Functorial constructions with vector bundles

Given an integern≥1, we denote byVecⁿthe category whose objects are n-tuples(Vi)ⁿ_i=1of real finite-dimensional vector spaces and whose morphisms from (V_i)ⁿ_i=1to(W_i)ⁿ_i=1aren-tuples(T_i)ⁿ_i=1 of linear isomorphismsT_i :V_i →W_i. We setVec = Vec¹. A functorF : Vecⁿ → Vecis calledsmooth if for any object (Vi)ⁿ_i=1ofVecⁿthe map:

(1.6.1) F: GL(V1)× · · · ×GL(Vn)−→GL F(V1, . . . , Vn)

is smooth. Observe that (1.6.1) is a Lie group homomorphism; its differential at the identity is a Lie algebra homomorphism that will be denoted by:

(1.6.2) f:gl(V1)⊕ · · · ⊕gl(Vn)−→gl F(V1, . . . , Vn) . We callfthedifferentialof the smooth functorF.

Letπ : E → M be a vector bundle with typical fiber E0. Given a smooth functorF:Vec→Vecwe set:

(1.6.3) F(E) = [

x∈M

F(E_x),

1.6. FUNCTORIAL CONSTRUCTIONS WITH VECTOR BUNDLES 59

where the union in (1.6.3) is understood to be disjoint⁵; we have an obviously defined projection map F(E) → M that sendsF(E_x)to x, for allx ∈ M. For each x ∈ M, the fiberF(Ex) ofF(E) over x has the structure of a real vector space having the same dimension asF(E0). In order to makeF(E)into a vector bundle with typical fiberF(E₀), we will describe a maximal atlas of local sections ofFR_F(E0) F(E)

→M. The map:

(1.6.4) F: FR_E0(E)3p7−→F(p)∈FR_F(E0) F(E)

is fiber preserving and its restriction to each fiber is a morphism of principal spaces whose subjacent Lie group homomorphism is:

(1.6.5) GL(E₀)3T 7−→F(T)∈GL F(E₀) .

Thus, Lemma 1.3.11 gives us a unique maximal atlas of local sections ofFR_F(E0) F(E) that makes (1.6.4) into a morphism of principal bundles. We will always consider F(E)to be endowed with the vector bundle structure described above.

Notice that ifs:U →FRE0(E)is a (smooth) localE0-frame ofEthenF◦s is a (smooth) localF(E₀)-frame ofF(E); we callF◦sthe local frameinducedby sonF(E).

REMARK1.6.1. LetF : Vec → Vec be a smooth functor andπ : E → M be a vector bundle with typical fiberE0. Since (1.6.4) is a morphism of principal bundles whose subjacent Lie group homomorphism is the representation (1.6.5), we are in the situation described in Definition 1.5.17 and thus we have the following isomorphism of vector bundles:

C^F=C^F(E)◦Fb: FR_E0(E)×_∼F(E₀)3[p,e]7−→F(p)·e∈F(E).

EXAMPLE 1.6.2. If F : Vec → Vec is the identity functor then for every vector bundleEthe vector bundleF(E)coincides withEitself. For any objectV ofVec, the mapfis the identity map ofgl(V).

EXAMPLE 1.6.3. Let Z be a fixed real finite-dimensional vector space and consider the constant functor F : Vec → Vec that sends any objectV of Vec to Z and any linear isomorphism T : V → W to the identity map of Z. For any object V ofVec, the mapf : gl(V) → gl(Z) is the identically zero map.

Given a vector bundleE over a differentiable manifoldM with typical fiber E0

thenF(E)is the trivial vector bundle M ×Z (recall Example 1.5.2); namely, if FR_Z(M ×Z) =M ×GL(Z)is endowed with the structure of a trivial GL(Z)-principal bundle (see Example 1.3.2) then the map:

F: FR_E0(E)3p7−→ Π(p),Id

∈M×GL(Z) = FR_Z(M×Z) is a morphism of principal bundles.

Now, a less trivial example.

5If the union is not disjoint, one can always replaceF(Ex)with{x} ×F(Ex), or else modify the functorFso thatF(V)is replaced with{V} ×F(V), for every real finite-dimensional vector spaceV.

EXAMPLE1.6.4. LetF:Vec →Vecbe the functor that sendsV to the dual space V^∗ and a linear isomorphism T : V → W toT^∗−1 : V^∗ → W^∗, where T^∗ :W^∗ → V^∗ denotes the transpose map ofT. The functorFis clearly smooth and for any objectV ofVec, the mapfis given by:

f:gl(V)3X 7−→ −X^∗ ∈gl(V^∗).

Given a vector bundleE then the vector bundle F(E)is denoted by E^∗ and it is called thedual bundleofE. IfE =T Mis the tangent bundle of the differentiable manifoldM then the dual bundleT M^∗ is also called thecotangent bundleofM.

EXAMPLE1.6.5. LetF:Vec→Vecbe the functor that sendsV to the space Lin(V) of linear endomorphisms ofV and a linear isomorphismT : V → W to the linear isomorphism:

I_T : Lin(V)3L7−→T◦L◦T⁻¹ ∈Lin(W).

The functorFis clearly smooth and for any objectV ofVec, the map:

f:gl(V)−→gl Lin(V) is given by:

f(X)·L= [X, L] =X◦L−L◦X,

for allX ∈ gl(V)and allL ∈ Lin(V). Given a vector bundleE then the vector bundleF(E)will be denoted byLin(E).

Given vector spaces V₁, . . . , V_k, W, we denote by Lin(V₁, . . . , V_k;W) the space ofk-linear mapsB : V1 × · · · ×Vk → W; byLink(V, W)we denote the space ofk-linear mapsB:V×· · ·×V →W. ByLin^s_k(V, W)(resp.,Lin^a_k(V, W)) we denote the subspace ofLin_k(V, W)consisting ofsymmetric(resp.,alternating) k-linear maps.

EXAMPLE1.6.6. Let k ≥ 1 be fixed and letF : Vec → Vecbe the functor that sendsV toLink(V,R) and a linear isomorphismT : V → W to the linear isomorphism:

Lin_k(V,R)3B7−→B(T⁻¹·, . . . , T⁻¹·)∈Lin_k(W,R).

The functorFis clearly smooth and for any objectV ofVec, the map:

f:gl(V)−→gl Link(V,R) is given by:

f(X)·L=−L(X·,·, . . . ,·)−L(·, X·, . . . ,·)− · · · −L(·,·, . . . , X·), for all X ∈ gl(V) and allL ∈ Link(V,R). Given a vector bundle E then the vector bundleF(E) is denoted byLin_k(E,R). IfM is a differentiable manifold then a section ofLin_k(T M,R)is called acovariantk-tensor fieldonM.

EXAMPLE 1.6.7. By replacing Link with Lin^s_k or Lin^a_k throughout Exam-ple 1.6.6 we obtain vector bundles Lin^s_k(E,R), Lin^a_k(E,R). The sections of Lin^s_k(T M,R) are calledsymmetric covariant k-tensor fieldson M and the sec-tions ofLin^a_k(T M,R)are calledk-formsordifferential forms of degreekonM.

1.6. FUNCTORIAL CONSTRUCTIONS WITH VECTOR BUNDLES 61

EXAMPLE 1.6.8. Let Z be a fixed real finite-dimensional vector space. By replacingRwithZ throughout Examples 1.6.6 and 1.6.7 we obtain vector bundles Link(E, Z), Lin^s_k(E, Z) andLin^a_k(E, Z). The sections of Link(T M, Z) (resp., Lin^s_k(T M, Z)) are calledZ-valued covariantk-tensor fields(resp.,symmetricZ -valued covariantk-tensor fields) on M; the sections ofLin^a_k(T M, Z) are called Z-valuedk-formsonM.

Let us now generalize the construction described above to the case of smooth functors of several variables. Letn ≥ 1be fixed and letF : Vecⁿ → Vecbe a smooth functor. Given vector bundlesE¹, . . . ,Eⁿ over a differentiable manifold M with typical fibersE₀¹, . . . ,E₀ⁿ, respectively, we set:

F(E¹, . . . , Eⁿ) = [

x∈M

F(E_x¹, . . . , E_xⁿ),

where the union is understood to be disjoint. We have an obviously defined pro-jectionF(E¹, . . . , Eⁿ) → M that sendsF(E_x¹, . . . , E_xⁿ) tox, for allx ∈ M; for eachxinMthe fiberF(E_x¹, . . . , E_xⁿ)has the structure of a real finite-dimensional vector space having the same dimension asF(E₀¹, . . . , E₀ⁿ). The fiberwise product FR_E1

0(E¹)?· · ·?FR_Eⁿ

0(Eⁿ)is a principal bundle overM with structural group GL(E¹₀)× · · · ×GL(Eⁿ₀); the map:

(1.6.6) FR_E1

0(E¹)?· · ·?FR_Eⁿ

0(Eⁿ)−−→^F FR_F(E1

0,...,Eⁿ₀) F(E¹, . . . , Eⁿ) (p₁, . . . , p_n)7−→F(p₁, . . . , p_n)

is fiber preserving and its restriction to each fiber is a morphism of principal spaces whose subjacent Lie group homomorphism is:

(1.6.7) GL(E₀¹)× · · · ×GL(E₀ⁿ)−→GL F(E₀¹, . . . , E₀ⁿ) (T₁, . . . , T_n)7−→F(T₁, . . . , T_n).

Lemma 1.3.11 gives us a unique maximal atlas of local sections of FR_F(E1

0,...,Eⁿ₀) F(E¹, . . . , Eⁿ)

−→M

that makes (1.6.6) into a morphism of principal bundles. We will always consider F(E¹, . . . , Eⁿ)to be endowed with the vector bundle structure described above.

If sⁱ : U → FR_Ei

0(Eⁱ) is a (smooth) localE₀ⁱ-frame of Eⁱ, i = 1, . . . , n, thenF◦(s¹, . . . , sⁿ)is a (smooth) localF(E₀¹, . . . , E₀ⁿ)-frame of the vector bun-dleF(E¹, . . . , Eⁿ); we callF◦(s¹, . . . , sⁿ)the frame inducedbys¹, . . . ,sⁿ on F(E¹, . . . , Eⁿ).

REMARK1.6.9. LetE¹, . . . ,Eⁿbe vector bundles over a differentiable mani-foldM with typical fibersE₀¹, . . . ,E₀ⁿrespectively, and letF:Vecⁿ →Vecbe a smooth functor. Since (1.6.6) is a morphism of principal bundles whose subjacent Lie group homomorphism is the representation (1.6.7), we are in the situation de-scribed in Definition 1.5.17 and thus we have the following isomorphism of vector

bundles:

(1.6.8)

FR_E¹

0(E¹)?· · ·?FR_Eⁿ

0(Eⁿ)

×_∼F(E¹₀, . . . , E₀ⁿ)

C^F

F(E¹, . . . , Eⁿ) which is given by:

C^F=C^F(E1,...,En)◦bF: [(p1, . . . , pn),e]7−→F(p1, . . . , pn)·e.

EXAMPLE 1.6.10. Let M be a differentiable manifold, E₀¹, . . . , E₀ⁿ be real finite-dimensional vector spaces and consider the trivial vector bundles:

Eⁱ=M×E₀ⁱ, i= 1, . . . , n.

If F : Vecⁿ → Vec is a smooth functor thenF(E¹, . . . , Eⁿ) can be identified as a set with the trivial vector bundle M ×F(E₀¹, . . . , E₀ⁿ). Let us show that F(E¹, . . . , Eⁿ) is a trivial vector bundle, i.e., such identification is a vector bun-dle isomorphism. To this aim, we look at the corresponding principal bunbun-dles of frames. The principal bundle ofF(E₀¹, . . . , E₀ⁿ)-frames ofF(E¹, . . . , Eⁿ)can be identified as a set with the trivial principal bundle:

M×GL F(E₀¹, . . . , E₀ⁿ) .

We have to check that such identification is an isomorphism of principal bundles.

This follows from the following two observations; first (see Exercise 1.56), the fiberwise product:

FR_E¹

0(E¹)?· · ·?FRE₀ⁿ(Eⁿ) = M×GL(E₀¹)

?· · ·? M ×GL(E₀ⁿ) is identified as a principal bundle with the trivial principal bundle:

M × GL(E₀¹)× · · · ×GL(E₀ⁿ) .

Second, the map (1.6.6) can be identified with the product of the identity map of M by the map (1.6.7) so that (1.6.6) is smooth when

FR_F(E¹

0,...,E₀ⁿ) F(E¹, . . . , Eⁿ)

is identified with the trivial principal bundleM×GL F(E₀¹, . . . , E₀ⁿ) .

EXAMPLE 1.6.11. LetF : Vec² → Vecbe the functor that sends an object (V1, V2)toV1⊕V2and a morphism(T1, T2)toT1⊕T2. The functorFis smooth and for any object(V1, V2)ofVec², the map:

f:gl(V₁)⊕gl(V₂)−→gl(V₁⊕V₂) is given by:

f(X₁, X₂) =X₁⊕X₂,

for allX₁ ∈gl(V₁),X₂ ∈ gl(V₂). Given vector bundlesE¹,E² over a differen-tiable manifoldM then the vector bundleF(E¹, E²)will be denoted byE¹⊕E² is will be called theWhitney sumofE¹andE².

1.6. FUNCTORIAL CONSTRUCTIONS WITH VECTOR BUNDLES 63

EXAMPLE1.6.12. LetF :Vec² → Vecbe the functor that sends(V1, V2)to Lin(V₁, V₂)and(T₁, T₂)to:

Lin(V1, V2)3L7−→T2◦L◦T₁⁻¹∈Lin(W1, W2),

whereT1 :V1 → W1 andT2 :V2 → W2 are linear isomorphisms. The functorF is smooth and for any object(V1, V2)ofVec², the map:

f:gl(V1)⊕gl(V2)−→gl Lin(V1, V2) is given by:

f(X1, X2)·L=X2◦L−L◦X1,

for allX1 ∈gl(V1),X2 ∈gl(V2)and allL ∈Lin(V1, V2). Given vector bundles E¹,E²overM, the vector bundleF(E¹, E²)will be denoted byLin(E¹, E²). A fiberwise linear mapL:E¹ →E² can be identified with a sectionx7→Lxof the vector bundleLin(E¹, E²). Ifsⁱ :U →FR_Ei

0(Eⁱ)is a smooth localE₀ⁱ-frame of Eⁱ,i= 1,2, and ifsdenotes the frame ofLin(E¹, E²)induced bys¹ands² then the representation of the fiberwise linear mapLwith respect tos¹ ands²is equal to the representation of the sectionx7→Lxwith respect tos. It follows thatLis a vector bundle morphism if and only ifx7→Lxis a smooth section ofLin(E¹, E²).

From now on we will systematically identify vector bundle morphisms fromE¹to E²with smooth sections of Lin(E¹, E²).

EXAMPLE 1.6.13. Let k ≥ 1 be fixed and let F : Vec^k+1 → Vec be the functor that sends (V₁, . . . , V_k, W) to Lin(V₁, . . . , V_k;W) and that sends linear isomorphismsTi : Vi → V_i⁰, i = 1, . . . , k, T : W → W⁰ to the linear isomor-phism:

Lin(V₁, . . . , V_k;W)−→Lin(V₁⁰, . . . , V_k⁰;W⁰) B7−→T ◦B(T₁⁻¹·, . . . , T_k⁻¹·).

The functorFis smooth and for any object(V₁, . . . , V_k, W)ofVec^k+1the map:

f:gl(V₁)⊕ · · · ⊕gl(V_k)⊕gl(W)−→gl Lin(V₁, . . . , V_k;W) is given by:

f(X1, . . . , X_k, X)·L=X◦L(·, . . . ,·)−L(X1·, . . . ,·)− · · ·

−L(·, . . . , X_k·), for allXi ∈ gl(Vi), i= 1, . . . , k,X ∈ gl(W), L ∈ Lin(V1, . . . , V_k;W). Given vector bundlesE¹, . . . ,E^k,FoverM, we will denote the vector bundleF(E¹, . . . , E^k, F) byLin(E¹, . . . , E^k;F). WhenE¹ =· · ·=E^k =E, we writeLink(E, F)rather thanLin(E¹, . . . , E^k;F). Sections of the vector bundleLink(T M, F)are called F-valued covariantk-tensor fieldsonM.

EXAMPLE1.6.14. Letk≥1be fixed and letF:Vec² →Vecbe the functor that sends(V₁, V₂)toLin^s_k(V₁, V₂)and(T₁, T₂)to:

Lin^s_k(V₁, V₂)3B 7−→T₂◦B(T₁⁻¹·, . . . , T₁⁻¹·)∈Lin^s_k(W₁, W₂),

whereT1 : V1 → W1,T2 :V2 → W2 are linear isomorphisms. The functorFis smooth and for any object(V1, V2)ofVec², the map:

f:gl(V₁)⊕gl(V₂)−→gl Lin^s_k(V₁, V₂) is given by:

f(X1, X2)·L=X2◦L(·, . . . ,·)−L(X1·, . . . ,·)− · · · −L(·, . . . , X₁·), for allX₁ ∈ gl(V₁), X₂ ∈ gl(V₂) and allL ∈ Lin^s_k(V₁, V₂). Given vector bun-dles E, F over M, the vector bundle F(E, F) will be denoted by Lin^s_k(E, F).

An analogous construction replacing Lin^s_k with Lin^a_k gives us the vector bundle Lin^a_k(E, F). The sections ofLin^s_k(T M, F)are calledsymmetricF-valued covari-ant k-tensor fieldson M and the sections of Lin^a_k(T M, F) are calledF-valued k-formsonM.

CONVENTION. From now on, when describing smooth functors we will only specify their actions on objects and leave as an exercise to the reader the task of describing their actions on morphisms.

PROPOSITION1.6.15. Letm, n≥1be fixed and let:

F= (F¹, . . . ,Fⁿ) :Vec^m −→Vecⁿ, G:Vecⁿ−→Vec

be smooth functors⁶; consider the smooth functor G◦F : Vec^m → Vec. Given vector bundlesE¹, . . . ,E^mover a differentiable manifoldM then:

(1.6.9) (G◦F)(E¹, . . . , E^m) =G F¹(E¹, . . . , E^m), . . . ,Fⁿ(E¹, . . . , E^m) . PROOF. Clearly both sides of (1.6.9) are equal as sets; we have to check that the principal bundle structure of their corresponding principal bundles of frames are also the same. Denote byE₀ⁱthe typical fiber of the vector bundleEⁱ,i= 1, . . . , m, by F^j = F^j(E₀¹, . . . , E₀^m) the typical fiber of F^j(E¹, . . . , E^m), j = 1, . . . , n and byG =G(F¹, . . . , Fⁿ)the typical fiber of(G◦F)(E¹, . . . , E^m). For each j = 1, . . . , n, letFR_Fj F^j(E¹, . . . , E^m)

be endowed with the unique principal fiber bundle structure that makes the map:

(1.6.10) F^j : FR_E1

0(E¹)?· · ·?FR_E^m

0 (E^m)−→FR_Fj F^j(E¹, . . . , E^m) a morphism of principal bundles and letFRG (G◦F)(E¹, . . . , E^m)

be endowed with the unique principal bundle structure that makes the map:

(1.6.11)

FR_F¹ F¹(E¹, . . . , E^m)

?· · ·?FR_Fⁿ Fⁿ(E¹, . . . , E^m)

G

FRG (G◦F)(E¹, . . . , E^m)

a morphism of principal bundles. To conclude the proof, we have to verify that the map:

(1.6.12) G◦F: FR_E1

0(E¹)?· · ·?FR_E^m

0 (E^m)−→FR_G (G◦F)(E¹, . . . , E^m)

6The smoothness ofFmeans that everyFⁱis smooth.

1.6. FUNCTORIAL CONSTRUCTIONS WITH VECTOR BUNDLES 65

is a morphism of principal bundles. This follows from the universal property of the fiberwise product of principal bundles (Corollary 1.3.27) and from the fact that the composition of morphisms of principal bundles is a morphism of principal bundles

(see Exercise 1.43).

PROPOSITION1.6.16. Letn≥1be fixed and letF:Vecⁿ→Vecbe a smooth functor. LetE¹,E¹, . . . ,Eⁿ,Eⁿbe vector bundles over a differentiable manifold M andLⁱ:Eⁱ →Eⁱ,i= 1, . . . , n, be vector bundle isomorphisms. The map:

F(L¹, . . . , Lⁿ) :F(E¹, . . . , Eⁿ)−→F(E¹, . . . , Eⁿ)

whose restriction to the fiber F(E_x¹, . . . , E_xⁿ) is equal to F(L¹_x, . . . , Lⁿ_x), for all x∈M is a vector bundle isomorphism.

PROOF. ClearlyF(L¹, . . . , Lⁿ)is fiber preserving, fiberwise linear and bijec-tive. Fori = 1, . . . , n, denote byEⁱ₀(resp., byE₀ⁱ) the typical fiber ofEⁱ (resp., ofEⁱ). Lets¹,s¯¹, . . . ,sⁿ, ¯sⁿ, be smooth local sections respectively of the prin-cipal bundles of framesFR_E¹

0(E¹),FR_E¹

0(E¹), . . . ,FR_Eⁿ₀(Eⁿ),FR_Eⁿ

0(Eⁿ), all defined in the same open subsetU ofM. Set:

s=F◦(s¹, . . . , sⁿ), ¯s=F◦(¯s¹, . . . ,s¯ⁿ), so that sis a smooth local section of FR_F(E¹

0,...,Eⁿ₀) F(E¹, . . . , Eⁿ)

and ¯sis a smooth local section ofFR_F(E¹

0,...,E₀ⁿ) F(E¹, . . . , Eⁿ)

. Fori= 1, . . . , n, let:

Leⁱ :U −→Lin(E₀ⁱ, E₀ⁱ)

denote the representation of Lⁱ with respect to sⁱ ands¯ⁱ (see Subsection 1.5.2);

since each Lⁱ is a morphism of vector bundles, the maps Leⁱ are smooth. Since eachLⁱ is a vector bundle isomorphism, the map Leⁱ actually takes values in the setIso(E₀ⁱ, E₀ⁱ)of linear isomorphisms from E₀ⁱ toE₀ⁱ. It is easy to see that the representation ofLwith respect tosand¯sis equal to the composition of the map (Le¹, . . . ,Leⁿ)with the map:

Iso(E₀¹, E₀¹)× · · · ×Iso(E₀ⁿ, E₀ⁿ)

F

Iso F(E₀¹, . . . , E₀ⁿ),F(E₀¹, . . . , E₀ⁿ)

Such map is smooth (see Exercise 1.66) and hence the representation ofL with respect tosand¯sis smooth. This concludes the proof.

EXAMPLE1.6.17. Letn≥ 1be fixed and letF :Vecⁿ → Vecbe a smooth functor. LetE¹, . . . ,Eⁿbe vector bundles over the differentiable manifoldMwith typical fibersE₀¹, . . . ,E₀ⁿ, respectively. Lets¹, . . . ,sⁿbe smooth local sections of the principal bundlesFR_E1

0(E¹), . . . ,FR_Eⁿ

0(Eⁿ)respectively, defined in an open

subsetU ofM. IfLⁱ = ˇsⁱ :U×E₀ⁱ →Eⁱ|_Udenotes the smooth local trivialization corresponding tosⁱthenLⁱis a vector bundle isomorphism and:

F(L¹, . . . , Lⁿ) = ˇs, wheres=F◦(s¹, . . . , sⁿ) :U →FR_F(E¹

0,...,E₀ⁿ) F(E¹, . . . , Eⁿ) .

LetF :Vecⁿ → Vecbe a smooth functor. Given vector bundlesE¹, . . . ,Eⁿ over a differentiable manifoldM and a smooth map f : M⁰ → M defined in a differentiable manifoldM⁰then there exists an obvious bijective map:

(1.6.13) f^∗F(E¹, . . . , Eⁿ)−→F(f^∗E¹, . . . , f^∗Eⁿ).

We have the following:

PROPOSITION1.6.18. Letn≥1be fixed and letF:Vecⁿ→Vecbe a smooth functor. Given vector bundlesE¹, . . . ,Eⁿover a differentiable manifoldM and a smooth map f :M⁰ → M defined in a differentiable manifoldM⁰ then the map (1.6.13)is an isomorphism of vector bundles.

PROOF. The map (1.6.13) induces a map:

(1.6.14)

FR_F(E¹

0,...,E₀ⁿ) f^∗F(E¹, . . . , Eⁿ)

FR_F(E1

0,...,E₀ⁿ) F(f^∗E¹, . . . , f^∗Eⁿ)

as in the statement of Lemma 1.5.18; the map (1.6.14) is fiber preserving and its restriction to each fiber is a morphism of principal spaces whose subjacent group homomorphism is the identity. We have to show that (1.6.14) is an isomorphism of principal bundles; in fact, by the result of Exercise 1.46, it suffices to show that (1.6.14) is a morphism of principal bundles. Recall from Subsection 1.5.3 that:

FR_F(E¹

0,...,E₀ⁿ) f^∗F(E¹, . . . , Eⁿ)

=f^∗FR_F(E¹

0,...,E₀ⁿ) F(E¹, . . . , Eⁿ) . By considering the pull-back by f of the morphism of principal bundles (1.6.6) (recall Example 1.3.23) we obtain a morphism of principal bundles:

f^∗ FR_E¹

0(E¹)?· · ·?FRE₀ⁿ(Eⁿ)

f^∗F

f^∗FR_F(E¹

0,...,Eⁿ₀) F(E¹, . . . , Eⁿ)

Using the isomorphism of principal bundles described in Lemma 1.3.29 we identify the principal bundles:

(1.6.15) f^∗ FR_E1

0(E¹)?· · ·?FR_Eⁿ

0(Eⁿ)

1.6. FUNCTORIAL CONSTRUCTIONS WITH VECTOR BUNDLES 67

and:

(1.6.16) f^∗FR_E1 0(E¹)

?· · ·? f^∗FR_Eⁿ

0(Eⁿ)

= FR_E1

0(f^∗E¹)?· · ·?FR_Eⁿ

0(f^∗Eⁿ).

We have a commutative diagram:

(1.6.17) FR_E¹

0(f^∗E¹)?· · ·?FREⁿ₀(f^∗Eⁿ)

f^∗F

F

''O

OO OO OO OO OO O

FR_F(E¹

0,...,E₀ⁿ) F(f^∗E¹, . . . , f^∗Eⁿ) FR_F(E1

0,...,E₀ⁿ) f^∗F(E¹, . . . , Eⁿ)

(1.6.14)pppppppppp77 pp

To conclude that (1.6.14) is a morphism of principal bundles, simply apply

Corol-lary 1.3.12 to such commutative diagram.

1.6.1. Smooth natural transformations.

DEFINITION1.6.19. Letn≥1be fixed and letF,Gbe smooth functors from VecⁿtoVec. By asmooth natural transformationfromFtoGwe mean a ruleN that associates to each object(V₁, . . . , V_n)ofVecⁿan open subsetDom(N_V1,...,Vn) ofF(V1, . . . , Vn)and a smooth map:

N_V1,...,Vn : Dom(N_V1,...,Vn)−→G(V₁, . . . , V_n)

in such a way that given objects(V₁, . . . , V_n),(W₁, . . . , W_n)ofVecⁿand a mor-phism(T1, . . . , Tn)from(V1, . . . , Vn)to(W1, . . . , Wn)then:

(a) F(T1, . . . , Tn) Dom(N_V1,...,Vn)

= Dom(N_W1,...,Wn);

(b) the following diagram is commutative:

Dom(N_V1,...,Vn) ^{NV1,...,Vn //}

F(T1,...,Tn)

G(V1, . . . , Vn)

G(T1,...,Tn)

Dom(N_W1,...,Wn)

N_W1,...,Wn //G(W₁, . . . , W_n)

A smooth natural transformationNfromF toGis said to be linear if for every object(V1, . . . , Vn)ofVecⁿ, we have:

Dom(N_V1,...,Vn) =F(V₁, . . . , V_n)

and the mapN_V1,...,Vn :F(V₁, . . . , V_n)→G(V₁, . . . , V_n)is linear.

EXAMPLE1.6.20. Consider the smooth functorsF,Gⁱ,i= 1,2, fromVec²to Vecdefined by:

F(V₁, V₂) =V₁⊕V₂, Gⁱ(V₁, V₂) =V_i, i= 1,2.

The rule that assigns to each object(V1, V2)ofVec²the map:

Nⁱ_V

1,V2 :V1⊕V2 3(v1, v2)7−→vi∈Vi, is a linear smooth natural transformation fromFtoGⁱ,i= 1,2.

EXAMPLE1.6.21. IfF,Gⁱare as in Example 1.6.20 then the rules that assign to each object(V₁, V₂)ofVec²the maps:

N¹_V

1,V2 :V₁ 3v 7−→(v,0)∈V₁⊕V₂, N²_V1,V2 :V2 3v 7−→(0, v)∈V1⊕V2,

are linear smooth natural transformations fromG¹ toFand fromG²toF, respec-tively.

EXAMPLE1.6.22. Consider the smooth functorsF,GfromVec² toVec de-fined by:

F(V₁, V₂) = Lin(V₁, V₂), G(V₁, V₂) = Lin(V₂^∗, V₁^∗).

The rule that assigns to each object(V₁, V₂)ofVec²the map:

N_V1,V2 : Lin(V1, V2)3T 7−→T^∗∈Lin(V₂^∗, V₁^∗) is a linear smooth natural transformation fromFtoG.

EXAMPLE1.6.23. Consider the smooth functorsF,GfromVec³ toVec de-fined by:

F(V1, V2, V3) = Lin(V2, V3)⊕Lin(V1, V2), G(V1, V2, V3) = Lin(V1, V3).

The rule that assigns to each object(V₁, V₂, V₃)ofVec³the map:

N_V1,V2,V3 : Lin(V₂, V₃)⊕Lin(V₁, V₂)3(T, T⁰)7−→T ◦T⁰ ∈Lin(V₁, V₃) is a smooth natural transformation fromFtoG.

EXAMPLE1.6.24. Letk≥1be fixed and consider the smooth functorsF,G fromVec^k+1toVecdefined by:

F(V1, . . . , V_k+1) = Lin(V1, . . . , V_k;V_k+1)⊕V1⊕ · · · ⊕V_k, G(V₁, . . . , V_k+1) =V_k+1.

The rule that assigns to each object(V1, . . . , V_k+1)ofVec^k+1the mapN_V1,...,Vk+1 defined by:

Lin(V1, . . . , Vk;Vk+1)⊕V1⊕ · · · ⊕Vk −→Vk+1

(B, v1, . . . , v_k)7−→B(v1, . . . , v_k) is a smooth natural transformation fromFtoG.

1.6. FUNCTORIAL CONSTRUCTIONS WITH VECTOR BUNDLES 69

EXAMPLE1.6.25. Letk≥1be fixed and consider the smooth functorsFfrom Vec^k+2 toVecdefined by:

F(V1, . . . , Vk+2) = Lin(Vk+1, Vk+2)⊕Lin(V1, . . . , Vk;Vk+1), G(V1, . . . , V_k+2) = Lin(V1, . . . , V_k;V_k+2).

The rule that assigns to each object(V1, . . . , Vk+2)ofVec^k+2the mapN_V1,...,Vk+2 defined by:

Lin(V_k+1, V_k+2)⊕Lin(V₁, . . . , V_k;V_k+1)−→Lin(V₁, . . . , V_k;V_k+2) (L, B)7−→L◦B

is a smooth natural transformation fromFtoG.

EXAMPLE1.6.26. Consider the smooth functorsF,GfromVec² toVec de-fined by:

F(V₁, V₂) = Lin(V₁, V₂), G(V₁, V₂) = Lin(V₂, V₁).

Given real vector spacesV₁, V₂, we denote by Iso(V₁, V₂) the (possibly empty) subset ofLin(V₁, V₂) consisting of linear isomorphisms. The rule that assigns to each object(V₁, V₂)ofVec²the map:

N_V1,V2 : Iso(V₁, V₂)3T 7−→T⁻¹∈Lin(V₂, V₁) is a smooth natural transformation fromFtoG.

EXAMPLE1.6.27. Consider the smooth functorsF,GfromVectoVecdefined by:

F(V) = Lin(V), G(V) =R. The rule that assigns to each objectV ofVecthe map:

N_V : Lin(V)3T 7−→det(T)∈R is a smooth natural transformation fromFtoG.

Given smooth functorsF,GfromVecⁿtoVec, a smooth natural transforma-tionNfromFtoGand vector bundlesE¹, . . . ,Eⁿover a differentiable manifold M thenNinduces a map:

(1.6.18) N_E1,...,Eⁿ : Dom(N_E¹_,...,Eⁿ)−→G(E¹, . . . , Eⁿ), where:

Dom(N_E¹_,...,Eⁿ) = [

x∈M

Dom(N_E¹

x,...,Eⁿ_x)⊂F(E¹, . . . , Eⁿ).

The mapN_E1,...,Eⁿ is defined by the requirement that for eachx ∈M, its restric-tion toDom(N_Ex¹_,...,Exⁿ)is equal toN_E1

x,...,Eⁿ_x.

PROPOSITION1.6.28. Letn≥ 1be fixed. Given smooth functorsF,Gfrom Vecⁿ to Vec, a smooth natural transformation N from F toG and vector bun-dles E¹, . . . , Eⁿ over a differentiable manifold M then Dom(N_E¹_,...,Eⁿ) is an open subset of the total space of the vector bundleF(E¹, . . . , Eⁿ) and the map N_E1,...,Eⁿ is smooth. In particular, ifNis linear thenN_E1,...,Eⁿis a vector bundle morphism.

PROOF. Denote byE₀ⁱ the typical fiber ofEⁱ,i= 1, . . . , n. The naturality of Nimplies that the open subsetDom(N_E¹

0,...,E₀ⁿ)of the vector spaceF(E₀¹, . . . , E₀ⁿ) is invariant under the representation (1.6.7) so that, by Lemma 1.4.11, the fiber product:

(1.6.19) FR_E¹

0(E¹)?· · ·?FRE₀ⁿ(Eⁿ)

×_∼Dom(N_E¹

0,...,E₀ⁿ) is an open submanifold of:

FR_E¹

0(E¹)?· · ·?FRE₀ⁿ(Eⁿ)

×_∼F(E₀¹, . . . , E₀ⁿ).

It follows easily from the naturality ofNthat the vector bundle isomorphism (1.6.8) carries (1.6.19) toDom(N_E¹_,...,Eⁿ), soDom(N_E¹_,...,Eⁿ)is indeed an open subset ofF(E¹, . . . , Eⁿ). The naturality ofNalso implies that the diagram:

(1.6.20)

P ×_∼Dom(N_E¹

0,...,E₀ⁿ)

C^{F ∼=}

Id_P×_∼N_E1 0,...,En

0 //P ×_∼G(E₀¹, . . . , E₀ⁿ)

∼= C^G

Dom(N_E1,...,Eⁿ)

N_E1,...,En

//G(E¹, . . . , Eⁿ) commutes, where P = FR_E1

0(E¹) ? · · ·? FR_Eⁿ

0(Eⁿ). The fact that the map N_E1,...,Eⁿ is smooth now follows from the fact that the map Id_P ×_∼ N_E1

0,...,Eⁿ₀ is

smooth (Lemma 1.4.11).

EXAMPLE1.6.29. LetE¹,E²be vector bundles over a differentiable manifold M and consider the Whitney sum E¹⊕E². Applying Proposition 1.6.28 to the linear smooth natural transformations described in Examples 1.6.20 and 1.6.21 we conclude that the projectionspr_i : E¹⊕E² → Eⁱ and the inclusionsι_i : Eⁱ → E¹ ⊕ E², i = 1,2, are vector bundle morphisms. This implies the following property: ifis a section ofE¹⊕E²andⁱ = pr_i◦,i= 1,2, are thecoordinates ofthenis smooth if and only if¹and²are smooth. Namely, ifis smooth then obviously¹and²are smooth, because the projections are smooth; conversely, if ¹ and² are smooth then= ι₁◦¹ +ι₂ ◦². See Exercises 1.70 and 1.71 for more basic results concerning Whitney sums.

REMARK1.6.30. Letπ :E → M be a vector bundle andE¹,E² be vector subbundles ofEsuch thatE_x =E_x¹⊕E_x², for allx∈M; denote byj_i :Eⁱ →E, i= 1,2, the inclusion maps. Consider the Whitney sumE¹⊕E² and denote by ι_i :Eⁱ → E¹⊕E²,i= 1,2, the inclusion maps. By the result of Exercise 1.70, there exists a unique vector bundle morphismj:E¹⊕E²→Esuch thatj◦ιi =j_i, i= 1,2. Clearlyjis a vector bundle isomorphism. We will use the isomorphismj to identify the vector bundleEwith the Whitney sumE¹⊕E². Thus, ifE¹,E²are vector subbundles of a vector bundleEsuch thatE_x =E_x¹⊕E_x², for allx ∈M, we will writeE =E¹⊕E².

EXAMPLE1.6.31. LetE¹, . . . ,E^k,F be vector bundles over a differentiable manifold M, B be a smooth section of Lin(E¹, . . . , E^k;F) and ⁱ be a smooth section ofEⁱ, i = 1, . . . , k. Applying Proposition 1.6.28 to the smooth natural

1.6. FUNCTORIAL CONSTRUCTIONS WITH VECTOR BUNDLES 71

transformation of Example 1.6.24 we obtain that the section B(¹, . . . , ^k) ofF defined by:

B(¹, . . . , ^k)(x) =Bx ¹(x), . . . , ^k(x)

, x∈M, is smooth. Namely, the map:

N_E1,...,E^k+1 : Lin(E¹, . . . , E^k;E^k+1)⊕E¹⊕ · · · ⊕E^k −→E^k+1 is smooth and:

B(¹, . . . , ^k) =N_E1,...,E^k,F ◦(B, ¹, . . . , ^k).

Recall also that(B, ¹, . . . , ^k) is smooth (Example 1.6.29). Thus, every smooth sectionB ofLin(E¹, . . . , E^k;F)defines aC^∞(M)-multilinear map:

Γ(E¹)× · · · ×Γ(E^k)3(¹, . . . , ^k)7−→B(¹, . . . , ^k)∈Γ(F).

The result of Exercises 1.63 and 1.72 says that, conversely, every C^∞ (M)-mul-tilinear map from Γ(E¹)× · · · ×Γ(E^k) toΓ(F)is defined by a unique smooth sectionB ofLin(E¹, . . . , E^k;F). In view of this correspondence we will be al-lowed to identify smooth sections ofLin(E¹, . . . , E^k;F) with the corresponding C^∞(M)-multilinear maps.

EXAMPLE1.6.32. LetE¹, . . . ,E^k,F,F⁰ be vector bundles over a differen-tiable manifold M, B be a section of Lin(E¹, . . . , E^k;F) andL : F → F⁰ be a vector bundle morphism. Recall from Example 1.6.12 that we identifyL with the smooth sectionx 7→ L_x ofLin(F, F⁰). We will denote (with some abuse of notation) byL◦B the section ofLin(E¹, . . . , E^k;F⁰)defined by:

(L◦B)(x) =L_x◦B(x),

for allx∈ M. We claim that ifB is smooth then alsoL◦B is smooth. Namely, by Example 1.6.29,(L, B)is a smooth section of the Whitney sum:

Lin(F, F⁰)⊕Lin(E¹, . . . , E^k;F).

IfNis the smooth natural transformation defined in Example 1.6.25 then:

L◦B =N_E1,...,E^k,F,F⁰◦(L, B), and thereforeL◦B is smooth by Proposition 1.6.28.

EXAMPLE 1.6.33. Given real finite-dimensional vector spaces V₁, . . . , V_k, V_k+1, . . . ,V_k+p,W then we have a linear isomorphism:

Lin V₁, . . . , V_k; Lin(V_k+1, . . . , V_k+p;W)

−→Lin(V₁, . . . , V_k+p;W) B7−→Be

(1.6.21) defined by:

Be(v1, . . . , v_k, v_k+1, . . . , v_k+p) =B(v1, . . . , v_k)·(v_k+1, . . . , v_k+p)∈W, for allv₁ ∈V₁, . . . ,v_k+p ∈V_k+p. The linear isomorphism (1.6.21) defines a linear smooth natural transformation between smooth functors and therefore, given vector

bundlesE¹, . . . ,E^k+p,F over a differentiable manifoldM, as an application of Proposition 1.6.28 we get an isomorphism of vector bundles:

Lin E¹, . . . , E^k; Lin(E^k+1, . . . , E^k+p;F)

−→Lin(E¹, . . . , E^k+p;F).

We will henceforth identify the vector bundles:

Lin E¹, . . . , E^k; Lin(E^k+1, . . . , E^k+p;F)

, Lin(E¹, . . . , E^k+p;F) using such isomorphism.

No documento The theory of connections and G-structures. Applications to affine and isometric immersions Paolo Piccione Daniel V. Tausk (páginas 66-80)