Vector bundles and the principal bundle of frames

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.

1.5. VECTOR BUNDLES AND THE PRINCIPAL BUNDLE OF FRAMES 45

is bijective and it restricts to a linear isomorphism fromFRE0(Ex)×_∼E0toEx, for allx ∈ M (recall Example 1.2.28). Thus, there is a unique differential structure on the set E that makes the contraction map C^E a smooth diffeomorphism. We will always consider the total spaceEof a vector bundle to be endowed with such differential structure. Clearly the topology ofEis Hausdorff and second countable, so thatEis a differentiable manifold. The following facts follow directly from the corresponding facts stated in Section 1.4 for general associated bundles and from the comments made in Example 1.4.5:

• the projectionπ :E→M is a smooth submersion;

• the mapFR_E0(E)×E0 3(p, e0)7→p(e0)∈Eis a smooth submersion;

• for everyx∈M the fiberE_xis a smooth submanifold ofE;

• for everyx ∈M the differential structure that the fiberEx inherits from Eas a submanifold coincides with the differential structure that is deter-mined by its real finite-dimensional vector space structure.

Lets:U →FRE0(E)be a smooth local section ofFRE0(E)and setˇs=C^E◦ˆs;

more explicitly, the mapˇsis given by:

ˇ

s:U×E₀3(x, e₀)7−→s(x)·e₀∈π⁻¹(U)⊂E.

The mapsˇis a smooth diffeomorphism and we will call it thelocal trivialization ofE corresponding to the smooth localE₀-frames. Notice that the differential structure of the total spaceE can also be characterized by the fact that for every smooth localE0-frames:U →FR_E0(E)the mapsˇis a smooth diffeomorphism onto the open subsetπ⁻¹(U)ofE.

EXAMPLE1.5.2 (the trivial vector bundle). LetMbe a differentiable manifold andE₀be a real finite-dimensional vector space. SetE=M×E₀and consider the mapπ :E → M given by projection onto the first coordinate. For everyx ∈M we identify the fiberEx ={x} ×E0withE0so thatExhas the structure of a real vector space and:

FRE0(M×E0) =M×GL(E0).

The setFRE0(M×E0)is thus naturally endowed with the structure of aGL(E0 )-principle bundle (see Example 1.3.2) and thereforeE is a vector bundle overM which we call the trivial vector bundle over M with typical fiber E0. Clearly the differential structure ofE = M×E0 coincides with the standard differential structure given to a cartesian product of differentiable manifolds.

EXAMPLE1.5.3. Letπ :E →M be a vector bundle with typical fiberE0. If U is an open subset ofM, we set:

E|_U =π⁻¹(U);

the projectionπ :E → M restricts to a map fromE|_U toM and for eachx ∈U the fiber E_x ofE|_U overxis endowed with the structure of a real vector space.

Clearly:

FR_E0(E|_U) = FR_E0(E)|_U,

so thatFR_E0(E|_U)is aGL(E₀)-principal bundle over the differentiable manifold U (see Example 1.3.3). ThusE|_U is a vector bundle overU which we call the

restrictionof the vector bundleEto the open setU. Clearly, the differential struc-ture ofE|_U coincides with the differential structure it inherits fromE as an open subset.

EXAMPLE 1.5.4 (the tangent bundle). LetM be an n-dimensional differen-tiable manifold, let

T M = [

x∈M

TxM

denote its tangent bundle and letπ:T M →Mdenote the standard projection that sendsT_xMtox, for allx∈M. For everyx∈M, the fiberT_xM has the structure of a real vector space isomorphic toRⁿ. Letϕ : U → Ue be a local chart ofM, whereU is an open subset ofM andUe is an open subset ofRⁿ. For everyx∈U the mapdϕ(x)⁻¹ :Rⁿ→TxM is a linear isomorphism and the map:

s^ϕ:U 3x7−→dϕ(x)⁻¹ ∈FR(T M)

is a local section ofFR(T M)→M. Ifψ:V →Veis another local chart ofMand ifα=ϕ◦ψ⁻¹ :ψ(U∩V)→ϕ(U∩V)denotes the transition map fromψtoϕthen the transition map froms^ϕtos^ψis given byU ∩V 3x7→dα ψ(x)

∈GL(Rⁿ) and therefore the set:

(1.5.2)

s^ϕ:ϕis a local chart ofM

is an atlas of local sections ofFR(T M) → M. We endow FR(T M) with the unique maximal atlas of local sections ofFR(T M) → M containing (1.5.2) and thenπ :T M →M is a vector bundle overMwith typical fiberRⁿ.

EXAMPLE1.5.5. LetPbe aG-principal bundle over a differentiable manifold M,E₀be a real finite-dimensional vector space andρ:G→GL(E₀)be a smooth representation ofGonE0. As explained in Example 1.4.5, the fibers of the associ-ated bundleP×_GE0have the structure of a real vector space isomorphic toE0. In order to makeP×_GE₀into a vector bundle overMwith typical fiberE₀, we have to describe a maximal atlas of local sections ofFRE0(P ×_GE0). This is done as follows. Consider the map (recall (1.2.16)):

(1.5.3) H:P 3p7−→pˆ∈FR_E0(P×_GE₀).

ClearlyHis fiber-preserving and for eachx∈M it restricts to a morphism of prin-cipal spaces fromP_xtoFR_E0(P_x×_GE₀)whose subjacent group homomorphism is the representationρ : G → GL(E0). By Lemma 1.3.11, there exists a unique maximal atlas of local sections ofFR_E0(P×_GE₀)→Mthat makesHinto a mor-phism of principal bundles.We will always regard the associated bundleP×_GE0

as a vector bundle withFR_E0(P ×_GE0)endowed with the maximal atlas of local sections that makesHa morphism of principal bundles.

Observe thatP×_GE0has, in principle,twodistinct differential structures: one that was defined in Section 1.4 for arbitrary associated bundles and the other that is assigned to the total space of vector bundles, i.e., the one for which the contraction map:

(1.5.4) C^P×GE0 : FR_E0(P×_GE₀)×_∼E₀ 3[%, e₀]7−→%(e₀)∈P×_GE₀

1.5. VECTOR BUNDLES AND THE PRINCIPAL BUNDLE OF FRAMES 47

is a smooth diffeomorphism. In order to check that these two differential structures coincide we endowP ×_GE₀ with the differential structure that makesC^P×GE0 into a smooth diffeomorphism and we show that for every smooth local section s:U →Pthe mapˆsis a smooth diffeomorphism onto an open subset ofP×_GE0

(recall that this is precisely the characterization of the differential structure of the total space of an associated bundle introduced in Section 1.4). Sets1 =H◦s, so thats₁ :U → FR_E0(P ×_GE₀)is a smooth localE₀-frame of the vector bundle P×_GE0. We claim that:

(1.5.5) ˆs= ˇs₁,

i.e., the local trivialization of the associated bundleP×_GE₀corresponding to the smooth local sectionsofP is equal to the local trivialization of the vector bundle P ×_GE₀ corresponding to the smooth localE₀-frames₁. Namely, givenx ∈U, e0∈E0then:

ˇ

s₁(x, e₀) =s₁(x)·e₀=H s(x)

·e₀ =s(x)(ed ₀) = [s(x), e₀] = ˆs(x, e₀).

Since the trivializationˇs₁ is a smooth diffeomorphism onto an open subset of the total space of the vector bundleP ×_GE0, it follows from (1.5.5) that ˆsis also a smooth diffeomorphism onto an open subset ofP×_GE₀. This concludes the proof that the two natural differential structures ofP×_GE0coincide. An alternative argu-ment to prove the coincidence of these two differential structures ofP×_GE₀is the following: we endowP×_GE0with the differential structure defined in Section 1.4 and we show that the contraction mapC^P×GE0 is a smooth diffeomorphism. Since His a morphism of principal bundles, we have an induced map:

Hb :P×_GE₀3[p, e₀]7−→[H(p), e₀]∈FR_E0(P×_GE₀)×_∼E₀.

By Lemma 1.4.10,Hb is a smooth diffeomorphism. To conclude the proof that the contraction map C^P×GE0 is a smooth diffeomorphism, we show that C^P×GE0 is equal to the inverse of H. Since bothb C^P×GE0 andHb are bijective, it suffices to check thatC^P×GE0 ◦Hb is the identity map ofP×_GE0; givenp∈P,e0∈E0, we compute:

C^P×GE0

Hb [p, e0]

=C^P×GE0 [H(p), e0]

=C^P×GE0 [ˆp, e0]

= ˆp(e₀) = [p, e₀].

DEFINITION1.5.6. Givenx∈M ande∈Ex then the tangent spaceTeExis a subspace ofT_eE and it is called thevertical spaceof the vector bundleE ate;

we write:

Ver_e(E) =T_eE_x. Clearly:

Vere(E) = Ker dπ(e) .

Since for everyx ∈ M the fiberEx is a real finite-dimensional vector space, we identify the tangent spaceT_eE_xat a pointe∈E_xwithE_xitself, i.e.:

Ver_e(E) =T_eE_x ∼=E_x.

For eachx ∈ M, the contraction mapC^E restricts to a linear isomorphism from FR_E0(E_x)×_∼E₀toE_xand thus its differential at a point[p, e₀]ofFR_E0(E_x)×_∼E₀ restricts to a linear isomorphism from the vertical spaceVer_[p,e0] FRE0(E)×_∼E0

toVer_p(e0)(E); recalling from (1.4.4) that the vertical space ofFRE0(E)×_∼E0at [p, e0]is identified with the fiber productFRE0(Ex)×_∼ E0 then the restriction to the vertical space of the differential ofC^E at[p, e0]is given by:

C^Ex : FR_E0(E_x)×_∼E₀ 3[p, e₀]7−→p(e₀)∈E_x.

1.5.1. Local sections of a vector bundle. Letπ:E →Mbe a vector bundle with typical fiberE₀.

DEFINITION1.5.7. By alocal sectionof the vector bundleEwe mean a map :U →E defined on an open subsetU ofM such thatπ◦is the inclusion map ofU inM, i.e., such that(x)∈Ex, for allx∈U.

If : U → E is a local section of E and ifs :U → FRE0(E)is a smooth localE₀-frame ofEthen the map˜:U →E₀defined by:

˜

(x) =s(x)⁻¹·(x)∈E0,

for allx∈Uis called therepresentationof the sectionwith respect to the smooth localE0-frames. Ifˇsis the local trivialization ofE corresponding tosthen:

(x) = ˇs x,˜(x) ,

for allx∈U; therefore the local sectionis smooth if and only if its representation

˜

is smooth.

A globally defined local section:M →Eof a vector bundleEwill be called aglobal section(or just asection) ofE. Notice that a local section:U → Eof E is the same as a global section of the restricted vector bundleE|_U. We denote by Γ(E) the set of all sections of E and by Γ(E) the set of all smooth sections ofE. Clearly Γ(E) is a real vector space endowed with the obvious operations of pointwise addition and multiplication by scalars; moreover, Γ(E) is a module over the ringR^M of all maps f : M → R. Ifs : U → FR_E0(E) is a smooth localE₀-frame ofEthen the map→ ˜that assigns to each section∈Γ(E|_U) its representation˜ : U → E0 with respect to sis a linear isomorphism of real vector spaces and also an isomorphism ofR^M-modules. Sinceis smooth if and only if˜is smooth, it follows thatΓ(E)is a subspace ofΓ(E); but it is obviously notanR^M-submodule in general. LetC^∞(M)denote the set of all smooth maps f :M →R; clearlyC^∞(M)is a subring ofR^M,Γ(E)is aC^∞(M)-module and Γ(E)is aC^∞(M)-submodule ofΓ(E).

EXAMPLE1.5.8. Let M be a differentiable manifold. A (smooth) section of the tangent bundleT Mis the same as a (smooth) vector field onM.

EXAMPLE 1.5.9. Let Π : P → M be a G-principal bundle, E₀ be a real finite-dimensional vector space andρ:G→GL(E0)be a smooth representation.

The associated bundleP ×_GE₀ is a vector bundle overM with typical fiberE₀ (recall Example 1.5.5) and the mapH:P →FR_E0(P ×_GE₀)defined by (1.5.3)

1.5. VECTOR BUNDLES AND THE PRINCIPAL BUNDLE OF FRAMES 49

is a morphism of principal bundles whose subjacent Lie group homomorphism is ρ. If s : U → P is a smooth local section ofP then the compositionH◦sis a smooth localE0-frame ofP×_GE0. Let:U → P ×_GE0be a local section of the associated bundleP×_GE0. In Subsection 1.4.1 we have defined the notion of representation ofwith respect tos(recall (1.4.6)). It is easily seen that the map

˜

:U → E0that represents the local sectionof the vector bundleP×_GE0 with respect to the smooth localE₀-frameH◦sis the same as the map that represents the local sectionof the associated bundleP ×_G E0 with respect to the smooth local sectionsofP.

EXAMPLE1.5.10. Letπ : E → M be a vector bundle with typical fiberE₀ and consider the contraction mapC^E : FRE0(E)×_∼E0 → E. If:U → E is a local section ofEthen then(C^E)⁻¹◦is a local section ofFR_E0(E)×_∼E₀and

(C^E)⁻¹◦

(x) = [s(x),(x)],˜

for all x ∈ U. Notice that the representation of the local section (C^E)⁻¹ ◦of the associated bundleFRE0(E)×_∼ E0 with respect to the smooth local sections ofFR_E0(E)(in the sense of Subsection 1.4.1) coincides with the representation of the local sectionof the vector bundleEwith respect to the smooth localE0-frame sofE.

1.5.2. Morphisms of vector bundles. We now define the natural morphisms of the category of vector bundles.

DEFINITION1.5.11. LetE,F be vector bundles over the same differentiable manifoldM. A mapL :E → F is calledfiber preservingifL(E_x) ⊂F_xfor all x ∈ M; we setL_x =L|_Ex : E_x → F_x. The mapL is calledfiberwise linearif Lis fiber preserving and ifLx is a linear map, for allx∈M. A smooth fiberwise linear mapL:E →F is called avector bundle morphism.

Denote byE₀,F₀ the typical fibers ofE andF, respectively and lets, s⁰ be smooth local sections ofFRE0(E)andFRF0(F)respectively, both defined in the same open subsetU ofM. IfL:E →F is a fiberwise linear map then we set:

L(x) =e s⁰(x)⁻¹◦L_x◦s(x)∈Lin(E₀, F₀),

for allx ∈U, whereLin(E0, F0)denotes the space of all linear maps fromE0to F₀. We callLe:U →Lin(E₀, F₀)therepresentationofLwith respect tosands⁰. We have a commutative diagram:

E|_U ^{L //}F|_U

U ×E0 ˇ s ^∼=

OO

(x,e0)7→(x,eL(x)·e₀)

//U ×F0

∼= sˇ⁰

OO

in which the vertical arrows are smooth diffeomorphisms. Clearly the bottom arrow of the diagram is smooth if and only if the mapLeis smooth. It follows that:

• ifLis a morphism of vector bundles then its representationLewith respect to arbitrary smooth local sectionssands⁰is smooth;

• if L is a fiberwise linear map and if every point of M is contained in the domainU of a pairs,s⁰ of smooth local sections for which the cor-responding representationLe is smooth then Lis a morphism of vector bundles.

LetL : E → F be a morphism of vector bundles. ObviouslyLis bijective if and only ifLx :Ex → Fx is a linear isomorphism, for allx ∈M. A bijective morphism of vector bundlesL :E → F will be called anisomorphism of vector bundles. IfL :E → F is an isomorphism of vector bundles thenLis a smooth diffeomorphism and the map L⁻¹ : F → E is also an isomorphism of vector bundles; namely,L⁻¹is clearly fiberwise linear and ifLeis the representation ofL with respect to local sections sands⁰ then x 7→ L(x)e ⁻¹ is the representation of L⁻¹with respect tosands⁰.

EXAMPLE 1.5.12. For any vector bundleπ : E → M, the contraction map C^Eis obviously an isomorphism of vector bundles fromFR_E0(E)×_∼E₀ontoE.

EXAMPLE 1.5.13. Ifs : U → FR_E0(E)is a smooth local E₀-frame of the vector bundleEthen the local trivializationˇs:U×E0→E|_Uis an isomorphism of vector bundles from the trivial bundleU ×E₀ontoE|_U.

EXAMPLE 1.5.14. LetP be aG-principal bundle over a differentiable man-ifoldM, E0 be a real finite-dimensional vector space and ρ : G → GL(E0) be a smooth representation. Ifs : U → P is a smooth local section then the map ˆ

s : U ×E₀ → (P|_U)×_GE₀ (recall (1.4.2)) is a vector bundle isomorphism.⁴ Notice that this example can also be seen as a particular case of Example 1.5.13.

Namely, by (1.5.5),sˆ= ˇs1, wheres1 =H◦sandH:P → FR_E0(P ×_GE0)is the morphism of principal bundles defined in (1.5.3).

Let us particularize Lemma 1.4.10 to the context of vector bundles.

LEMMA 1.5.15. 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 real finite-dimensional vector space E₀ and a smooth representationρ : H → GL(E0), we consider the smooth representation ofGin E0given byρ◦φ₀ :G→GL(E0). Then the induced mapφˆ:P×_GE0→Q×_HE0

is an isomorphism of vector bundles.

PROOF. The restriction ofφˆto each fiber ofP×_GE0is a linear isomorphism (Example 1.2.33). Moreover, Lemma 1.4.10 implies thatφˆis smooth.

We also have a version of Lemma 1.4.11 for vector bundles.

4Recall from Example 1.5.5 that the differential structure ofP×GE0that makes the mapˆsa smooth diffeomorphism coincides with the differential structure thatP×GE0has as the total space of a vector bundle.

1.5. VECTOR BUNDLES AND THE PRINCIPAL BUNDLE OF FRAMES 51

LEMMA 1.5.16. Let P, Qbe principal bundles over the same differentiable manifold M with structural groups Gand H, respectively. Letφ : P → Q be a morphism of principal bundles and letφ0 : G → H denote its subjacent Lie group homomorphism. LetE0,F0be real finite-dimensional vector spaces and let ρ :G→ GL(E₀),ρ⁰ :H →GL(F₀)be smooth representations. Assume that we are given a linear mapT0 :E0 →F0such thatT0◦ρ(g) =ρ⁰ φ0(g)

◦T0, for all g∈G. Then the induced mapφ×_∼T₀ :P×_GE₀ →Q×_H F₀ is a vector bundle morphism.

PROOF. Clearlyφ×_∼T₀is fiber preserving and, by Example 1.2.33,φ×_∼T₀is fiberwise linear. Finally, Lemma 1.4.11 implies thatφ×_∼T0 is smooth.

DEFINITION1.5.17. LetP be aG-principal bundle over a differentiable man-ifold M, E0 be a real finite-dimensional vector space, ρ : G → GL(E0) be a smooth representation, E be a vector bundle overM with typical fiber E₀ and φ:P →FRE0(E)be a morphism of principal bundles whose subjacent Lie group homomorphism is the representationρ. We set:

C^φ=C^E◦φˆ:P ×_GE03[p, e0]7−→φ(p)·e0∈E, and we callC^φtheφ-contraction map.

It follows from Lemma 1.5.15 and Example 1.5.12 thatC_φ^E is an isomorphism of vector bundles.

There is a relation between isomorphisms of vector bundles and isomorphisms of the corresponding principal bundles of frames. Let E, E⁰ be vector bundles over a differentiable manifoldM, with the same typical fiberE₀. Given a bijective fiberwise linear mapL:E →E⁰then the map:

L∗ : FR_E0(E)3p7−→L◦p∈FR_E0(E⁰)

is fiber preserving and its restriction to each fiber is a morphism of principal spaces whose subjacent Lie group homomorphism is the identity (recall Examples 1.2.17 and 1.2.23). We callL∗the mapinducedbyLon the frame bundles. We have the following:

LEMMA1.5.18. LetE,E⁰be vector bundles over the same differentiable man-ifoldM, with the same typical fiber E0. IfL : E → E⁰ is a bijective fiberwise linear map thenLis smooth if and only if the induced mapL∗is smooth; in other words,Lis an isomorphism of vector bundles if and only ifL∗is an isomorphism of principal bundles whose subjacent Lie group homomorphism is the identity.

PROOF. Lets:U →FR_E0(E),s⁰:U →FR_E0(E⁰)be smooth local sections and denote byLe:U → Lin(E₀, E₀)the representation ofLwith respect tosand s⁰. SinceL is an isomorphism of vector bundles, the mapLe takes values on the general linear groupGL(E₀); we have:

(L∗◦s)(x) =L_x◦s(x) =s⁰(x)◦L(x),e

for allx ∈ U. Since boths⁰ andLe are smooth, it follows thatL∗◦sis a smooth local section ofFR_E0(E⁰). Hence, by the result of Exercise 1.45,L∗is a morphism of principal bundles whose subjacent Lie group homomorphism is the identity.

Conversely, assume thatL∗is an isomorphism of principal bundles whose sub-jacent Lie group homomorphism is the identity. We have an induced map:

Lc∗ : FR_E0(E)×_∼E₀ −→FR_E0(E⁰)×_∼E₀

which is a smooth diffeomorphism, by Lemma 1.4.10. It is easily seen that the diagram:

(1.5.6)

FRE0(E)×_∼E0

Lc∗ //

C^E

FRE0(E⁰)×_∼E0 C^E0

E L //E⁰

commutes. This proves thatLis smooth.

EXAMPLE 1.5.19. If s : U → FRE0(E) is a smooth local E0-frame of a vector bundleE then the mapβ_s:U×GL(E₀)→ FR_E0(E|_U)(recall (1.3.2)) is an isomorphism of principal bundles whose subjacent Lie group homomorphism is the identity map ofGL(E₀)(see Example 1.3.10). Clearlyβ_s= (ˇs)∗.

1.5.3. Pull-back of vector bundles. Letπ :E →M be a vector bundle over a differentiable manifoldMwith typical fiberE₀and letf :M⁰→Mbe a smooth map defined on a differentiable manifoldM⁰. Thepull-back ofE byf is the set f^∗Edefined by:

f^∗E= [

y∈M⁰

{y} ×E_f(y) .

The setf^∗Eis a subset of the cartesian productM⁰×E. The restriction tof^∗Eof the projection onto the first coordinate is a mapπ1 :f^∗E→M⁰and the restriction tof^∗Eof the projection onto the second coordinate is a mapf¯: f^∗E → E; we have a commutative diagram:

f^∗E

f¯ //

π1

E

π

M⁰ _f ^//M

For eachy ∈ M⁰, the fiber(f^∗E)y is equal to {y} ×E_f(y); we will identify the fiber (f^∗E)y of f^∗E with the fiber E_f(y) of E. Since every fiber of f^∗E is a fiber ofE, each fiber off^∗Eis endowed with the structure of a real vector space isomorphic toE0. The setFRE0(f^∗E) can be naturally identified with the pull-backf^∗FR_E0(E); this identification makesFR_E0(f^∗E)into aGL(E₀)-principal bundle and thusf^∗Einto a vector bundle with typical fiberE₀.

1.5. VECTOR BUNDLES AND THE PRINCIPAL BUNDLE OF FRAMES 53

EXAMPLE1.5.20. Letπ :E→M be a vector bundle with typical fiberE0. If U is an open subset ofM andi:U →Mdenotes the inclusion map then the pull-backi^∗Ecan be identified with the restrictionE|_U (see Example 1.5.3); namely, by Example 1.3.17, we havei^∗FR_E0(E) = FR_E0(E)|_U = FR_E0(E|_U).

EXAMPLE1.5.21. Letπ :E → M be a vector bundle with typical fiberE0, f :M⁰ → M,g :M⁰⁰ → M⁰ be smooth maps, whereM⁰,M⁰⁰ are differentiable manifolds. Bothg^∗f^∗E and(f ◦g)^∗E are vector bundles over M⁰⁰; there exists an obvious mapL:g^∗f^∗E →(f◦g)^∗E, which is the identity on each fiber. The corresponding map:

L∗: FRE0(g^∗f^∗E)−→FRE0 (f◦g)^∗E

is the isomorphism of principal bundles considered in Example 1.3.24; thus, by Lemma 1.5.18,Lis an isomorphism of vector bundles. We use such isomorphism to identify the vector bundlesg^∗f^∗Eand(f◦g)^∗E.

The following lemma is the analogue of Lemma 1.3.16 for vector bundles.

LEMMA 1.5.22. Let π : E → M be a vector bundle with typical fiber E0, M⁰ be a differentiable manifold andf : M⁰ → M be a smooth map. Denote by π1 : f^∗E → M⁰ the pull-back ofE byf. The map(π1,f¯) : f^∗E → M⁰×E is a smooth embedding whose image is the set of pairs(y, e) ∈ M⁰ ×E such that f(y) =π(e). In particular, the mapf¯:f^∗E →Eis smooth.

Notice that the map(π₁,f¯)is just the inclusion map off^∗Einto the cartesian productM⁰×E.

PROOF. Clearly the image of(π₁,f¯) consists of the pairs(y, e) ∈ M⁰ ×E such thatf(y) = π(e). To prove that(π₁,f¯) is an embedding, we consider the commutative diagram:

FRE0(f^∗E)×_∼E0 inclusion //

C^f∗E

M⁰× FRE0(E)×_∼E0

Id×C^E

f^∗E

(π1,f)¯

//M⁰×E

The vertical arrows of the diagram are smooth diffeomorphisms and the top arrow of the diagram is a smooth embedding, by Lemma 1.4.12. Hence (π₁,f¯) is a

smooth embedding.

COROLLARY 1.5.23 (universal property of the pull-back). Under the condi-tions of Lemma 1.5.22, letX be a differentiable manifold and letφ1 :X → M⁰, φ₂ : X → E be maps withπ ◦φ₂ = f ◦φ₁. Then there exists a unique map φ:X →f^∗Esuch thatπ₁◦φ=φ₁andf¯◦φ=φ₂. The mapφis smooth.

PROOF. The hypothesis π◦φ₂ = f ◦φ₁ means that the image of the map (φ1, φ2) :X →M⁰×Eis contained in the image of the injective map(π1,f¯); thus there exists a unique mapφ :X → f^∗Esuch that(π1,f¯)◦φ= (φ1, φ2). Since (π₁,f¯)is an embedding and(φ₁, φ₂)is smooth, it follows thatφis smooth.

DEFINITION 1.5.24. By alocal section of the vector bundle E along f we mean a map : U⁰ → P defined on an open subset U⁰ of M⁰ satisfying the conditionπ◦=f|_U⁰.

EXAMPLE1.5.25. If:U → E is a local section ofE then the composition ◦f :f⁻¹(U)→E is a local section ofEalongf.

Given a local section : U⁰ → E ofE along f there exists a unique local section ¯ : U⁰ → f^∗E off^∗E such that f¯◦¯ = ; the following commutative diagram illustrates this situation:

(1.5.7)

f^∗E ^{f¯ //}E

π

U⁰

¯

OO

==z

zz zz zz z

f|_U0

//M

Thus, composition on the left withf¯induces a bijection between the set of local sections off^∗E and the set of local sections ofEalongf.

COROLLARY1.5.26. Under the conditions of Lemma 1.5.22, if:U⁰ →Eis a smooth local section ofEalongf then the unique local section¯:U⁰ →f^∗Eof f^∗Esuch thatf¯◦¯=is also smooth.

PROOF. Apply Corollary 1.5.23 withX =U⁰,φ₁the inclusion map ofU⁰ in M⁰ andφ2 = . The mapφ given by the thesis of Corollary 1.5.23 is precisely

¯

.

Corollary 1.5.26 tells us that composition on the left withf¯induces a bijection between the set of smooth local sections off^∗Eand the set of smooth local sections ofEalongf.

EXAMPLE1.5.27. LetM⁰,M be differentiable manifolds andf :M⁰ → M be a smooth map. Denote byπ : T M → M, π⁰ : T M⁰ → M the projections.

Applying the universal property of pull-backs (Corollary 1.5.23) withX =T M⁰, φ1 = π⁰, φ2 = df : T M → T M⁰ and E = T M, we obtain a smooth map

←−

df :T M⁰ → f^∗T M such thatf¯◦←−

df = df andπ₁◦←−

df = π⁰. Clearly,←− df is a morphism of vector bundles. More generally, given vector bundlesπ : E → M, π⁰ :E⁰ →M⁰and smooth mapsL:E⁰ →E,f :M⁰ →Msuch that the diagram:

E^{0 L //}

π⁰

E

π

M⁰ _f ^//M

commutes then the universal property of pull-backs gives us a smooth mapL : E⁰ →f^∗Esuch thatf¯◦L=Landπ1◦L=π⁰. If for ally∈M⁰, the restriction L|_E⁰

y :E_y⁰ →E_f(y)is linear thenLis a morphism of vector bundles.

1.5. VECTOR BUNDLES AND THE PRINCIPAL BUNDLE OF FRAMES 55

1.5.4. Vector subbundles. Letπ : E → M be a vector bundle with typical fiberE₀,F₀be a subspace ofE₀andFbe a subset ofEsuch that for everyx∈M, the setFx=F∩E_xis a subspace of the fiberExhaving the same dimension asF0. Givenx ∈M, then anE0-framep ∈FR_E0(Ex)is said to beadaptedto(F0, F) ifpis adapted to(F₀, F_x), i.e., ifp(F₀) = F_x (recall Example 1.1.10). Consider the set:

FR_E0(E;F₀, F) = [

x∈M

FR_E0(E_x;F₀, F_x)

of allE₀-frames ofEadapted to(F₀, F). For eachx∈M, the setFR_E0(E_x;F₀, F_x) is a principal subspace ofFRE0(Ex) whose structural group is the Lie subgroup GL(E₀;F₀)ofGL(E₀).

DEFINITION1.5.28. Letπ :E→M be a vector bundle with typical fiberE₀. A subsetF ⊂ E is called avector subbundleif there exists a subspaceF₀ ofE₀ such that:

(a) for eachx ∈ M, the setF_x = F ∩E_x is a subspace ofE_x having the same dimension asF0;

(b) FR_E0(E;F₀, F) is a principal subbundle of FR_E0(E) with structural groupGL(E0;F0).

Condition (b) in Definition 1.5.28 means that every point ofM belongs to the domain U of a smooth local E₀-frames : U → FR_E0(E) ofE such thats(x) mapsF₀toF_x, for allx∈U.

REMARK1.5.29. IfF₀ is a subspace ofE₀such that conditions (a) and (b) in Definition 1.5.28 are satisfied then every subspaceF₀⁰ ofE0 having the same di-mension asF₀satisfies conditions (a) and (b). Namely, letg∈GL(E₀)be a linear isomorphism ofE0 such thatg(F₀⁰) =F0. The mapγg : FRE0(E) → FRE0(E) is an isomorphism of principal bundles whose subjacent Lie group isomorphism is the inner automorphismI_g⁻¹ ofGL(E0)(Exercise 1.44); since:

γg FRE0(E;F0, F)

= FRE0(E;F₀⁰, F),

it follows that ifFR_E0(E;F₀, F)is a principal subbundle ofFR_E0(E)with struc-tural groupGL(E0;F0)thenFRE0(E;F₀⁰, F)is a principal subbundle ofFRE0(E) with structural groupI_g⁻¹ GL(E;F0)

= GL(E;F₀⁰)(Exercise 1.47). It follows that ifF is a vector subbundle ofEand ifF₀is a subspace ofE₀such that condi-tion (a) in Definicondi-tion 1.5.28 is satisfied then also condicondi-tion (b) is satisfied.

Let us now show how a vector subbundleF of a vector bundleπ :E → M can be regarded as a vector bundle in its own right. LetF0 be a subspace of the typical fiber E₀ ofE such that condition (a) in Definition 1.5.28 is satisfied (by Remark 1.5.29, also condition (b) is then satisfied). First of all, the projection π : E → M restricts to a mapπ|_F : F → M and for all x ∈ M the fiber Fx is endowed with the structure of a real vector space isomorphic to the real finite-dimensional vector spaceF₀. In order to makeF a vector bundle overM with typical fiber F₀, we have to describe a maximal atlas of local sections of

FRF0(F)→ M. Ifp ∈FRE0(E;F0, F)is anE0-frame ofEadapted to(F0, F) thenp|_F0 is anF₀-frame ofF; we have therefore a map:

(1.5.8) FR_E0(E;F₀, F)3p7−→p|_F0 ∈FR_F0(F)

that is fiber preserving and whose restriction to each fiber is a morphism of princi-pal spaces whose subjacent Lie group homomorphism is:

(1.5.9) GL(E₀;F₀)3T 7−→T|_F0 ∈GL(F₀).

Thus, by Lemma 1.3.11, there exists a unique maximal atlas of local sections of FRF0(F) such that (1.5.8) is a morphism of principal bundles. We will always consider a vector subbundle to be endowed with the structure of vector bundle described above.

PROPOSITION1.5.30. IfEis a vector bundle andF is a vector subbundle of EthenF is an embedded submanifold ofEand the differential structure ofF (as a total space of a vector bundle) coincides with the differential structure it inherits fromEas an embedded submanifold. In particular, the inclusion map ofFinEis smooth and hence a morphism of vector bundles.

PROOF. Denote by F0, E0 the typical fibers of F and E, respectively. If r denotes the morphism of principal bundles (1.5.8) then by Lemma 1.4.10 the map:

ˆr: FR_E0(E;F₀, F)×_∼F₀ −→FR_F0(F)×_∼F₀

is a smooth diffeomorphism, where we consider the smooth representation of the structural groupGL(E0;F0)ofFRE0(E;F0, F)on F0 given by (1.5.9). Ifi de-notes the inclusion map of FR_E0(E;F₀, F) in FR_E0(E) and if i₀ denotes the inclusion map ofF0inE0then the map:

i×_∼i0 : FR_E0(E;F0, F)×_∼F0 −→FR_E0(E)×_∼E0

is a smooth embedding, by Lemma 1.4.11. It is easy to see that the diagram:

FR_F0(F)×_∼F0

(i×_∼i0)◦ˆr⁻¹

//

C^{F ∼=}

FR_E0(E)×_∼E0 C^E

∼=

F inclusion

//E

commutes. Thus, the inclusion map ofFinEis a smooth embedding.

PROPOSITION1.5.31. LetE,E⁰be vector bundles over the same differentiable manifoldM and letL:E→E⁰ be a morphism of vector bundles. Then:

(a) ifLis injective then its imageL(E)is a vector subbundle ofE⁰; (b) ifL is surjective then itskernel Ker(L) = S

x∈MKer(Lx) is a vector subbundle ofE.

To prove Proposition 1.5.31, we need the following:

LEMMA1.5.32. LetE₀,E₀⁰ be real finite-dimensional vector spaces,M be a differentiable manifold,Le :M → Lin(E₀, E₀⁰)be a smooth map andx₀ ∈M be a fixed point.

1.5. VECTOR BUNDLES AND THE PRINCIPAL BUNDLE OF FRAMES 57

(a) Assume thatL(xe 0) is injective. Given a subspaceF0 ofE₀⁰ having the same dimension asE₀ then there exists a smooth mapg:U →GL(E⁰₀) defined in an open neighborhoodU ofx0 inM such that the linear iso-morphism g(x) : E₀⁰ → E₀⁰ carries F₀ to the image of L(x), for alle x∈U.

(b) Assume thatL(xe ₀)is surjective. Given a subspaceF₀ofE₀withdim(F₀) = dim(E0)−dim(E₀⁰)then there exists a smooth mapg : U → GL(E0) defined in an open neighborhoodU ofx₀ inM such that the linear iso-morphism g(x) : E0 → E0 carries F0 to the kernel of L(x), for alle x∈U.

PROOF. Let us prove (a). Choose a subspaceZofE₀⁰ such thatE₀⁰ =L(xe 0)(E0)⊕

Z. For eachx∈M, let¯g(x) :E₀⊕Z →E₀⁰ be the linear map such that¯g(x)|_E0 equalsL(x)e andg(x)|¯ _Zequals the inclusion. Theng¯:M →Lin(E₀⊕Z, E₀⁰)is a smooth map and¯g(x0)is a linear isomorphism; thus, there exists an open neighbor-hoodUofx₀inMsuch thatg(x)¯ is a linear isomorphism, for allx∈U. SinceF₀ has the same dimension asE0, there exists a linear isomorphismT :E⁰₀→E0⊕Z withT(F0) =E0⊕ {0}. Settingg(x) = ¯g(x)◦T, for allx∈U, then:

g(x)(F₀) = ¯g(x) E₀⊕ {0}

=L(x)(Ee ₀).

This concludes the proof of (a). Now let us prove (b). Choose a subspaceZ ofE₀ such thatE₀ = Ker L(xe ₀)

⊕Z; denote byp:E₀ →Ker L(xe ₀)

the projection onto the first coordinate corresponding to such direct sum decomposition. For each x ∈M, let¯g(x) :E0 → E₀⁰ ⊕Ker L(xe 0)

be the linear mapg(x) =¯ L(x),e p . Then:

¯

g:M −→Lin

E0, E₀⁰ ⊕Ker L(xe 0)

is a smooth map and g(x¯ ₀) is a linear isomorphism; thus, there exists an open neighborhoodU ofx0inM such that¯g(x)is a linear isomorphism, for allx∈U. It is easy to see that:

¯ g(x)

Ker L(x)e

={0} ⊕Ker L(xe 0) ,

for all x ∈ U. Now, since F0 has the same dimension as Ker L(xe 0) , there exists a linear isomorphism T : E0 → E₀⁰ ⊕Ker L(xe 0)

such that T(F0) = {0}⊕Ker L(xe ₀)

. Settingg(x) = ¯g(x)⁻¹◦Tfor allx∈Utheng:U →GL(E₀) is a smooth map and:

g(x)(F₀) = Ker L(x)e ,

for allx∈U. This concludes the proof.

PROOF OFPROPOSITION1.5.31. Denote byE0, E₀⁰ respectively the typical fibers ofE andE⁰. Let us prove (a). Assume thatLis injective and letF₀ be a subspace of E₀⁰ having the same dimension as E0. Given x0 ∈ M, we have to find a smooth local section ofFR_E⁰

0(E⁰) defined in an open neighborhood ofx₀ inM with image contained inFR_E⁰

0 E⁰;F0, L(E)

. Lets:V → FRE0(E),s⁰ : V →FR_E⁰

0(E⁰)be smooth local sections, defined in the same open neighborhood

No documento The theory of connections and G-structures. Applications to affine and isometric immersions Paolo Piccione Daniel V. Tausk (páginas 52-66)