<< . .

. 17
( : 71)



. . >>

i.e. the space „¦hor (P ; W ) := {ψ ∈ „¦(P ; W ) : iX ψ = 0 for X ∈ V P }. The notion
of horizontal form is independent of the choice of a connection.
The projection χ— has the following properties where in the ¬rst assertion one
of the two forms has values in R:

χ— (• § ψ) = χ— • § χ— ψ,
χ— —¦ χ— = χ— ,
χ— —¦ (rg )— = (rg )— —¦ χ— for all g ∈ G,

χ ω=0
χ— —¦ L(ζX ) = L(ζX ) —¦ χ— .

They follow easily from the corresponding properties of χ, the last property uses
ζ(X)
= rexp tX .
that Flt
Now we de¬ne the covariant exterior derivative dω : „¦k (P ; W ) ’ „¦k+1 (P ; W )
by the prescription dω := χ— —¦ d.

Theorem. The covariant exterior derivative dω has the following properties.
dω (• § ψ) = dω (•) § χ— ψ + (’1)deg • χ— • § dω (ψ) if • or ψ is real valued.
(1)
L(ζX ) —¦ dω = dω —¦ L(ζX ) for each X ∈ g.
(2)
(rg )— —¦ dω = dω —¦ (rg )— for each g ∈ G.
(3)
dω —¦ p— = d —¦ p— = p— —¦ d : „¦(M ; W ) ’ „¦hor (P ; W ).
(4)
(5)dω ω = „¦, the curvature form.
(6)dω „¦ = 0, the Bianchi identity.
dω —¦ χ— ’ dω = χ— —¦ i(R), where R is the curvature.
(7)
dω —¦ dω = χ— —¦ i(R) —¦ d.
(8)
Let „¦hor (P, g)G be the algebra of all horizontal G-equivariant g-valued
(9)
forms, i.e. (rg )— ψ = Ad(g ’1 )ψ. Then for any ψ ∈ „¦hor (P, g)G we have
dω ψ = dψ + [ω, ψ]§ .
(10) The mapping ψ ’ ζψ , where ζψ (X1 , . . . , Xk )(u) = ζψ(X1 ,... ,Xk )(u) (u), is
an isomorphism between „¦hor (P, g)G and the algebra „¦hor (P, V P )G of
all horizontal G-equivariant forms with values in the vertical bundle V P .
Then we have ζdω ψ = ’[¦, ζω ].

Proof. (1) through (4) follow from the properties of χ— .

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
104 Chapter III. Bundles and connections


(5) We have

(dω ω)(ξ, ·) = (χ— dω)(ξ, ·) = dω(χξ, χ·)
= (χξ)ω(χ·) ’ (χ·)ω(χξ) ’ ω([χξ, χ·])
= ’ω([χξ, χ·]) and
’ζ(„¦(ξ, ·)) = R(ξ, ·) = ¦[χξ, χ·] = ζω([χξ,χ·]) .

(6) Using 11.2 we have

dω „¦ = dω (dω + 1 [ω, ω]§ )
2
= χ— ddω + 1 χ— d[ω, ω]§
2
= 2 χ— ([dω, ω]§ ’ [ω, dω]§ ) = χ— [dω, ω]§
1

= [χ— dω, χ— ω]§ = 0, since χ— ω = 0.

(7) For • ∈ „¦(P ; W ) we have

(dω χ— •)(X0 , . . . , Xk ) = (dχ— •)(χ(X0 ), . . . , χ(Xk ))
(’1)i χ(Xi )((χ— •)(χ(X0 ), . . . , χ(Xi ), . . . , χ(Xk )))
=
0¤i¤k

(’1)i+j (χ— •)([χ(Xi ), χ(Xj )], χ(X0 ), . . .
+
i<j

. . . , χ(Xi ), . . . , χ(Xj ), . . . )
(’1)i χ(Xi )(•(χ(X0 ), . . . , χ(Xi ), . . . , χ(Xk )))
=
0¤i¤k

(’1)i+j •([χ(Xi ), χ(Xj )] ’ ¦[χ(Xi ), χ(Xj )], χ(X0 ), . . .
+
i<j

. . . , χ(Xi ), . . . , χ(Xj ), . . . )
= (d•)(χ(X0 ), . . . , χ(Xk )) + (iR •)(χ(X0 ), . . . , χ(Xk ))
= (dω + χ— iR )(•)(X0 , . . . , Xk ).

(8) dω dω = χ— dχ— d = (χ— iR + χ— d)d = χ— iR d holds by (7).
(9) If we insert one vertical vector ¬eld, say ζX for X ∈ g, into dω ψ, we
get 0 by de¬nition. For the right hand side we use iζX ψ = 0 and LζX ψ =
ζX — exp tX —
‚ ‚ ‚
) ψ = ‚t 0 Ad(exp(’tX))ψ = ’ ad(X)ψ to get
‚t 0 (Flt ) ψ = ‚t 0 (r

iζX (dψ + [ω, ψ]§ ) = iζX dψ + diζX ψ + [iζX ω, ψ] ’ [ω, iζX ψ]
= LζX ψ + [X, ψ] = ’ ad(X)ψ + [X, ψ] = 0.

Let now all vector ¬elds ξi be horizontal, then we get

(dω ψ)(ξ0 , . . . , ξk ) = (χ— dψ)(ξ0 , . . . , ξk ) = dψ(ξ0 , . . . , ξk ),
(dψ + [ω, ψ]§ )(ξ0 , . . . , ξk ) = dψ(ξ0 , . . . , ξk ).

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
11. Principal and induced connections 105


So the ¬rst formula holds.
(10) We proceed in a similar manner. Let Ψ be in the space „¦hor (P, V P )G
of all horizontal G-equivariant forms with vertical values. Then for each X ∈ g
we have iζX Ψ = 0; furthermore the G-equivariance (rg )— Ψ = Ψ implies that
LζX Ψ = [ζX , Ψ] = 0 by 8.16.(5). Using formula 8.11.(2) we have

iζX [¦, Ψ] = [iζX ¦, Ψ] ’ [¦, iζX Ψ] + i([¦, ζX ])Ψ + i([Ψ, ζX ])¦
= [ζX , Ψ] ’ 0 + 0 + 0 = 0.

Let now all vector ¬elds ξi again be horizontal, then from the huge formula 8.9
for the Fr¨licher-Nijenhuis bracket only the following terms in the third and ¬fth
o
line survive:
(’1)
[¦, Ψ](ξ1 , . . . , ξ +1 ) = sign σ ¦([Ψ(ξσ1 , . . . , ξσ ), ξσ( +1) ])
!
σ
1
+ sign σ ¦(Ψ([ξσ1 , ξσ2 ], ξσ3 , . . . , ξσ( +1) ).
( ’1)! 2!
σ


For f : P ’ g and horizontal ξ we have ¦[ξ, ζf ] = ζξ(f ) = ζdf (ξ) since it is
C ∞ (P, R)-linear in ξ. So the last expression becomes

’ζ(dψ(ξ0 , . . . , ξk )) = ’ζ(dω ψ(ξ0 , . . . , ξk )) = ’ζ((dψ + [ω, ψ]§ )(ξ0 , . . . , ξk ))

as required.
11.6. Theorem. Let (P, p, M, G) be a principal ¬ber bundle with principal
connection ω. Then the parallel transport for the principal connection is globally
de¬ned and G-equivariant.
In detail: For each smooth curve c : R ’ M there is a smooth mapping
Ptc : R — Pc(0) ’ P such that the following holds:
d
(1) Pt(c, t, u) ∈ Pc(t) , Pt(c, 0) = IdPc(0) , and ω( dt Pt(c, t, u)) = 0.
(2) Pt(c, t) : Pc(0) ’ Pc(t) is G-equivariant, i.e. Pt(c, t, u.g) = Pt(c, t, u).g
holds for all g ∈ G and u ∈ P . Moreover we have Pt(c, t)— (ζX |Pc(t) ) =
ζX |Pc(0) for all X ∈ g.
(3) For any smooth function f : R ’ R we have
Pt(c, f (t), u) = Pt(c —¦ f, t, Pt(c, f (0), u)).

Proof. By 11.4 the Christo¬el forms “± ∈ „¦1 (U± , X(G)) of the connection ω with
respect to a principal ¬ber bundle atlas (U± , •± ) are given by “± (ξx ) = Rω± (ξx ) ,
so they take values in the Lie subalgebra XR (G) of all right invariant vector
¬elds on G, which are bounded with respect to any right invariant Riemannian
metric on G. Each right invariant metric on a Lie group is complete. So the
connection is complete by the remark in 9.9.
Properties (1) and (3) follow from theorem 9.8, and (2) is seen as follows:

ω( dt Pt(c, t, u).g) = Ad(g ’1 )ω( dt Pt(c, t, u)) = 0
d d


Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
106 Chapter III. Bundles and connections


implies that Pt(c, t, u).g = Pt(c, t, u.g). For the second assertion we compute for
u ∈ Pc(0) :

Pt(c, t)— (ζX |Pc(t) )(u) = T Pt(c, t)’1 ζX (Pt(c, t, u)) =
= T Pt(c, t)’1 ds |0 Pt(c, t, u). exp(sX) =
d

= T Pt(c, t)’1 ds |0 Pt(c, t, u. exp(sX)) =
d

’1
d
ds |0 Pt(c, t)
= Pt(c, t, u. exp(sX))
d
ds |0 u. exp(sX) = ζX (u).
=


11.7. Holonomy groups. Let (P, p, M, G) be a principal ¬ber bundle with
principal connection ¦ = ζ —¦ ω. We assume that M is connected and we ¬x
x0 ∈ M .
In 9.10 we de¬ned the holonomy group Hol(¦, x0 ) ‚ Di¬(Px0 ) as the group
of all Pt(c, 1) : Px0 ’ Px0 for c any piecewise smooth closed loop through
x0 . (Reparametrizing c by a function which is ¬‚at at each corner of c we may
assume that any c is smooth.) If we consider only those curves c which are
nullhomotopic, we obtain the restricted holonomy group Hol0 (¦, x0 ).
Now let us ¬x u0 ∈ Px0 . The elements „ (u0 , Pt(c, t, u0 )) ∈ G form a subgroup
of the structure group G which is isomorphic to Hol(¦, x0 ); we denote it by
Hol(ω, u0 ) and we call it also the holonomy group of the connection. Considering
only nullhomotopic curves we get the restricted holonomy group Hol0 (ω, u0 ) a
normal subgroup of Hol(ω, u0 ).
Theorem. The main results for the holonomy are as follows:
(1) We have Hol(ω, u0 .g) = conj(g ’1 ) Hol(ω, u0 ) and
Hol0 (ω, u0 .g) = conj(g ’1 ) Hol0 (ω, u0 ).
(2) For each curve c in M with c(0) = x0 we have Hol(ω, Pt(c, t, u0 )) =
Hol(ω, u0 ) and Hol0 (ω, Pt(c, t, u0 )) = Hol0 (ω, u0 ).
(3) Hol0 (ω, u0 ) is a connected Lie subgroup of G and the quotient group
Hol(ω, u0 )/ Hol0 (ω, u0 ) is at most countable, so Hol(ω, u0 ) is also a Lie
subgroup of G.
(4) The Lie algebra hol(ω, u0 ) ‚ g of Hol(ω, u0 ) is linearly generated by
{„¦(Xu , Yu ) : Xu , Yu ∈ Tu P }, and it is isomorphic to the holonomy Lie
algebra hol(¦, x0 ) we considered in 9.10.
(5) For u0 ∈ Px0 let P (ω, u0 ) be the set of all Pt(c, t, u0 ) for c any (piecewise)
smooth curve in M with c(0) = x0 and for t ∈ R. Then P (ω, u0 ) is
a sub ¬ber bundle of P which is invariant under the right action of
Hol(ω, u0 ); so it is itself a principal ¬ber bundle over M with structure
group Hol(ω, u0 ) and we have a reduction of structure group, cf. 10.6 and
10.13. The pullback of ω to P (ω, u0 ) is then again a principal connection
form i— ω ∈ „¦1 (P (ω, u0 ); hol(ω, u0 )).
(6) P is foliated by the leaves P (ω, u), u ∈ Px0 .
(7) If the curvature „¦ = 0 then Hol0 (ω, u0 ) = {e} and each P (ω, u) is a
covering of M .

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
11. Principal and induced connections 107


(8) If one uses piecewise C k -curves for 1 ¤ k < ∞ in the de¬nition, one gets
the same holonomy groups.
In view of assertion 5 a principal connection ω is called irreducible if Hol(ω, u0 )
equals the structure group G for some (equivalently any) u0 ∈ Px0 .
Proof. 1. This follows from the properties of the mapping „ from 10.2 and from
the G-equivariance of the parallel transport.
The rest of this theorem is a compilation of well known results, and we refer
to [Kobayashi-Nomizu I, 63, p. 83¬] for proofs.
11.8. Inducing connections on associated bundles. Let (P, p, M, G) be a
principal bundle with principal right action r : P — G ’ P and let : G — S ’
S be a left action of the structure group G on some manifold S. Then we
consider the associated bundle P [S] = P [S, ] = P —G S, constructed in 10.7.
Recall from 10.18 that its tangent and vertical bundle are given by T (P [S, ]) =
T P [T S, T ] = T P —T G T S and V (P [S, ]) = P [T S, T2 ] = P —G T S.
Let ¦ = ζ —¦ ω ∈ „¦1 (P ; T P ) be a principal connection on the principal bundle
¯
P . We construct the induced connection ¦ ∈ „¦1 (P [S], T (P [S])) by the following
diagram:
w w
¦ — Id =
TP — TS TP — TS T (P — S)


u u u
Tq = q q Tq

w TP — w T (P —
¯
¦ =
T P —T G T S TS S).
TG G

Let us ¬rst check that the top mapping ¦ — Id is T G-equivariant. For g ∈ G and
X ∈ g the inverse of Te (»g )X in the Lie group T G is denoted by (Te (»g )X)’1 ,
see lemma 10.17. Furthermore by 5.13 we have
T r(ξu , Te (»g )X) = Tu (rg )ξu + T r((0P — LX )(u, g))
= Tu (rg )ξu + Tg (ru )(Te (»g )X)
= Tu (rg )ξu + ζX (ug).
We may compute
(¦ — Id)(T r(ξu , Te (»g )X), T ((Te (»g )X)’1 , ·s ))
= (¦(Tu (rg )ξu + ζX (ug)), T ((Te (»g )X)’1 , ·s ))
= (¦(Tu (rg )ξu ) + ¦(ζX (ug)), T ((Te (»g )X)’1 , ·s ))
= ((Tu (rg )¦ξu ) + ζX (ug), T ((Te (»g )X)’1 , ·s ))
= (T r(¦(ξu ), Te (»g )X), T ((Te (»g )X)’1 , ·s )).
¯
So the mapping ¦ — Id factors to ¦ as indicated in the diagram, and we have
¯¯ ¯ ¯
¦ —¦ ¦ = ¦ from (¦ — Id) —¦ (¦ — Id) = ¦ — Id. The mapping ¦ is ¬berwise linear,
¯
since ¦ — Id and q = T q are. The image of ¦ is
q (V P — T S) = q (ker(T p : T P — T S ’ T M ))
= ker(T p : T P —T G T S ’ T M ) = V (P [S, ]).

Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
108 Chapter III. Bundles and connections

¯
Thus ¦ is a connection on the associated bundle P [S]. We call it the induced
connection.
From the diagram it also follows, that the vector valued forms ¦ — Id ∈
¯
„¦ (P — S; T P — T S) and ¦ ∈ „¦1 (P [S]; T (P [S])) are (q : P — S ’ P [S])-related.
1

So by 8.15 we have for the curvatures

R¦—Id = 1 [¦ — Id, ¦ — Id] = 1 [¦, ¦] — 0 = R¦ — 0,
2 2
1¯ ¯
R¦ = [¦, ¦],
¯ 2

that they are also q-related, i.e. T q —¦ (R¦ — 0) = R¦ —¦ (T q —M T q).
¯
By uniqueness of the solutions of the de¬ning di¬erential equation we also get
that
Pt¦ (c, t, q(u, s)) = q(Pt¦ (c, t, u), s).
¯

11.9. Recognizing induced connections. We consider again a principal
¬ber bundle (P, p, M, G) and a left action : G — S ’ S. Suppose that Ψ ∈
„¦1 (P [S]; T (P [S])) is a connection on the associated bundle P [S] = P [S, ]. Then
the following question arises: When is the connection Ψ induced from a principal
connection on P ? If this is the case, we say that Ψ is compatible with the G-
bundle structure on P [S]. The answer is given in the following
Theorem. Let Ψ be a (general) connection on the associated bundle P [S]. Let
us suppose that the action is in¬nitesimally e¬ective, i.e. the fundamental
vector ¬eld mapping ζ : g ’ X(S) is injective.
Then the connection Ψ is induced from a principal connection ω on P if and
only if the following condition is satis¬ed:
In some (equivalently any) ¬ber bundle atlas (U± , ψ± ) of P [S] belong-
ing to the G-bundle structure of the associated bundle the Christo¬el
forms “± ∈ „¦1 (U± ; X(S)) have values in the sub Lie algebra Xf und (S) of
fundamental vector ¬elds for the action .

Proof. Let (U± , •± : P |U± ’ U± — G) be a principal ¬ber bundle atlas for P .
Then by the proof of theorem 10.7 it is seen that the induced ¬ber bundle atlas
(U± , ψ± : P [S]|U± ’ U± — S) is given by

ψ± (x, s) = q(•’1 (x, e), s),
’1
(1) ±

(ψ± —¦ q)(•’1 (x, g), s) = (x, g.s).
(2) ±

¯
Let ¦ = ζ —¦ω be a principal connection on P and let ¦ be the induced connection
on the associated bundle P [S]. By 9.7 its Christo¬el symbols are given by

¯ ’1
(0x , “± (ξx , s)) = ’(T (ψ± ) —¦ ¦ —¦ T (ψ± ))(ξx , 0s )
¯
¦
¯
= ’(T (ψ± ) —¦ ¦ —¦ T q —¦ (T (•’1 ) — Id))(ξx , 0e , 0s ) by (1)
±

= ’(T (ψ± ) —¦ T q —¦ (¦ — Id))(T (•’1 )(ξx , 0e ), 0s ) by 11.8
±

= ’(T (ψ± ) —¦ T q)(¦(T (•’1 )(ξx , 0e )), 0s )
±


Electronic edition of: Natural Operations in Differential Geometry, Springer-Verlag, 1993
11. Principal and induced connections 109


= (T (ψ± ) —¦ T q)(T (•’1 )(0x , “± (ξx , e)), 0s ) by 11.4.(3)
± ¦

= ’T (ψ± —¦ q —¦ (•’1 — Id))(0x , ω± (ξx ), 0s ) by 11.4.(7)
±
= ’Te ( s )ω± (ξx ) by (2)
= ’ζω± (ξx ) (s).
So the condition is necessary. Now let us conversely suppose that a connection
Ψ on P [S] is given such that the Christo¬el forms “± with respect to a ¬ber
Ψ
bundle atlas of the G-structure have values in Xf und (S). Then unique g-valued
forms ω± ∈ „¦1 (U± ; g) are given by the equation “± (ξx ) = ζ(ω± (ξx )), since the
Ψ
action is in¬nitesimally e¬ective. From the transition formulas 9.7 for the “± Ψ
follow the transition formulas 11.4.(5) for the ω ± , so that they give a unique
principal connection on P , which by the ¬rst part of the proof induces the given
connection Ψ on P [S].
11.10. Inducing connections on associated vector bundles.
Let (P, p, M, G) be a principal ¬ber bundle and let ρ : G ’ GL(W ) be a
representation of the structure group G on a ¬nite dimensional vector space W .
We consider the associated vector bundle (E := P [W, ρ], p, M, W ), from 10.11.
Recall from 6.11 that T (E) = T P —T G T W has two vector bundle structures
with the projections

πE : T (E) = T P —T G T W ’ P —G W = E,
T p —¦ pr1 : T (E) = T P —T G T W ’ T M.

Now let ¦ = ζ —¦ ω ∈ „¦1 (P ; T P ) be a principal connection on P . We consider
¯
the induced connection ¦ ∈ „¦1 (E; T (E)) from 11.8. Inserting the projections
of both vector bundle structures on T (E) into the diagram in 11.8 we get the
following diagram

xw
&& ¦ — Id
TP — TW TP — TW TP — W — W
&& x
xxπ
(
& 
x
π
P —W

u
q
Tq Tq

x
x
P —G W = E


u  xu

<< . .

. 17
( : 71)



. . >>