongation formula [10, 15, 24, 26]. (As implied by the
c
discussioninsection3.3above,noprolongationofαm componentsappear). For
the evolutionaryrepresentativeY :=(X(k)) =X(k) we get, with Qa =D Qa,
v v J J
∂ ∂
Y = Qa + Pm . (8)
J ∂ua ∂αm
J
4.2 Prolongation of gauged vector fields
We are specially interested in a particular class of vector fields in (M), i.e.
v
X
those for which
f
Qa(x,u,g;u ) = [Ψ(g)]a Θb(x,u;u ) . (9)
x b x
In the following we will refer to these as gauged vector fields.
Restricting gauged vector fields to M, and using local coordinates (x,u,α),
eq. (9) becomes
c
Qa(x,u,α;u ) = [K(α)]a Θb(x,u;u ) ; (10)
x b x
hereK(α)istherepresentationofthegroupelementg(α)=exp(α),i.e. K(α)=
Ψ[exp(α)]. (Westressthat(9)and(10)donotconstraininanywaythecompo-
nents Pm of the vector fields along the αm variables; this will be of use below.)
Keeping in mind our discussion above about the total derivative operators
in M(k), see section 3.3, we obtain immediately that
c Qa = D Qa = [K(α)]a D Θb . (11)
J J b J
This implies that – writing Θ =D Θ – the prolongationY of the vector field
J J
X, see (8), is given by
∂ ∂
Y = [K(α)]a Θb + Pm . (12)
b J ∂ua ∂αm
J
Remark 3. Let X0 = ρ∗X and Y0 = ρ(∗k)Y be the projection of the vector
fields X and Y to the bundles, respectively, M and JkM. Then we can state
formally that Y is the µ-prolongation of X for a suitable µ. In fact, we have
0 0
∂ ∂
X0 =ρ∗X =[K(α)]abΘb ∂ua ; Y0 =ρ(∗k)Y =[K(α)]abΘbJ ∂ua . (13)
J
ThusX andY arethegaugetransformed–viathesamegaugetransformation
0 0
– of vector fields X and Y such that Y is the ordinary prolongation of X ;
0 0 0 0
By Proposition 1 above, Y is the µ-prolongationof X for a suitable one-form
0 0
µ. Notethis statementis onlyformal,asthe αvariableshavenomeaningwhen
8
weworkinM andJkM;inordertomakethisintoarealtheorem,wewillneed
to “fix the gauge”, as discussed below. Note also the relation between X and
0
Y depends substantially on the assumption X is a gauged vector field.
0
⊙
Remark 4. The reasonfor the name “gauged”of vector fields considered here
is quite clear: the horizontal(for the gauge fibration)componentQa∂ of these
a
corresponds to vector fields in M on which we operate with an element of the
Lie group G, element which may vary for varying x and u.6
⊙
5 Standard prolongation of vector fields in M
and µ-prolongation of vector fields in M d
We showed above that the natural prolongation in gauge bundles is natu-
rally related (via the results in [8]) to a µ-prolongation. However, λ- and µ-
prolongations are usually [6, 8, 12, 17, 18, 19, 21, 20] defined with no use of
auxiliarygaugevariables;inourlanguagethiswillcorrespondtoagaugefixing.
In this section we will discuss how gauge fixing affects the prolongation op-
eration,andtherelationbetweengauge-fixedprolongationandµ-prolongations.
5.1 Sub-bundles defined by sections of the gauge bundle
Earlier on we considered the augmented bundle M and correspondingly JkM.
Herewe wantto considerthe subbundle M :=γ(M) M definedby asection
γ
c ⊂ c
γ Γ( ) of the gauge bundle (hence by a section of the global gauge bundle
⊂ GB c c
closetothereferencesection̟);wewillalsoconsiderM(k) :=γ(JkM) JkM.
γ
⊂
Inthe(x,u,α)coordinates,M isthesetofpoints(x,u,α)withα=A(x,u);
γ c
similarly M(k) is the set of points (x,u,α,u(1),...,u(k)) againwith α=A(x,u).
γ c
Note that M M, and correspondinglyM(k) JkM.
c γ ≃ γ ≃
These submanifolds of the gauge bundle and of the jet gauge bundle have
c c
a natural structure of fiber bundles (over B) themselves, and can be seen as
sub-bundles of M and JkM7; we will thus also write JkM for M(k). It will
γ γ
thus make sense to speak of vertical vector fields in M and JkM (referring
c c γ c cγ
implicitly to these fiber bundle structures).
c c
Givenasectionγ Γ( ),wewilldenotebyω(γ) theoperatorofrestriction
∈ GB
fromM to M , andby ρ(γ) the restrictionofthe projectionρ:M M to M .
γ γ
→
We also denote by ω(γ) : JkM M(k) and by ρ(γ) : M(k) JkM the lift of
c c k → γ k γ →c c
the maps ω(γ) and ρ(γ) to maps between corresponding jet spaces of order k.
c c c
6Note that if we operate on e by a x-dependent change of frame, i.e. pass to a frame
f =(f1,...,fn)withfa=Tabeb whereT =(KT)−1,thenwegetΦ=ϕaea=ϕa(T−1)abfb:=
ϕafa, i.e. the components of Φ in the new frame are ϕa = Kabϕb. This point of view is
discussedelsewhere[11].
e 7We stress this refers to the structure of bundles oveer B, i.e. – referring to diagram (6)
–totheπk projections; morecarewillbeneeded forwhat concerns thestructureofbundles
overM,seesection5.2below.
b
b
9
Note that while ρ is of course not invertible, it follows from M M that ρ(γ)
γ
is invertible, with (ρ(γ))−1 =γ. similarly, ρ(γ) is invertible. ≃
We willsummarizerelationsandmapsbetweenrelevantfiberbundles inthe
following diagram:
ω(γ) ρ(γ)
M M M
γ
−→ −→
c c
jk[M] jk[Mγ] jk[M] (14)
  
JkyM bωk(γ) My(k) bρ(kγ) JkyM
γ
−→ −→
c c
Remark 5. In physical terms, passing to consider M rather than the full M,
γ
and M(k) rather than the full M(k), corresponds to a gauge fixing.
γ c c⊙
c c
5.2 Prolongations and gauge fixing
WhendealingwithM ,i.e. workinginafixedgauge,weshouldthinkthegauge
γ
variablesαasexplicitfunctionsofxandu(givenbyA(x,u)identifyingγ);note
this means D αm will now read as D Am(x,u) and thus will in general give a
i i
nonzerofunction. This entailsthe jet structure ofM(k) is notthe one inherited
γ
from the jet structure of M(k) (which we denote by j [M ], the associated
c k γ
prolongationoperator being Pr(k)[M ]).
c γ c
Let us now consider vector fields. Consider X (M) written as in (7),
c ∈ Xv
and Xγ =ω∗(γ)X be its restriction to Mγ ⊂M. Then Xγcis
X = Qa (∂/∂cua)+Pcm(∂/∂αm) (15)
γ γ γ
where the coefficients Q ,P are of course given by
γ γ
Qa =[Qa] , Pm =[Pm] . (16)
γ α=A(x,u) γ α=A(x,u)
ItshouldbenotedthatforarbitraryX andγ,thesubmanifoldγ isisgeneral
not invariant under X . More precisely, with the notation (16), we have:
γ
Lemma 1. Let X be in the form (7). Then the submanifold M M identified
γ
⊂
by αm =Am(x,u) is invariant under X if and only if
γ
c c
Pm = (∂Am/∂ua) Qa . (17)
γ γ
Proof. By standard computation. Note that using the notation in section 4.1,
this also reads as Pm =X(A) on M .
γ
♦
c
Corollary 1. Given arbitrary smooth functions Qa(x,u,α), and an arbitrary
section γ Γ(M), there is always a vector field X (M) of the form
v v
∈ ∈ X
X =Qa∂ +Pm∂ and such that X leaves M invariant.
v a cm v γ c
c
10