Principal fiber bundles - Principal and associated fiber bundles

Chapter 1. Principal and associated fiber bundles

1.3. Principal fiber bundles

Let M be a differentiable manifold, G be a Lie group, P be a set and let Π :P →M be a map; for eachx∈M we denote byP_xthe subsetΠ⁻¹(x)ofP and we call it thefiberofP overx. Assume that for eachx ∈ M we are given a right action ofGon the fiberP_xthat makes it into a principal space with structural groupG; equivalently, assume that the mapΠis surjective and that we are given a right action

(1.3.1) P×G3(p, g)7−→p·g∈P

ofGon P such that Π(p·g) = Π(p)for allp ∈ P, g ∈ Gand such that for all p, q∈P withΠ(p) = Π(p)there exists a uniqueg∈Gwithp·g=q.

By alocal sectionofΠwe mean a maps:U →P defined on an open subset U ofM such that Π◦ sis the inclusion map of U in M; this means that s(x) is a point of the fiber P_x, for all x ∈ U. A local sections ofΠ whose domain is the entire manifoldM will be called asection(orglobal section) ofΠ. Given local sectionss₁ : U₁ → P,s₂ : U₂ → P ofΠ then there exists a unique map g:U1∩U2 →Gsuch thats2(x) =s1(x)·g(x), for allx∈U1∩U2. The mapg is called thetransition mapfroms1 tos2. The local sectionss1 ands2are called compatibleif the mapgis smooth (this is the case, for instance, ifU₁∩U₂ =∅).

Anatlas of local sectionsofΠis a setAof local sections ofΠsuch that:

• the union of the domains of the local sections belonging toAis the whole manifoldM;

• any two local sections belonging toAare compatible.

It is easy to see that any atlasA of local sections ofΠ is contained in a unique maximal atlasA_maxof local sections ofΠ(see Exercise 1.41).

DEFINITION1.3.1. Aprincipal fiber bundle(or, more simply, aprincipal bun-dle) consists of:

• a setP, called thetotal space;

• a differentiable manifoldM, called thebase space;

• a mapΠ :P →M, called theprojection;

• a Lie groupG, called thestructural group;

• a right action (1.3.1) ofGonP that makes the fiberP_x= Π⁻¹(x)into a principal space with structural groupG, for allx∈M;

1.3. PRINCIPAL FIBER BUNDLES 23

• a maximal atlasA_maxof local sections ofΠ. The elements ofA_maxare called theadmissible local sectionsof the principal bundle.

When working with principal fiber bundles we will usually refer to the projec-tionΠ :P →Mor to the total spacePas if it were the collection of all the objects listed in Definition 1.3.1. We will also say thatP is a principal bundleoverM or thatP (orΠ :P →M) is aG-principal bundle.

Let P be a G-principal bundle over M. For every admissible local section s:U →P the map:

(1.3.2) βs:U ×G3(x, g)7−→s(x)·g∈Π⁻¹(U)⊂P

is a bijection. It follows from the result of Exercise A.1 that there exists a unique differential structure on the set P such that for every admissible local section s : U → P the set Π⁻¹(U) is open inP and the map βs is a smooth diffeo-morphism. We will always regard the total spaceP of a principal bundle to be endowed with such differential structure. The fact that the topologies ofM andG are Hausdorff and second countable implies that the topology ofP is also Haus-dorff and second countable, so thatP is a differentiable manifold. One can easily check the following facts:

• the right action (1.3.1) ofGonP is a smooth map;

• the projectionΠ :P →M is a smooth submersion;

• for everyx∈M the fiberPxis a smooth submanifold ofP;

• for everyx∈M and everyp∈P_xthe mapβ_p :G→P_x(recall (1.1.4)) is a smooth diffeomorphism;

• every admissible local sections:U →Pis a smooth map;

• if a local sections:U → P is a smooth map then it is compatible with every admissible local section and therefore (by the maximality ofA_max) it is itself an admissible local section.

Thus, the admissible local sections ofP are precisely the same as the smooth local sections ofP. Observe also that ifs:U → P is a smooth local section ofP and ifg :U → Gis a smooth map then, since the action (1.3.1) is smooth, it follows that:

s⁰:U 3x7−→s(x)·g(x)∈P is also a smooth local section ofP.

EXAMPLE1.3.2 (trivial principal bundle). LetM be a differentiable manifold and let P0 be a principal space whose structural groupG is a Lie group (for in-stance, we can takeP0 = G). Set P = M ×P0. LetΠ : P → M denote the projection onto the first coordinate and define a right action ofGonP by setting (x, p)·g = (x, p·g), for allx ∈ M, p ∈ P0 and allg ∈ G. For every p ∈ P0

the maps^p :M 3 x 7→ (x, p) ∈ P is a (globally defined) local section ofΠand the set

s^p : p ∈ P0 is an atlas of local sections ofΠ. ThusP is aG-principal bundle overMwhich we call thetrivial principal bundle overMwith typical fiber P0.indexprincipal bundle!trivial LetP0 be endowed with the differential structure that makes the mapβ_p :G→ P₀ a smooth diffeomorphism, for everyp∈P (the existence of such differential structure follows from commutative diagram (1.2.1)).

Clearly the differential structure ofP =M×P0coincides with the standard dif-ferential structure defined on a cartesian product of differentiable manifolds.

EXAMPLE1.3.3. Let Π : P → M be aG-principal bundle. IfU is an open subset ofM, we set:

P|_U = Π⁻¹(U)⊂P.

The right action ofGonPrestricts to a right action ofGonP|_Uand the projection Πrestricts to a map (also denoted byΠ) fromP|_U toU. The setP|_U is then a G-principal bundle over the manifold U endowed with the maximal atlas of local sections consisting of all the smooth local sections ofP with domain contained in U. We callP|_U the restrictionof the principal bundle P to the open set U. Obviously,P|_U is an open subset ofP; moreover, the differential structure ofP|_U coincides with the differential structure it inherits fromP as an open subset.

EXAMPLE1.3.4. LetGbe a Lie group andHa closed subgroup ofG. Con-sider the quotient mapΠ : G → G/H and the action of H onGby right trans-lations. For each x ∈ G/H, the fiber Π⁻¹(x) is a left coset of H in Gand it is therefore a principal space with structural groupH(see Example 1.2.3). Since Gis a manifold, we can talk about smooth local sections of Π. Ifs1 : U → G, s₂:V →Gare smooth local sections ofΠthen the transition maph:U∩V →H is given by:

h(x) =s1(x)⁻¹s2(x),

for allx ∈ U ∩V, and therefore it is smooth. Hence the set of all smooth local sections ofΠis an atlas of local sections ofΠandΠ :G→G/His anH-principal bundle endowed with atlas of all smooth local sections ofΠ. It is easily seen that the differential structure on G induced by such atlas coincides with the original differential structure ofG.

DEFINITION1.3.5. Givenx∈M andp∈P_x, then the tangent spaceT_pP_xis a subspace ofTpP and it is called thevertical spaceofP atp; we write:

Verp(P) =TpPx. Clearly,Ver_p(P)is equal to the kernel ofdΠ(p), i.e.:

Verp(P) = Ker dΠ(p) .

Since the mapβpis a smooth diffeomorphism fromGonto the fiber containingp, its differential at the unit element1∈Gis an isomorphism

(1.3.3) dβ_p(1) :g−→Ver_p(P)

from the Lie algebragof the structural groupGonto the vertical spaceVerp(P).

We call (1.3.3) thecanonical isomorphismfromgtoVer_p(P).

By differentiating the right action (1.3.1) ofGon P with respect to the first variable we obtain a right actionT P ×G→T P ofGon the tangent bundleT P; more explicitly, for everyg∈Gand everyζ ∈T P we set:

ζ·g= dγ_g(ζ)∈T P,

1.3. PRINCIPAL FIBER BUNDLES 25

whereγg : P → P is the diffeomorphism given by the action ofg onP. Since the diffeomorphismγ_g takes fibers to fibers, the action ofGonT P takes vertical spaces to vertical spaces, i.e.:

(1.3.4) dγ_g Ver_p(P)

= Verp·g(P),

for allp ∈ P and allg ∈ G. Let us look at the action ofGon vertical spaces by identifying them with the Lie algebragvia the canonical isomorphisms; for every p∈P,g∈G, we have the following commutative diagram:

(1.3.5)

Ver_p(P) ^{action of}^g ^//Verp·g(P)

Ad_g−1

dβp(1) ^∼=

dβp·g(1)

∼=

whereAddenotes theadjoint representationofGongdefined by (recall (1.1.3)):

Adg = dI_g(1) :g−→g,

for allg ∈G. The commutativity of diagram (1.3.5) follows from the commuta-tivity of diagram (1.2.5) by differentiation.

DEFINITION1.3.6. LetP be aG-principal bundle overM and letHbe a Lie subgroup ofG. Aprincipal subbundleofP with structural groupHis a subsetQ ofP satisfying the following conditions:

• for allx∈M,Q_x=P_x∩Qis a principal subspace ofP_xwith structural groupH, i.e.,Qxis anH-orbit;

• for allx ∈M, there exists a smooth local sections:U → P such that x∈U ands(U)⊂Q.

We consider the restriction of the right action ofGonPto a right action ofH on Qand we consider the restriction of the projectionΠ : P → M toQ. Then Qis anH-principal bundle overM endowed with the maximal atlas consisting of all local sections s : U → QofQfor whichi◦s : U → P is smooth³, where i:Q→P denotes the inclusion map.

Being the total space of a principal bundle, the setQis endowed with a differ-ential structure. Let us take a look at the relation between the differdiffer-ential structure ofQand ofP. Ifs:U →Qis a smooth local section ofQtheni◦s:U →P is

3To prove the compatibility between the local sections ofQthe reader should recall the follow-ing important result from the theory of Lie groups: ifGis a Lie group andHis a Lie subgroup ofG then a smooth map havingGas its counter-domain and having its image contained inH remains a smooth map if we replace its counter-domain byH.

a smooth local section ofP and we have a commutative diagram:

U ×G ^β^i◦s∼= //P|_U

U ×H

βs

∼= //

inclusion

Q|_U

inclusion

in which the horizontal arrows are smooth diffeomorphisms. It follows that the inclusion map i : Q → P is a smooth immersion. Unfortunately, it is not in general an embedding; in fact, the inclusion mapi : Q → P is an embedding if and only ifH is an embedded Lie subgroup ofG(recall that a subgroupH ofG is an embedded Lie subgroup ofGif and only ifH is closed inG). AlthoughQ is in general just an immersed submanifold ofP, it has the followingreduction of counter-domain property: if X is a locally connected topological space (resp., a differentiable manifold) and if φ : X → Qis a map such thati◦φ : X → P is continuous (resp., smooth) then the mapφ :X → Qis also continuous (resp., smooth). In fact, the principal subbundleQis an almost embedded submanifold of P.

Let us now define the natural morphisms of the category of principal bundles with base spaceM.

DEFINITION1.3.7. LetP,Qbe principal bundles over the same differentiable manifoldM, with structural groupsGandH respectively. A mapφ :P → Qis calledfiber preserving ifφ(P_x) ⊂ Q_x, for allx ∈ M. Amorphism of principal bundlesfromPtoQis a smooth fiber preserving mapφ:P →Qfor which there exists a group homomorphism φ0 : G → H such that for all x ∈ M, the map φ_x = φ|_P_x : P_x → Q_x is a morphism of principal spaces with subjacent group homomorphismφ0.

The group homomorphismφ₀:G→His uniquely determined from the mor-phism of principal bundlesφ:P → Q; the commutativity of diagram (1.2) (with P andQreplaced by fibersP_x andQ_x, respectively) shows that the smoothness ofφimplies the smoothness of the group homomorphismφ0. Thus,φ0 is indeed a Lie group homomorphism. We call it theLie group homomorphism subjacent to the morphism of principal bundlesφ.

The compositionψ◦φof morphisms of principal bundlesφandψwith sub-jacent Lie group homomorphismsφ₀ andψ₀ is a morphism of principal bundles with subjacent Lie group homomorphismψ0◦φ0(see Exercise 1.43). A morphism of principal bundles φ is bijective if and only if its subjacent Lie group homo-morphismφ0 is bijective. A bijective morphism of principal bundles is called an isomorphism of principal bundles. If φ is an isomorphism of principal bundles with subjacent Lie group homomorphism φ₀ thenφis a smooth diffeomorphism andφ⁻¹ is also an isomorphism of principal bundles with subjacent Lie group ho-momorphismφ⁻¹₀ (see Exercise 1.46).

EXAMPLE1.3.8. IfP is aG-principal bundle,H is a Lie subgroup ofGand Q is an H-principal subbundle of P then the inclusion map from Q to P is a

1.3. PRINCIPAL FIBER BUNDLES 27

morphism of principal bundles whose subjacent Lie group homomorphism is the inclusion map fromHtoG(compare with Example 1.2.22).

EXAMPLE1.3.9. LetM be a differentiable manifold andP₀,Q₀be principal spaces whose structural groups are Lie groups G, H, respectively; consider the trivial principal bundlesM ×P₀ andM ×Q₀. Letφ:P₀ → Q₀ be a morphism of principal spaces whose subjacent group homomorphismφ0 :G → H is a Lie group homomorphism. Then Id×φ : M ×P0 → M ×Q0 is a morphism of principal bundles whose subjacent Lie group homomorphism isφ₀.

EXAMPLE 1.3.10. LetP be aG-principal bundle over a differentiable man-ifoldM and lets : U → P be a smooth local section ofP. The map βs is an isomorphism of principal bundles from the trivialG-principal bundleU ×Gonto P|_U. The Lie group homomorphism subjacent toβsis the identity map ofG.

A fiber preserving mapφ:P → Qthat is a morphism of principal spaces on each fiber can be used to push-forward the principal bundle structure of the domain P to the counter-domainQ; more precisely, we have the following:

LEMMA1.3.11. LetΠ :P →Mbe aG-principal bundle over a differentiable manifold M. LetQ be a set, Π⁰ : Q → M be a map, H be a Lie group and assume that it is given right action of H on Q that makes the fiber Qx into a principal space with structural groupH, for all x ∈ M. Letφ₀ : G → H be a Lie group homomorphism and letφ : P → Q be a fiber preserving map such that φ|_P_x : P_x → Q_x is a morphism of principal spaces with subjacent group homomorphismφ0, for all x ∈ M. Then there exists a unique maximal atlas of local sections of Π⁰that makesφ:P →Qa morphism of principal bundles.

PROOF. Consider the following set of local sections ofΠ⁰:

(1.3.6)

φ◦s:sis a smooth local section ofP .

Let us show that (1.3.6) is an atlas of local sections ofΠ⁰. Obviously, the domains of the local sections belonging to (1.3.6) constitute a covering ofM. Moreover, if s₁ : U₁ → P, s₂ : U₂ → P are smooth local sections ofP with transition map g:U1∩U₂ →Gthen the transition map fromφ◦s₁toφ◦s₂isφ0◦g:U1∩U₂ →H;

thusφ◦s₁ andφ◦s₂are compatible and (1.3.6) is an atlas of local sections ofΠ⁰. To conclude the proof, observe that a maximal atlasA_max of local sections ofΠ⁰ makesφ : P → Q a morphism of principal bundles if and only ifA_max is the maximal atlas of local sections ofΠ⁰containing (1.3.6) (see Exercise 1.45).

COROLLARY1.3.12. LetP,P⁰,Qbe principal bundles over a differentiable manifold M with structural groups G, G⁰ andH respectively. Letφ : P → Q, ψ : P → P⁰ be morphisms of principal bundles with subjacent Lie group ho-momorphisms φ₀ : G → H and ψ₀ : G → G⁰. Let φ⁰₀ : G⁰ → H be a Lie group homomorphism and letφ⁰ : P⁰ → Q be a fiber preserving map such that

φ|_P⁰

x : P_x⁰ → Qx is a morphism of principal spaces with subjacent group homo-morphismφ⁰₀, for allx∈M. Assume that the diagram:

ψ P⁰

φ⁰

//Q

commutes. Thenφ⁰ is a morphism of principal bundles with subjacent Lie group homomorphismφ⁰₀.

PROOF. LetA_maxbe the maximal atlas of local sections of the principal bun-dleQand letA⁰_maxbe the unique maximal atlas of local sections ofQthat makes φ⁰ a morphism of principal bundles. BothA_max andA⁰_max make φ = φ⁰ ◦ψ a morphism of principal bundles; by the uniqueness part of Lemma 1.3.11, we have

A_max=A⁰_max. This concludes the proof.

1.3.1. Pull-back of principal bundles. AG-principal bundle over a differen-tiable manifoldM can be though of as a “smoothly varying” family(P_x)x∈M of principal spacesPxwith structural groupGparameterized by the points ofM. If M⁰is another differentiable manifold andf :M⁰→M is a smooth map then it is natural to consider a reparametrization(P_f_(y))y∈M⁰ of the family(Px)x∈M by the mapf. This idea motivates the definition of the pull-back of a principal bundle.

Let us now give the precise definitions.

LetΠ :P → M be aG-principal bundle and letf : M⁰ → M be a smooth map defined on a differentiable manifoldM⁰. Thepull-back ofP byf is the set f^∗P defined by:

f^∗P = [

y∈M⁰

{y} ×P_f(y) .

Thus, the set f^∗P is a subset of the cartesian product M⁰ ×P. The restriction to f^∗P of the projection onto the first coordinate is a map Π₁ : f^∗P → M⁰ and the restriction tof^∗P of the projection onto the second coordinate is a map f¯:f^∗P →P; the following diagram commutes:

(1.3.7)

f^∗P ^f^¯ ^//

Π1

M⁰ _f ^//M

We callf¯:f^∗P →Pthecanonical mapassociated to the pull-backf^∗P; when it is necessary to make the principal bundleP explicit, we will also writef¯^P instead of justf¯.

Notice that the pull-back f^∗P is precisely the subset ofM⁰ ×P where the mapsΠ◦f¯andf ◦Π1 coincide; moreover, the map(Π1,f¯) : f^∗P → M⁰×P is just the inclusion map. From this two simple observations, we get the following set-theoretical lemma:

1.3. PRINCIPAL FIBER BUNDLES 29

LEMMA 1.3.13. LetΠ : P → M be a G-principal bundle, M⁰ be a differ-entiable manifold andf :M⁰ → M be a smooth map. Given a setX and maps τ1 :X → M⁰,τ2 :X → P withΠ◦τ2 =f ◦τ1 then there exists a unique map τ :X→f^∗P such thatΠ1◦τ =τ1andf¯◦τ =τ2.

PROOF. The condition Π◦ τ2 = f ◦τ1 means that the image of the map (τ₁, τ₂) :X →M⁰×P is contained inf^∗P; since(Π₁,f¯)is the inclusion map of f^∗P intoM⁰×P, there exists a unique mapτ :X→f^∗P such that:

(1.3.8) (Π1,f¯)◦τ = (τ1, τ2).

But this last equality is equivalent toΠ₁◦τ =τ₁andf¯◦τ =τ₂. The situation in Lemma 1.3.13 is illustrated by the following commutative diagram:

(1.3.9)

X _τ₂

τ1

f^∗P _¯

Π1

M⁰ _f ^//M

In Exercise 1.53 we define the general notion of pull-back in arbitrary categories and in Exercise 1.54 we ask the reader to generalize Lemma 1.3.13 by presenting the notion of pull-back in the category of sets and maps.

Our goal now is to makeΠ₁ :f^∗P →M⁰ into aG-principal bundle overM⁰. For eachy∈M⁰, the fiber(f^∗P)yis equal to{y}×P_f(y);we will identify the fiber (f^∗P)y off^∗P with the fiberP_f(y)ofP. Under such identification, every fiber of f^∗P is a fiber ofP and thus each fiber off^∗P is endowed with a right action of Gthat makes it into a principal space with structural groupG. Our next step is to define an atlas of local sections ofΠ₁.

DEFINITION1.3.14. By alocal section of the principal bundleP alongf we mean a map σ : U⁰ → P defined on an open subset U⁰ of M⁰ satisfying the conditionΠ◦σ=f|_U⁰.

EXAMPLE1.3.15. Ifs:U → P is a local section ofP then the composition s◦f :f⁻¹(U)→P is a local section ofP alongf.

Clearly, if we compose a local section of Π₁ : f^∗P → M⁰ on the left with f¯, we obtain a local section of P along f; moreover, ifσ : U⁰ → P is a local section of P along f then there exists a unique local section ←−σ : U⁰ → f^∗P of Π1 : f^∗P → M⁰ such that f¯◦ ←−σ = σ. Namely, taking X = U⁰, τ1 to be the inclusion map of U⁰ in M⁰ and τ₂ = σ then ←−σ is the map τ given by the thesis of Lemma 1.3.13. The following commutative diagram illustrates the

relation between←−σ andσ:

f^∗P ^f^¯ ^//P

U⁰

←−σ

==z

zz zz zz z

f|_U0

//M

We have thus established that composition on the left with f¯induces a bijection between the set of local sections ofΠ₁ :f^∗P → M⁰ and the set of local sections ofP alongf.

Lets1 :U1 → P,s2 :U2 →P be smooth local sections ofP with transition mapg : U₁∩U₂ → G. Setσ_i = s_i◦f,i= 1,2, and consider the local section

←σ−i : f⁻¹(Ui) → f^∗P of Π1 : f^∗P → M⁰ such thatf¯◦ ←σ−i = σi, i = 1,2.

Evidently, the transition map from←σ−₁ to←σ−₂ is g◦f : f⁻¹(U₁ ∩U₂) → Gand therefore the local sections ←σ−1 and←σ−2 are compatible. This observation implies that the set:

(1.3.10) ←σ−:σ =s◦f andsis a smooth local section ofP

is an atlas of local sections of Π₁ : f^∗P → M⁰. If we endow f^∗P with the unique maximal atlas of local sections containing (1.3.10) then f^∗P becomes a G-principal bundle overM⁰. We will always consider the pull-back f^∗P to be endowed with such maximal atlas of local sections.

The following lemma allows us to understand better the manifold structure of the total spacef^∗P.

LEMMA1.3.16. LetΠ : P → M be aG-principal bundle,M⁰ be a differen-tiable manifold andf :M⁰ → M be a smooth map. LetΠ₁ :f^∗P → M⁰denote the pull-back of P by f. Then the map(Π1,f¯) : f^∗P → M⁰ ×P is a smooth embedding; in particular, the canonical mapf¯:f^∗P →P is smooth.

PROOF. By the result of Exercise A.2, in order to prove that (Π1,f¯) is a smooth embedding, it suffices to show that for every smooth local sections:U → PofPthe restriction of the map(Π1,f¯)to the open set(Π1,f¯)⁻¹ f⁻¹(U)×P

= (f^∗P)|_f−1(U)is a smooth embedding. Setσ =s◦fand consider the local section

←σ−off^∗P such thatf¯◦ ←σ−=σ. We have a commutative diagram:

(f^∗P)|_f−1(U)

(Π1,f)¯ //f⁻¹(U)×P|_U

f⁻¹(U)×G

β^←⁻_σ ^∼=

(y,g)7→(y,f(y),g) //f⁻¹(U)×(U ×G)

Id×β_s

∼=

in which the vertical arrows are smooth diffeomorphisms. The proof is concluded by observing that the bottom arrow of the diagram is a smooth embedding.

Lemma 1.3.16 says that the pull-back of principal bundles is a particular case of the notion of pull-back in the category of differentiable manifolds and smooth maps (see Exercise 1.55).

1.3. PRINCIPAL FIBER BUNDLES 31

EXAMPLE 1.3.17. Let Π : P → M be a G-principal bundle. If U is an open subset ofM andi : U → M denotes the inclusion map then the canonical map ¯ı : i^∗P → P is injective and its image is equal to P|_U. Moreover, the map¯ı :i^∗P → P|_U is an isomorphism of principal bundles whose subjacent Lie group homomorphism is the identity map ofG(the fact that¯ı is smooth follows from Lemma 1.3.16). We will use the map¯ıto identify the pull-backi^∗P with the restricted principal bundleP|_U.

Using Lemma 1.3.16 we can prove the following important property of pull-backs.

PROPOSITION1.3.18 (universal property of the pull-back). Under the condi-tions of Lemma 1.3.13, ifXis a differentiable manifold then the mapτ is smooth if and only if bothτ1andτ2are smooth.

PROOF. Follows directly from the equality (1.3.8) and from the fact that the map(Π1,f¯)is a smooth embedding (Lemma 1.3.16).

COROLLARY1.3.19. LetΠ :P → M be a principal bundle, M⁰ be a differ-entiable manifold andf :M⁰ →Mbe a smooth map. A local sectionσ :U⁰→P ofP alongf is smooth if and only if the local section←σ−: U⁰ → f^∗P off^∗P is smooth.

PROOF. If we takeX=U⁰,τ₁to be the inclusion map ofU⁰inM⁰andτ₂ =σ then←σ−is the mapτ given by the thesis of Lemma 1.3.13. The conclusion follows

from Proposition 1.3.18.

Corollary 1.3.19 implies that composition on the left withf¯induces a bijection between the set of smooth local sections off^∗Pand the set of smooth local sections ofP alongf.

DEFINITION 1.3.20. LetΠ : P → M, Π⁰ : P⁰ → M⁰ be principal bundles with structural groupsGandG⁰, respectively and letf : M⁰ → M be a smooth map. A mapϕ:P⁰ → P is said to befiber preserving alongf ifϕ(P_y⁰) ⊂P_f_(y), for all y ∈ M⁰. By a morphism of principal bundles along f from P⁰ to P we mean a smooth mapϕ:P⁰→P such that:

• ϕis fiber preserving alongf;

• there exists a group homomorphismϕ0 :G⁰ → Gsuch that for allyin M⁰ the mapϕ_y =ϕ|_P⁰

y :P_y⁰ → P_f_(y)is a morphism of principal spaces with subjacent group homomorphismϕ0.

As we have previously observed for morphisms of principal bundles (recall Definition 1.3.7), ifϕis a morphism of principal bundles alongf then the group homomorphismϕ₀ is uniquely determined byϕand the smoothness of ϕimplies the smoothness ofϕ₀. We callϕ₀theLie group homomorphism subjacent toϕ.

Clearly a mapϕ:P⁰ → P is fiber preserving alongf :M⁰ → M if and only if the diagram:

(1.3.11)

P⁰ ^ϕ ^//

Π⁰

M⁰ _f ^//M commutes.

EXAMPLE1.3.21. IfΠ : P → M is a principal bundle with structural group Gand iff : M⁰ → M is a smooth map defined in a differentiable manifoldM⁰ then the canonical mapf¯:f^∗P →P is fiber preserving alongf(compare (1.3.7) with (1.3.11)); moreover, f¯is a morphism of principal bundles along f whose subjacent Lie group homomorphism is the identity map of G. If Π⁰ : P⁰ → M⁰ is aG⁰-principal bundle overM⁰then the composition off¯with a fiber preserving map fromP⁰tof^∗P is a fiber preserving map alongf fromP⁰toP. Conversely, if a mapϕ:P⁰ →f^∗P is fiber preserving alongf then there exists a unique fiber preserving map←ϕ− :P⁰ → f^∗P such thatf¯◦ ←ϕ−=f; namely, the map←ϕ−is the map τ given by the thesis of Lemma 1.3.13, if we takeX = P⁰, τ₁ = Π⁰ and τ2 =ϕ. The relation betweenϕand←ϕ−is illustrated by the following commutative diagram:

P⁰ _ϕ

Π⁰

←ϕ−

f^∗P

f¯

Π1

M⁰ _f ^//M

We can now state another corollary of Proposition 1.3.18.

COROLLARY1.3.22. LetΠ : P → M, Π⁰ : P⁰ → M⁰ be principal bundles with structural groups G andG⁰, respectively, f : M⁰ → M be a smooth map andϕ : P⁰ → P be a fiber preserving map alongf. Then ϕis smooth if and only if the fiber preserving map ←ϕ− : P⁰ → f^∗P is smooth. Moreover, ϕ is a morphism of principal bundles alongf with subjacent Lie group homomorphism ϕ₀ : G⁰ → Gif and only if←ϕ−is a morphism of principal bundles with subjacent Lie group homomorphismϕ0.

PROOF. The fact thatϕis smooth if and only if ←ϕ−is smooth follows from Proposition 1.3.18 and Example 1.3.21. The rest of the thesis follows from the observation that for ally∈M⁰the maps:

ϕ_y :P_y⁰ −→P_f(y), ←ϕ−_y :P_y⁰ −→(f^∗P)_y =P_f_(y)

are the same.

1.3. PRINCIPAL FIBER BUNDLES 33

EXAMPLE 1.3.23. Let Π : P → M, Π⁰ : Q → M be principal bundles with structural groupsGandH, respectively; letM⁰be a differentiable manifold and letf : M⁰ → M be a smooth map. Given a morphism of principal bundles φ:P →Qwith subjacent Lie group homomorphismφ0 :G→Hthen:

φ◦f¯^P :f^∗P −→Q

is a morphism of principal bundles alongf with subjacent Lie group homomor-phismφ0; we set:

f^∗φ=

←−−−−

φ◦f¯^P,

so thatf^∗φ:f^∗P →f^∗Qis the unique fiber preserving map such that the diagram:

(1.3.12)

P ^φ ^//Q

f^∗P

f¯^P

f^∗φ //f^∗Q

f¯^Q

commutes. By Corollary 1.3.22, the mapf^∗φis a morphism of principal bundles with subjacent Lie group homomorphism φ₀. We call f^∗φ the pull-back of the morphism φby f. As a particular case of this construction, notice that ifP is a principal subbundle ofQandi:P →Qdenotes the inclusion map thenf^∗P is a principal subbundle off^∗Qandf^∗i:f^∗P →f^∗Qis the inclusion map.

EXAMPLE 1.3.24. Let M, M⁰, M⁰⁰ be differentiable manifolds, P be a G-principal bundle overM and letf : M⁰ → M,g : M⁰⁰ → M⁰ be smooth maps.

The compositionf¯◦¯gof the canonical maps:

f¯:f^∗P −→P, g¯:g^∗f^∗P −→f^∗P

is a morphism of principal spaces alongf ◦gwhose subjacent Lie group homo-morphism is the identity map ofG. Thus, by Corollary 1.3.22, the map:

(1.3.13) ←−−−

f¯◦g¯ :g^∗f^∗P −→(f◦g)^∗P characterized by the equality:

f ◦g◦←−−−

f¯◦g¯ = ¯f◦g¯

is an isomorphism of principal bundles whose subjacent Lie group homomorphism is the identity map ofG.We use the map(1.3.13)to identify the principal bundles g^∗f^∗P and(f ◦g)^∗P. Under such identification, we have:

f ◦g= ¯f◦¯g.

1.3.2. The fiberwise product of principal bundles. LetΠ : P → M and Π⁰ : Q → M be principal fiber bundles with structural groupsGandH, respec-tively. Thefiberwise productofP withQis the setP ? Qdefined by:

P ? Q= [

x∈M

(P_x×Q_x).

No documento The theory of connections and G-structures. Applications to afﬁne and isometric immersions Paolo Piccione Daniel V. Tausk (páginas 30-44)