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 =