Chapter 1. Principal and associated fiber bundles
1.4. Associated bundles
Associated bundles are constructed from principal bundles by a fiberwise ap-plication of the notion of fiber product discussed in Subsection 1.2.1. We begin by stating a smooth version of Definition 1.2.24.
DEFINITION1.4.1. By adifferentiableG-spacewe mean a differentiable man-ifoldN endowed with a smooth left actionG×N →N of a Lie groupG.
Notice that the effective groupGef of a differentiableG-space is a subgroup of the group Diff(N) of all smooth diffeomorphisms of N. The kernel of the homomorphismG3 g7→ γg ∈Diff(N)corresponding to the left action ofGon N is a closed normal subgroup ofGandGefis isomorphic to the quotient ofGby such kernel; we can therefore endowGefwith the structure of a Lie group.
LetΠ :P →Mbe aG-principal bundle and letNbe a differentiableG-space.
For eachx∈M we consider the fiber productPx×GN of the principal spacePx by theG-spaceN and we set:
P×GN = [
x∈M
(Px×GN);
we have aprojection map:
π :P×GN −→M
that sendsPx×GN to the pointx∈M and aquotient mapqdefined by:
q:P×N 3(p, n)7−→[p, n]∈P×GN.
The following commutative diagram illustrates the relation between the mapsΠ,π andq:
(1.4.1)
P ×N
first projection
q
##G
GG GG GG G
P
Π
P ×GN
{{wwwwwwπww
M
We callπ : P ×GN → M (or just P ×G N) theassociated bundle to the G-principal bundleP and to the differentiableG-spaceN. The setP ×GN is also called thetotal spaceof the associated bundle. For eachx∈M, the set
Px×GN =π−1(x)
1.4. ASSOCIATED BUNDLES 37
is called thefiberofP ×GN overx.
Notice that each fiberPx×GN is naturally endowed with theGef-structure Pcx =
ˆ
p : p ∈ Px . Since Gef is a subgroup of Diff(N), such Gef-structure can be weakened to aDiff(N)-structure that corresponds to the structure of a dif-ferentiable manifold smoothly diffeomorphic to N on the fiberPx×GN (recall Example 1.1.4).
Our goal now is to endow the entire total spaceP×GN with the structure of a differentiable manifold. Given a smooth local sections:U → P ofP then we have an associated bijective map:
(1.4.2) ˆs:U ×N 3(x, n)7−→[s(x), n] =s(x)(n)d ∈π−1(U)⊂P×GN, which we call thelocal trivializationof the associated bundleP×GN correspond-ing to the smooth local sections. Ifs1 :U1 →P,s2 :U2 → P are smooth local sections of P and ifg : U1∩U2 → Gdenotes the transition map froms1 tos2
then thetransition mapsˆ−11 ◦sˆ2fromsˆ1toˆs2 is given by:
ˆ
s−11 ◦sˆ2 : (U1∩U2)×N 3(x, n)7−→ x, g(x)·n
∈(U1∩U2)×N and is therefore a smooth diffeomorphism between open sets. It follows from the result of Exercise A.1 that there exists a unique differential structure on the set P×GN such that for every smooth local sections:U →P ofP the setπ−1(U) is open inP×GNand the local trivializationsˆis a smooth diffeomorphism.We will always regard the total spaceP×GNof an associated bundle to be endowed with such differential structure. The fact that the topologies ofM andN are Hausdorff and second countable implies that the topology ofP ×GN is also Hausdorff and second countable, so that P ×GN is a differentiable manifold. One can easily check the following facts:
• the projectionπ :P×GN →Mis a smooth submersion;
• the quotient mapq:P ×N →P×GN is a smooth submersion;
• for everyx∈M the fiberPx×GNis a smooth submanifold ofP×GN;
• for everyx∈M and everyp∈Px, if the fiberPx×GN is endowed with the differential structure inherited fromP ×GN as a submanifold then the mappˆ:N →Px×GN is a smooth diffeomorphism.
The last item above implies that the differential structure ofPx×GN that is ob-tained by weakening theGef-structurePcxcoincides with the differential structure thatPx×GN inherits fromP×GN.
EXAMPLE1.4.2 (the trivial associated bundle). LetMbe a differentiable man-ifold,P0 be a principal space whose structural groupGis a Lie group and letN be a differentiable G-space. Consider the trivial principal bundleP = M ×P0 (recall Example 1.3.2). The associated bundle P ×G N can be naturally iden-tified with the cartesian product M × (P0 ×G N) of M by the fiber product P0×GN. The fiber productP0 ×GN is endowed with a Gef-structure that can be weakened into aDiff(N)-structure that corresponds to the structure of a differ-entiable manifold smoothly diffeomorphic toN. Clearly the differential structure
ofP ×GN =M ×(P0×GN)coincides with the standard differential structure defined in a cartesian product of differentiable manifolds.
EXAMPLE1.4.3. LetΠ :P →M be aG-principal bundle andN be a differ-entiableG-space. IfUis an open subset ofMthen the total space of the associated bundle(P|U)×GN is equal to the open subsetπ−1(U) of the total space of the associated bundle π : P ×G N → M. Clearly, the differential structure of the associated bundle(P|U)×GN coincides with the differential structure it inherits fromP×GN as an open subset.
DEFINITION1.4.4. Givenx ∈M,p ∈Pxandn∈N then the tangent space T[p,n](Px×GN)is a subspace ofT[p,n](P×GN)and it is called thevertical spacef P×GN at[p, n]; we write:
Ver[p,n](P×GN) =T[p,n](Px×GN).
Clearly:
Ver[p,n](P×GN) = Ker dπ([p, n]) .
Notice that, sincepˆis a smooth diffeomorphism fromN onto the fiberPx×GN, its differential atn∈N is an isomorphism:
(1.4.3) dˆp(n) :TnN −→Ver[p,n](P×GN) from the tangent spaceTnN onto the vertical space.
In the example below we look at a case that is of particular interest to us.
EXAMPLE1.4.5. LetE0 be a real finite-dimensional vector space and assume that we are given a smooth representation ρ : G → GL(E0) of a Lie group G on E0. Then E0 is a differentiableG-space and the effective groupGef is a Lie subgroup of the general linear group GL(E0). IfΠ : P → M is a G-principal bundle then we can consider the associated bundleP×GE0. For eachx∈M, the fiberPx×GE0has the structure of a real vector space such that for everyp ∈Px
the mappˆ : E0 → Px×GE0 is a linear isomorphism (recall Example 1.2.26).
Since eachpˆis both a smooth diffeomorphism and a linear isomorphism, it follows that the differential structure of the fiberPx×GE0(inherited from the total space P ×G E0) coincides with the differential structure that is determined by its real finite-dimensional vector space structure. We can therefore identify the vertical space at any point of the fiberPx×GE0with the fiber itself, i.e.:
(1.4.4) Ver[p,e0](P ×GE0) =T[p,e0](Px×GE0)∼=Px×GE0,
for allp∈Pxand alle0 ∈E0. Moreover, the linear isomorphism (1.4.3) is justp,ˆ i.e.:
(1.4.5) dˆp(e0) = ˆp:E0−→Px×GE0, for allp∈Pxand alle0 ∈E0.
1.4. ASSOCIATED BUNDLES 39
1.4.1. Local sections of an associated bundle. LetΠ : P → M be a G-principal bundle,N be a differentiableG-space and consider the associated bundle π:P×GN →M and the quotient mapq:P×N →P×GN.
DEFINITION 1.4.6. By a local sectionof the associated bundle P ×GN we mean a map : U → P ×G N defined on an open subset U ofM such that π◦= IdU, i.e., such that(x)∈Px×GN, for allx∈U.
If : U → P ×G N is a local section of P ×GN and if s : U → P is a smooth local section ofP then there exists a unique map˜ : U → N such that =q◦(s,˜), i.e., such that:
(1.4.6) (x) = [s(x),˜(x)],
for allx ∈U; namely,˜is just the second coordinate of the mapsˆ−1◦. We call
˜
therepresentationofwith respect tos. Clearlyis smooth if and only if˜is smooth.
1.4.2. The differential of the quotient map. Let Π : P → M be a G-principal bundle,N be a differentiableG-space and consider the associated bundle π:P×GN →M and the quotient mapq:P×N →P×GN. For everyx∈M, the mapqcarriesPx×N onto the fiberPx×GNoverxand therefore, for allp∈Px and alln∈N, the differentialdq(p, n)carriesT(p,n)(Px×N) = Verp(P)⊕TnN to the vertical space Ver[p,n](P ×G N). We wish to compute the restriction of the differential dq(p, n) to Verp(P)⊕TnN. To this aim, we identify Verp(P) with the Lie algebra g via the canonical isomorphism (1.3.3) and we identify Ver[p,n](P ×G N) with TnN via the isomorphism (1.4.3). Recall from Defini-tion A.2.3 that for everyX ∈g, we denote byXN the induced vector field on the differentiable manifoldN and byXP the induced vector field on the differentiable manifoldP.
LEMMA1.4.7. LetΠ :P →Mbe aG-principal bundle,Nbe a differentiable G-space and consider the associated bundleπ :P ×GN → M and the quotient mapq :P ×N → P ×GN. Givenp ∈ P,n ∈N then the dotted arrow in the commutative diagram:
Verp(P)⊕TnN dq(p,n) //Ver(p,n)(P×GN)
g⊕TnN
dβp(1)⊕Id ∼=
OO //TnN
d ˆp(n)
∼=
OO
is given by:
g⊕TnN 3(X, u)7−→u+XN(n)∈TnN.
PROOF. Setx= Π(p). The mapq(p,·) :N →Px×GN is the same aspˆand therefore the dotted arrow on the diagram carries(0, u)tou, for allu∈TnN. To conclude the proof, we show that the dotted arrow on the diagram carries(X,0)to
XN(n), for allX ∈g; this follows from the commutative diagram:
Px q(·,n) //Px×GN
G
βp
OO
βn
//N
ˆ p
OO
by differentiation.
COROLLARY1.4.8. LetΠ :P →M be aG-principal bundle,N be a differ-entiableG-space and consider the associated bundleπ :P ×GN → M and the quotient mapq : P ×N → P ×GN. Given p ∈ P,n ∈ N then the kernel of dq(p, n)is equal to the image under the isomorphism:
dβp(1)⊕Id :g⊕TnN −→Verp(P)⊕TnN of the space:
X,−XN(n)
:X∈g ⊂g⊕TnN; more explicitly:
Ker dq(p, n)
=
(XP(p),−XN(n)
:X∈g ⊂Verp(P)⊕TnN.
PROOF. By differentiating (1.4.1), we see that the diagram:
(1.4.7) TpP ⊕TnN
dΠLpL◦prLLL1LLLLL&&
dq(p,n)
//T[p,n](P×GN)
dπ[p,n]
xxppppppppppp
TxM
commutes. Thus the kernel of dq(p, n) is contained in Verp(P) ⊕TnN. The
conclusion now follows easily from Lemma 1.4.7.
EXAMPLE 1.4.9. Let us go back to the context of Example 1.4.5 and let us take a closer look at the statement of Lemma 1.4.7. Denote byρ¯:g→gl(E0)the differential at1 ∈Gof the representationρ : G→ GL(E0); bygl(E0)we have denoted the Lie algebra of the general linear groupGL(E0), which is just the space of all linear endomorphisms ofE0, endowed with the standard Lie bracket of linear operators. GivenX ∈gthen the induced vector fieldXE0 on the manifoldE0 is given byXE0(e0) = ¯ρ(X)·e0, for alle0 ∈ E0; thusXE0 :E0 →E0is just the linear mapρ(X). Keeping in mind (1.4.5), Lemma 1.4.7 tells us that the restriction¯ toVerp(P)⊕E0of the differential of the quotient mapq:P×E0→P×GE0at a point(p, e0)∈P×E0is given by:
dq(p,e0) dβp(1)·X, u
= ˆp u+ ¯ρ(X)·e0
= [p, u+ ¯ρ(X)·e0]∈Px×GE0, for allX ∈ gand all u ∈ E0. For instance, ifG is a Lie subgroup ofGL(E0) andρ : G → GL(E0)is the inclusion map thengis a Lie subalgebra ofgl(E0),
¯
ρ:g→gl(E0)is the inclusion map and thusρ(X)¯ is justXitself; hence:
dq(p,e0) dβp(1)·X, u
= ˆp u+X(e0)
= [p, u+X(e0)]∈Px×GE0, for allX∈gand allu∈E0.
1.4. ASSOCIATED BUNDLES 41
1.4.3. Induced maps on associated bundles. In Subsection 1.2.1 we have defined the notion of induced maps on fiber products. Such notion can be applied fiberwise to get a notion of induced map on an associated bundle. More precisely, let P, Q be principal bundles over a differentiable manifold M with structural groupsGandH, respectively. Letφ:P →Qbe a morphism of principal bundles and let φ0 : G → H denote its subjacent Lie group homomorphism. Given a differentiableH-spaceN we consider the smooth left action ofGonN defined in (1.2.20), so thatNis also a differentiableG-space. For eachx∈M, the morphism of principal spacesφx :Px →Qxinduces a mapφˆx :Px×GN →Qx×HN and thus there is a map:
φˆ:P×GN −→Q×H N
whose restriction to the fiber Px ×G N is equal to φˆx, for all x ∈ M. More explicitly, we have:
φˆ [p, n]
= [φ(p), n],
for all p ∈ P and alln ∈ N. We callφˆthe map inducedindexassociated bun-dle!induced map on by φ on the associated bundles. Notice that the following diagram:
(1.4.8)
P ×N φ×Id //
q
Q×N
q0
P×GN
φˆ
//Q×H N
commutes, whereq,q0denote the quotient maps.
Since for eachx ∈M the mapφˆx is bijective (Corollary 1.2.32) then clearly the mapφˆis also bijective. Moreover, we have the following:
LEMMA 1.4.10. Let P, Qbe principal bundles over the same differentiable manifoldM with structural groupsGandH, respectively. Letφ: P → Qbe a morphism of principal bundles and letφ0 :G→Hdenote its subjacent Lie group homomorphism. Given a differentiable H-spaceN we consider the smooth left action ofGonN defined by(1.2.20). The induced mapφˆ:P×GN −→Q×HN is a smooth diffeomorphism.
PROOF. Lets:U →P be a smooth local section ofP and sets0 =φ◦s, so thats0 :U →Qis a smooth local section ofQ. We have a commutative diagram analogous to diagram (1.2.23):
(1.4.9)
(P|U)×GN φˆ //(Q|U)×H N
U ×N
ˆ s
∼=
eeKKKKK
KKKKK sb0
∼=
99s
ss ss ss ss s
Sincesˆandsb0 are smooth diffeomorphisms, we conclude thatφˆis a smooth local diffeomorphism. Since φˆis bijective, it follows thatφˆis indeed a global smooth
diffeomorphism.
As in Subsection 1.2.1 we have also a more general notion of induced map.
Let P, Q be principal bundles over a differentiable manifold M with structural groupsGandH, respectively. Letφ:P →Qbe a morphism of principal bundles with subjacent Lie group homomorphism φ0 : G → H. LetN be a differen-tiable G-space and N0 be a differentiable H-space; assume that we are given a φ0-equivariant map κ : N → N0. For eachx ∈ M the map φx : Px → Qx is a morphism of principal spaces with subjacent group homomorphism φ0 and therefore we have an induced map:
φx×∼κ:Px×GN −→Qx×H N0. We can therefore consider theinduced map:
φ×∼κ:P ×GN −→Q×HN0
whose restriction to the fiberPx×GN is equal toφx×∼κ, for allx∈M. Notice that the following diagram:
(1.4.10)
P×N φ×κ //
q
Q×N0
q0
P×GN
φ×∼κ //Q×H N0
commutes, whereq,q0 denote the quotient maps. IfN = N0 andN is endowed with the action ofGdefined in (1.2.20) then the induced mapφ×∼Idis the same asφ.ˆ
The induced mapφ×∼κretains many properties of the mapκas is shown by the following:
LEMMA1.4.11. LetP,Qbe principal bundles over a nonempty differentiable manifoldM with structural groupsGandH, respectively. Letφ: P → Qbe a morphism of principal bundles with subjacent Lie group homomorphismφ0 :G→ H, letN be a differentiableG-space andN0 be a differentiableH-space; assume that we are given aφ0-equivariant mapκ :N → N0. Consider the induced map φ×∼κ:P×GN →Q×H N0. Then:
(a) φ×∼κis smooth if and only ifκis smooth;
(b) φ×∼ κ is injective (resp., surjective) if and only if κ is injective (resp., surjective);
(c) φ×∼κis a smooth immersion (resp., a smooth submersion) if and only if κis a smooth immersion (resp., a smooth submersion);
(d) φ×∼κis a smooth embedding if and only ifκis a smooth embedding;
(e) φ×∼κis an open mapping if and only ifκis an open mapping.
1.4. ASSOCIATED BUNDLES 43
PROOF. Lets:U →P be a smooth local section ofP and sets0 =φ◦s, so thats0 :U →Qis a smooth local section ofQ. We have a commutative diagram similar to (1.2.22):
(P|U)×GN φ×∼κ //(Q|U)×H N0
U ×N
ˆ s ∼=
OO
Id×κ //U×N0
∼= sb0
OO
The conclusion follows then easily from the fact that the mapsˆsandsb0are smooth diffeomorphisms (for the proof of item (d), use also the result of Exercise A.2).
Notice that ifΠ : P → M is aG-principal bundle,N is a differentiable G-space andN0 is a smooth submanifold ofN invariant by the action ofGthen the inclusion mapi : N0 → N is a smoothG-equivariant embedding and therefore, by Lemma 1.4.11, the induced mapId×∼ i : P ×GN0 → P ×GN is a smooth embedding. We use the mapId×∼ito identifyP×GN0with a smooth submanifold ofP ×GN. Notice that ifN0 is an open submanifold ofN thenP ×GN0 is an open submanifold ofP×GN (item (e) of Lemma 1.4.11).
1.4.4. The associated bundle to a pull-back. LetP be aG-principal bundle over a differentiable manifoldMand letf :M0 →Mbe a smooth map defined in a differentiable manifoldM0. Given a differentiableG-spaceNthen the associated bundle (f∗P)×GN can be identified with the following subset of the cartesian productM0×(P ×GN):
(1.4.11) [
y∈M0
{y} ×(Pf(y)×GN) .
We have the following:
LEMMA1.4.12. LetP be aG-principal bundle over a differentiable manifold M and letf :M0→M be a smooth map defined in a differentiable manifoldM0. LetNbe a differentiableG-space. If we identify the associated bundle(f∗P)×GN with the set(1.4.11)then the inclusion map of(f∗P)×GN inM0×(P×GN)is a smooth embedding.
PROOF. By the result of Exercise A.2, in order to prove that the inclusion map (f∗P)×G N → M0 ×(P ×G N) is a smooth embedding it suffices to verify that for every smooth local sections : U → P ofP the inclusion map from the open subset (f∗P)×GN
∩ f−1(U)×(P×GN)
= (f∗P)|f−1(U)
×GNof (f∗P)×GN toM0×(P×GN)is a smooth embedding. Setσ =s◦fand consider the smooth local section←σ−: f−1(U) → f∗P off∗P such thatf¯◦ ←σ−= σ. We
have a commutative diagram:
(f∗P)|f−1(U)×GN inclusion //f−1(U)× (P|U)×GN
f−1(U)×N
(y,n)7→(y,f(y),n) //
c←σ− ∼=
OO
f−1(U)×(U×N)
∼= Id׈s
OO
in which the vertical arrows are smooth diffeomorphisms. Clearly the bottom arrow of the diagram is a smooth embedding and the conclusion follows.