<< . .

. 25
( : 71)



. . >>


W r P := {j r ψ(0, e); ψ ∈ PB m (G)(Rm — G, P )}.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
15. Prolongations of principal ¬ber bundles 151


In particular, W r (Rm — G) is identi¬ed with Rm — Wm G by the rule
r

Rm — W m G
r
(x, j r •(0, e)) ’ j r („x —¦ •)(0, e) ∈ W r (Rm — G)
where „x = tx —idG , and so there is a well de¬ned structure of a smooth manifold
on W r (Rm — G). Furthermore, if we de¬ne the action of W r on PB m (G)-
morphisms by the composition of jets, i.e.
W r χ(j r ψ(0, e)) := j r (χ —¦ ψ)(0, e),
W r becomes a functor. Now, taking any principle atlas on a principal bundle
P , the application of the functor W r to the local trivializations yields a ¬bered
atlas on W r . Finally, there is the right action of Wm G on W r P de¬ned for
r

every j r •(0, e) ∈ Wm G and j r ψ(0, e) ∈ W r P by (j r ψ(0, e))(j r •(0, e)) = j r (ψ —¦
r

•)(0, e). Since all the jets in question are invertible, this action is free and
transitive on the individual ¬bers and therefore we have got principal bundle
(W r P, p —¦ β, M, Wm G) called the r-th principal prolongation of the principal
r

bundle (P, p, M, G). By the de¬nition, for a morphism • the mapping W r •
r
always commutes with the right principal action of Wm G and we have de¬ned the
functor W r : PBm (G) ’ PB m (Wm G) of r-th principal prolongation of principal
r

bundles.
15.4. Every PBm (G)-morphism ψ : Rm — G ’ P is identi¬ed with a couple
(ψ0 , ψ1 ), see 15.1.(2). This yields the identi¬cation
W r P = P r M —M J r P
(1)
and also the smooth structures on both sides coincide. Let us express the corre-
sponding action of Gr Tm G on P r M —M J r P . If (u, v) = (j0 ψ0 , j r ψ1 (ψ0 (0))) ∈
r r
m
P r M —M J r P and (A, B) = (j0 •0 , j0 •) ∈ Gr
r r r
¯ Tm G, then 15.2.(2) implies
m
ψ —¦ •(x, a) = ψ(•0 (x), •(x).a) = ψ1 (ψ0 —¦ •0 (x)).•(x).a
¯ ¯
= (ρ —¦ (ψ1 , • —¦ •’1 —¦ ψ0 ) —¦ (ψ0 —¦ •0 )(x)).a
’1
¯ 0
where ρ is the principal right action on P . Hence we have
(u, v)(A, B) = (u —¦ A, v.(B —¦ A’1 —¦ u’1 ))
(2)
where ™.™ is the multiplication
m : J r P —M J r (M, G) ’ J r P, r r r
(jx σ, jx s) ’ jx (ρ —¦ (σ, s)).
The decomposition (1) is natural in the following sense. For every PBm (G)-
morphism ψ : (P, p, M, G) ’ (P , p , M, G), the PB m (Wm G)-morphism W r ψ
r

has the form (P r ψ0 , J r ψ). Indeed, given • : Rm — G ’ P , we have (ψ —¦ •)0 =
’1
ψ0 •0 , (ψ —¦ •)1 = ψ —¦ • —¦ (ψ0 —¦ •0 )’1 = ψ —¦ •1 —¦ ψ0 . Therefore, in the category
˜
of functors and natural transformations, the following diagram is a pullback
wJ
Wr r



u u
wB
Pr —¦ B
Here B : PBm (G) ’ Mfm is the base functor, the upper and left-hand natural
transformations are given by the above decomposition and the right-hand and
bottom arrows are the usual projections.

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
152 Chapter IV. Jets and natural bundles


15.5. For every associated bundle E = P [S; ] to a principal bundle (P, p, M, G)
there is a canonical left action r : Wm G — Tm S ’ Tm S of Wm G = Gr
r r r r r
Tm G
m
r r r r
on Tm S. We simply compose the prolonged action Tm of Tm G on Tm S, see
12.13, with the canonical left action of Gr on both Tm G and Tm S, i.e. we set
r r
m


(j r •(0, e), j0 s) = j0 ( —¦ (• —¦ •’1 , s —¦ •’1 ))
r r r
(1) ¯ 0 0


for every j r •(0, e) = (j0 •0 , j0 •) ∈ Gr
r r r
¯ Tm G.
m

Proposition. For every associated bundle E = P [S; ], there is a canonical
structure of the associated bundle W r P [Tm S; r ] on the r-th jet prolongation
r

J r E.

Proof. Similarly to 14.6, every action : G — S ’ S determines the functor L
on PBm (G), P ’ P [S, ] and • ’ {•, idS }, with values in the category of
the associated bundles with standard ¬ber S and structure group G. We shall
essentially use the identi¬cation

Tm S ∼ J0 (Rm — S) ∼ J0 ((Rm — G)[S; ])
r
=r =r
r r r
j0 s ’ j0 (idRm , s) ’ j0 {ˆ, s}
(2) e

where e : Rm ’ Rm — G, e(x) = (x, e). Then the action r
ˆ ˆ becomes the form

(j r •(0, e), j0 {ˆ, s}) = j0 {ˆ, —¦ (• —¦ •’1 , s —¦ •’1 )}
r r r
(3) e e ¯ 0 0
= J r (L•)(j0 {ˆ, s}).
r
e

Now we can de¬ne a map q : W r P — Tm S ’ J r E determining the required
r

structure on J r E. Given u = j r ψ(0, e) ∈ W r P and B = j0 s ∈ Tm S, we set
r r



q(u, B) = J r (Lψ)(j0 {ˆ, s}).
r
e

Since the map ψ is a local trivialization of the principal bundle P , the restriction
r r
qu = q(u, ) : Tm S ’ Jψ0 (0) E is a di¬eomorphism. Moreover, for every A =
j r •(0, e) ∈ Wm G, formula (3) implies
r



(A’1 , B)) = J r (L(ψ —¦ •)) J r (L•’1 )(j0 {ˆ, s}) = q(u, B)
r r
q(u.A, e

and the proposition is proved.

For later purposes, let us express the corresponding map „ : W r P —M J r E ’
r
Tm S. It holds
r r
„ (u, jx s) = j0 („E —¦ (ψ —¦ e, s —¦ ψ0 ))
ˆ

where „E : P —M E ’ S is the canonical map of E and u = j r ψ(0, e) ∈ Wx P .
r


Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
15. Prolongations of principal ¬ber bundles 153


15.6. First order principal prolongation. We shall point out some special
properties of the groups Wm G and the bundles W 1 P . Let us start with the group
1

Tm G. Every map s : Rm ’ G can be identi¬ed with the couple (s(0), »s(0)’1 —¦s),
r

and for a second map s : Rm ’ G we have (we recall that »a and ρa are the left
and right translations by a in G, µ is the multiplication on G)

µ —¦ (s , s)(x) = s (0)s (0)’1 s (x)s(0)s(0)’1 s(x)
(1)
= s (0)s(0) conjs(0)’1 (s (0)’1 s (x)) s(0)’1 s(x) .

It follows that Tm G is the semi direct product G J0 (Rm , G)e . This can be
r r

described easily in more details in the case r = 1. Namely, the ¬rst order jets
are identi¬ed with linear maps between the tangent spaces, so that (1) implies
Tm G = G (g — Rm— ) with the multiplication
1



(a , Z ).(a, Z) = (a a, Ad(a’1 )(Z ) + Z),
(2)

where a, a ∈ G, Z, Z ∈ Hom(Rm , g). Taking into account the decomposition
15.2.(1) and formula 15.2.(2), we get

1
(g — Rm— )
Wm G = (GL(m) — G)

with multiplication

(A , a , Z ).(A, a, Z) = (A —¦ A, a a, Ad(a’1 )(Z ) —¦ A + Z).
(3)

Now, let us view ¬bers Px M as subsets in Hom(Rm , Tx M ) and elements
1
1
in Jx P as homomorphisms in Hom(Tx M, Ty P ), y ∈ Px . Given any (u, v) ∈
P 1 M —M J 1 P = W 1 P and (A, a, Z) ∈ (G1 — G) (g — Rm— ), 15.4.(2) implies
m


(u, v)(A, a, Z) = (u —¦ A, T ρ(v, T »a —¦ Z —¦ A’1 —¦ u’1 ))
(4)

where ρ is the principal right action on P .
15.7. Principal prolongations of frame bundles. Consider the r-th prin-
cipal prolongation W r (P s M ) of the s-th order frame bundle P s M of a manifold
M . Every local di¬eomorphism • : Rm ’ M induces a principal ¬ber bundle
morphism P s • : P s Rm ’ P s M and we can construct j(0,es ) (P s •) ∈ W r (P s M ),
r

where es denotes the unit of Gs . One sees directly that this element de-
m
r+s r+s
pends on the (r + s)-jet j0 • only. Hence the map j0 • ’ j(0,es ) (P s •)
r

de¬nes an injection iM : P r+s M ’ W r (P s M ). Since the group multiplication
in both Gr+s and Wm Gs is de¬ned by the composition of jets, the restriction
r
m m
i0 : Gr+s ’ Wm Gs of iRm to the ¬bers over 0 ∈ Rm is a group homomor-
r
m m
phism. Thus, the (r + s)-order frames on a manifold M form a natural reduction
iM : P r+s M ’ W r (P s M ) of the r-th principal prolongation of the s-th order
frame bundle of M to the subgroup i0 (Gr+s ) ‚ Wm Gs .r
m m


Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
154 Chapter IV. Jets and natural bundles


15.8. Coordinate expression of i0 : Gr+s ’ Wm Gs . The canonical coordi-
r
m m
r+s
nates xi on Rm induce coordinates ai , 0 < |±| ¤ r + s, on Gr+s , ai (j0 f ) =
± m ±
|±| i
1‚ f rs r
±! ‚x± (0), and the following coordinates on Wm Gm : Any element j •(0, e) ∈
Wm Gs is given by j0 •0 ∈ Gr and j0 • ∈ Tm Gs , see 15.2. Let us denote the
r r r r
¯
m m m
i
coordinate expression of • by bγ (x), 0 < |γ| ¤ s, so that we have the coordi-
¯
‚ |δ| bi
1
nates bi , 0 < |γ| ¤ s, 0 ¤ |δ| ¤ r on Tm Gs , bi (j0 •) = δ! ‚xδγ (0), and the
r r
¯
m
γ,δ γ,δ
coordinates (ai ; bi ), 0 < |β| ¤ r, 0 < |γ| ¤ s, 0 ¤ |δ| ¤ r, on Wm Gs . By
r
m
β γ,δ
de¬nition, we have
i0 (ai ) = (ai ; ai ).
(1) ± β γ+δ

In the ¬rst order case, i.e. for r = 1, we have to take into account a further
s+1
structure, namely Tm Gs = Gs
1
(gs — Rm— ), cf. 15.6. So given i0 (j0 f ) =
m m m
(j0 f, j0 q), where q : Rm ’ Gs , we are looking for b = q(0) ∈ Gs and Z =
1 1
m m
s m—
T »b’1 —¦ T0 q ∈ gm — R . Let us perform this explicitly for s = 2.
In G2 we have (ai , ai )’1 = (˜i , ai ) with ai aj = δk and ai = ’˜i al as ap .
i
aj ˜jk j ˜k ˜jk al ps ˜k ˜j
m j jk
i i i i 2 i i 2
Let X = (ak , ajk , Aj , Ajk ) ∈ T Gm and b = (bk , bjk ) ∈ Gm . It is easy to compute

T »b (X) = (bi ak , bi al + bi ap as , bi Ap , bi Ap + bi Ap as + bi ap As ).
k j l jk ps j k p j p jk ps j k ps j k

Taking into account all our identi¬cations we get a formula for i0 : G3 ’ Wm G2
1
m m

i0 (ai , ai , ai ) = (ai ; ai , ai ; ai ap , ai ap + ai ap as + ai ap as ).
jk ˜p jl ˜p jkl ˜ps jl k ˜ps j kl
j jk jkl j j

If we perform the above consideration up to the ¬rst order terms only, we get
i0 : G2 ’ Wm G1 , i0 (ai , ai ) = (ai ; ai ; ai ap ).
1
j ˜p jl
m m j j
jk



16. Canonical di¬erential forms

16.1. Consider a vector bundle E = P [V, ] associated to a principal bundle
(P, p, M, G) and the space of all E-valued di¬erential forms „¦(M ; E). By theo-
rem 11.14, there is the canonical isomorphism q between „¦(M ; E) and the space
of horizontal G-equivariant V -valued di¬erential forms on P . According to 10.12,
the image ¦ = q (•) ∈ „¦k (P ; V )G is called the frame form of • ∈ „¦k (M ; E).
hor
We have
¦(X1 , . . . , Xk ) = „ (u, ) —¦ •(T pX1 , . . . , T pXk )
(1)
where Xi ∈ Tu P and „ : P —M E ’ V is the canonical map. Conversely, for
¯ ¯
every X1 , . . . , Xk ∈ Tx M , we can choose arbitrary vectors X1 , . . . , Xk ∈ Tu P
¯
with u ∈ Px and T pXi = Xi to get
¯ ¯
•(X1 , . . . , Xk ) = q(u, ) —¦ ¦(X1 , . . . , Xk )
(2)
where q : P —V ’ E is the other canonical map. The elements ¦ ∈ „¦hor (P ; V )G
are sometimes called the tensorial forms of type , while the di¬erential forms
in „¦(P ; V )G are called pseudo tensorial forms of type .

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
16. Canonical di¬erential forms 155


16.2. The canonical form on P 1 M . We de¬ne an Rm -valued one-form θ =
θM on P 1 M for every m-dimensional manifold M as follows. Given u = j0 g ∈
1

P 1 M and X = j0 c ∈ Tu P 1 M we set
1


θM (X) = u’1 —¦ T p(X) = j0 (g ’1 —¦ p —¦ c) ∈ T0 Rm = Rm .
1


In words, the choice of u ∈ P 1 M determines a local chart at x = p(u) up to the
¬rst order and the form θM transforms X ∈ Tu P 1 M into the induced coordinates
of T pX. If we insert • = idT M into 16.1.(1) we get immediately
Proposition. The canonical form θM ∈ „¦1 (P 1 M ; Rm ) is a tensorial form which
is the frame form of the 1-form idT M ∈ „¦1 (M ; T M ).
Consider further a principal connection “ on P 1 M . Then the covariant ex-
terior di¬erential d“ θM is called the torsion form of “. By 11.15, d“ θM is
identi¬ed with a section of T M — Λ2 T — M , which is called the torsion tensor of
“. If d“ θM = 0, connection “ is said to be torsion-free.
16.3. The canonical form on W 1 P . For every principal bundle (P, p, M, G)
we can generalize the above construction to an (Rm • g)-valued one-form on
W 1 P . Consider the target projection β : W 1 P ’ P , an element u = j 1 ψ(0, e) ∈
W 1 P and a tangent vector X = j0 c ∈ Tu (W 1 P ). We de¬ne the form θ = θP by
1


θ(X) = u’1 —¦ T β(X) = j0 (ψ ’1 —¦ β —¦ c) ∈ T(0,e) (Rm — G) = Rm • g.
1


Let us notice that if G = {e} is the trivial structure group, then we get P = M ,
W 1 P = P 1 M and θP = θM .
The principal action ρ on P induces an action of G on the tangent space
T P . We claim that the space of orbits T P/G is the associated vector bundle
E = W 1 P [Rm • g; ] with the left action of Wm G on T(0,e) (Rm — G) = Rm • g,
1

’1
(j 1 •(0, e), j0 c) = j0 (ρ•(0)
1 1 ¯
—¦ • —¦ c).

Indeed, every PBm (G)-morphism commutes with the principal actions, so that
is a left action which is obviously linear and the map q : W 1 P —T(0,e) (Rm —G) ’
E transforming every couple j 1 ψ(0, e) ∈ W 1 P and j0 c ∈ T(0,e) (Rm — G) into the
1
1
orbit in T P/G determined by j0 (ψ —¦ c) describes the associated bundle structure
on E.
Proposition. The canonical form θP on W 1 P is a pseudo tensorial one-form
of type .
1
Proof. We have to prove θP ∈ „¦1 (W 1 P ; Rm • g)Wm G . Let ρ and ρ be the
¯
1 1 1 1
principal actions on P and W P , X = j0 c ∈ Tu W P , u = j ψ(0, e), A =

<< . .

. 25
( : 71)



. . >>