Differential forms in a principal bundle

f^∗∇is identified with a smooth section ofLin2(T M, f^⊥). Since theι-torsion of f^∗∇is zero, it follows from Example 2.8.2 that α is actually a smooth section ofLin^s₂(T M, f^⊥), i.e., for everyx ∈ M, α_x : T_xM ×T_xM → f_x^⊥ is a sym-metric bilinear form. We callαthesecond fundamental formand∇^⊥thenormal connectionof the isometric immersionf.

2.9. DIFFERENTIAL FORMS IN A PRINCIPAL BUNDLE 145

DEFINITION 2.9.4. A (possibly vector-valued) differentialk-form λon P is said to behorizontalif:

λ_p(ζ₁, . . . , ζ_k) = 0,

for allp ∈ P,ζ1, . . . ,ζk ∈ TpP, provided that at least one of the vectorsζi is in Ver_p(P).

EXAMPLE 2.9.5. The covariant exterior differential of a smooth differential formλ on P is always horizontal, even ifλis not horizontal. In particular, the curvature form of a connection is always horizontal.

Given aG-principal bundleΠ : P → M then, since we are given a smooth right action ofGonP, one can define for everyAin the Lie algebragthe smooth vector fieldA^P ∈ Γ(T P)on P induced byA (recall Definition A.2.3). Clearly A^P_p ∈Verp(P)and:

(2.9.1) ωp(A^P_p) =A,

for allp∈P.

LEMMA 2.9.6. LetΠ : P → M be a principal bundle endowed with a con-nectionHor(P). Given a vector fieldXonM then for everyg∈Gthe horizontal liftX^horisγg-related to itself, i.e.:

X^hor(p·g) =X^hor(p)·g, for allp∈P.

PROOF. SinceHor(P)isG-invariantX^hor(p)·gis inHorp·g(P); moreover, the result of Exercise 1.42 implies that:

dΠp·g X^hor(p)·g

= dΠp X^hor(p)

=X Π(p) .

This proves thatX^hor(p·g) =X^hor(p)·g.

COROLLARY2.9.7. LetΠ :P →Mbe aG-principal bundle endowed with a connectionHor(P). Given a vector fieldXonMandA∈gthen:

[A^P, X^hor] = 0.

PROOF. The flow ofA^P at time t is equal to γ_exp(tA), for all t ∈ R. The

conclusion follows.

PROPOSITION2.9.8. The curvature formΩis given by:

(2.9.2) Ω = dω+¹₂ω∧ω,

where the wedge product is considered with respect to the Lie bracket ofg. More explicitly,(2.9.2)means that:

(2.9.3) Ωp(ζ1, ζ2) = dωp(ζ1, ζ2) + [ωp(ζ1), ωp(ζ2)], for allp∈P and allζ₁, ζ₂ ∈T_pP.

PROOF. Since both sides of equality (2.9.3) are bilinear and antisymmetric in (ζ₁, ζ₂), it suffices to verify the equality in the cases:

(a) ζ1, ζ2 ∈Horp(P);

(b) ζ₁∈Hor_p(P),ζ₂∈Ver_p(P);

(c) ζ1, ζ2 ∈Verp(P).

Equality (2.9.3) is obvious in case (a). To prove the equality in case (b), let X be an arbitrary smooth vector field onM such thatX Π(p)

= dΠ_p(ζ₁)and set A =ω(ζ2) ∈g; clearlyX^hor(p) = ζ1 andA^P(p) = ζ2. Using Cartan’s formula for exterior differentiation we compute:

dω(X^hor, A^P) =X^hor ω(A^P)

−A^P ω(X^hor)

−ω [X^hor, A^P] . Sinceω(A^P)is a constant map (see (2.9.1)) andω(X^hor)≡0, then:

dω(X^hor, A^P) =−ω [X^hor, A^P] .

Moreover, by Corollary 2.9.7,[X^hor, A^P] = 0and thusdωp(ζ1, ζ2) = 0. Clearly all the other terms in equality (2.9.3) are also equal to zero, proving the equality in case (b). To prove the equality in case (c), setAi=ω(ζi)∈g, so thatA^P_i (p) =ζi, i= 1,2. Using again Cartan’s formula for exterior differentiation, we obtain:

dωp(ζ1, ζ2) =−ω_p [A^P₁, A^P₂]p

. By the result of Exercise A.4,[A^P₁, A^P₂] = [A1, A2]^P, so that:

dωp(ζ1, ζ2) =−[A₁, A2] =−[ω(ζ₁), ω(ζ2)],

proving equality (2.9.3) in case (c).

LetE₀ be a real finite-dimensional vector space and letρ :G → GL(E₀) be a smooth representation of Gon E0. Consider the associated bundle P ×_G E0. Let`be ak-form onM with values on the vector bundleP×_GE₀. We define an E0-valuedk-formλonP by setting:

(2.9.4) λp(ζ1, . . . , ζ_k) = ˆp⁻¹·`x dΠp(ζ1), . . . ,dΠp(ζ_k)

∈E0,

for all x ∈ M and all p ∈ Px. Clearlyλ is horizontal and it is smooth if` is smooth. We claim thatλisρ-pseudoG-invariant. Letp ∈P andg ∈Gbe given and setq=p·g. We compute:

(γ_g^∗λ)p(ζ1, . . . , ζk) =λq(ζ1·g, . . . , ζk·g) = ˆq⁻¹·`x(ζ1, . . . , ζk)

=ρ(g)⁻¹·λp(ζ1, . . . , ζ_k), where in the second equality we have used the result of Exercise 1.42 and in the last equality we have used thatqˆ= ˆp◦ρ(g)(recall (1.2.17)).

DEFINITION2.9.9. Let`be aP×<sub>G</sub>E<sub>0</sub>-valuedk-form onM. TheE<sub>0</sub>-valued k-formλdefined by (2.9.4), for allp ∈P and allζ<sub>1</sub>, . . . , ζ<sub>k</sub> ∈T<sub>p</sub>P, is called the differential form associated to`.

LEMMA2.9.10. LetΠ :P →Mbe aG-principal bundle,E0 be a real finite-dimensional vector space andρ :G→ GL(E₀)be a smooth representation ofG onE₀. Letλbe anE₀-valued horizontalρ-pseudoG-invariantk-form onP. Then

2.9. DIFFERENTIAL FORMS IN A PRINCIPAL BUNDLE 147

there exists a uniqueP ×_GE0-valuedk-form`onM such thatλis associated to

`. Ifs:U →P is a smooth local section then the following equality holds:

(2.9.5) [s(x),(s^∗λ)x(v1, . . . , vk)] =`x(v1, . . . , vk),

for allx∈U and allv₁, . . . , v_k ∈T_xM. Moreover,`is smooth ifλis smooth.

PROOF. Givenx∈M,v1, . . . , v_k∈TxM, we set:

(2.9.6) `x(v1, . . . , vk) = ˆp·λp(ζ1, . . . , ζk) = [p, λp(ζ1, . . . , ζk)]∈Px×_GE0, wherep is arbitrarily chosen inP_x and the vectorsζ₁, . . . , ζ_k ∈ T_pP are chosen with dΠp(ζi) = vi, i = 1, . . . , k. We have to check that the righthand side of (2.9.6) does not depend on the choices of pandζ₁, . . . ,ζ_k. Independence of the choice of theζi’s amounts to proving that:

λ_p(ζ₁, . . . , ζ_k) =λ_p(ζ₁+A₁, . . . , ζ_k+A_k),

whereA1, . . . , Ak∈Verp(P)are vertical; this follows immediately from the mul-tilinearity of λ_p and from the horizontality of λ. Once the independence of the ζi’s has been established, the independence of thepwill follow once we prove the equality:

(2.9.7) qˆ·λq(ζ1·g, . . . , ζk·g) = ˆp·λp(ζ1, . . . , ζk),

where q = p·g (recall from Exercise 1.42 that dΠ_q(ζ_i ·g) = dΠ_p(ζ_i) = v_i, fori = 1, . . . , k). To prove (2.9.7) we useqˆ= ˆp◦ρ(g) (recall (1.2.17)) and the ρ-pseudoG-invariance ofλas follows:

ˆ

q·λq(ζ1·g, . . . , ζk·g) = ˆq·(γ_g^∗λp)(ζ1, . . . , ζk)

= ˆq◦ρ(g)⁻¹

·λp(ζ1, . . . , ζ_k) = ˆp·λp(ζ1, . . . , ζ_k).

Obviously equality (2.9.6) is equivalent toλbeing associated to`(equality (2.9.4)), so that` is indeed the unique P ×_GE₀-valuedk-form on M such that λis as-sociated to `. If s : U → P is a smooth local section then equality (2.9.5) is proven by taking p = s(x)and ζi = dsx(vi), i = 1, . . . , k, in (2.9.6), keeping in mind that dΠ_s(x) ds_x(v_i)

= v_i. Now assume that λ is smooth. The map ˆ

s:U×E₀ →(P|_U)×_GE₀defined in (1.4.2) is an isomorphism of vector bundles (see Example 1.5.14) and therefore`is smooth if and only ifsˆ<sup>−1</sup>◦`is smooth (see Example 1.6.32). The smoothness ofsˆ⁻¹◦`is proven by observing that equality (2.9.5) is the same as:

ˆ

s⁻¹◦`=s^∗λ.

REMARK 2.9.11. Let Π : P → M be a G-principal bundle and ρ : G → GL(E0)be a smooth representation. Letλbe a horizontalρ-pseudoG-invariant k-form onPand`be aP×_GE₀-valuedk-form onM. If every point ofMis in the domain of a smooth local sections:U →P such that equality (2.9.5) holds then

`is associated toλ. Namely, if`⁰ is theP×_GE0-valuedk-form onM associated toλthen equality (2.9.5) holds with`replaced with`⁰. Thus,`|<sub>U</sub> =`⁰|_U.

LEMMA 2.9.12. Let P be a G-principal bundle endowed with a connection Hor(P), E₀ be a real finite-dimensional vector space andρ : G → GL(E₀) be a smooth representation of G on E0. If λ is a smooth horizontal ρ-pseudo G-invariant differential form onP then its exterior covariant derivative is given by:

(2.9.8) Dλ= dλ+ω∧λ,

whereω denotes theg-valued connection form ofHor(P) and the wedge product is taken with respect to the bilinear pairing:

g×E₀ 3(X, e)7−→ρ(X)¯ ·e, andρ¯= dρ(1) :g→gl(E₀).

PROOF. Formula (2.9.8) is equivalent to:

(2.9.9) Dλ_p(ζ₀, . . . , ζ_k) = dλ_p(ζ₀, ζ₁, . . . , ζ_k) + 1

k!

X

σ∈S_k+1

sgn(σ) ¯ρ ωp(ζ_σ(0))

·λp(ζ_σ(1), . . . , ζ_σ(k)),

for all p ∈ P and all ζ₀, . . . , ζ_k ∈ T_pP. By multilinearity, it suffices to prove formula (2.9.9) when each ζi is either horizontal or vertical. If all the ζi’s are horizontal, the equality is obvious. Assume that at least two of theζi’s are vertical, sayζ₀ andζ₁. Clearly, both the lefthand side and the sum on the righthand side of (2.9.9) vanish; we have to check that, in this case, also the term withdλvanishes.

SetA_i = ω_p(ζ_i) ∈ gandZ_i = A^P_i , so thatZ_i(p) = ζ_i, for i = 0,1. Choose arbitrary smooth vector fieldsZi onP withZi(p) = ζi, for i = 2, . . . , k. Using Cartan’s formula for exterior differentiation (A.3.2), it is clear thatdλ(Z₀, . . . , Z_k) vanishes; namely, sinceZ0,Z1and[Z0, Z1]are vertical andλis horizontal, all the terms in the righthand side of Cartan’s formula vanish. Now assume that exactly one of theζ_i’s is vertical; by antisymmetry, we may assume thatζ₀ is vertical. Set A0 = ωp(ζ0) ∈ g, Z0 = A^P₀; fori = 1, . . . , k, letXi be a smooth vector field on M withX_i Π(p)

= dΠ_p(ζ_i) and setZ_i = X_i^hor. Then Z_i(p) = ζ_i, for all i= 0, . . . , k. Using again Cartan’s formula for exterior differentiation, keeping in mind Corollary 2.9.7 and the fact thatλis horizontal, we obtain:

dλ(Z₀, . . . , Z_k) =Z₀ λ(Z₁, . . . , Z_k) . SinceZ₀is vertical, in order to computeZ₀ λ(Z₁, . . . , Z_k)

(p), it suffices to con-sider the restriction ofλ(Z1, . . . , Z_k)to the fiberPx, wherex = Π(p). Denoting byf :P_x →E₀such restriction, we obtain:

dλ_p(ζ₀, . . . , ζ_k) = df_p(ζ₀) = df_p dβ_p(1)·A₀

= d(f ◦β_p)(1)·A₀.

2.9. DIFFERENTIAL FORMS IN A PRINCIPAL BUNDLE 149

But:

(f◦βp)(g) =f(p·g) =λp·g(X₁^hor(p·g), . . . , X_k^hor(p·g)

=λp·g(X₁^hor(p)·g, . . . , X_k^hor(p)·g

=λp·g(ζ₁·g, . . . , ζ_k·g)

= (γ_g^∗λ)_p(ζ₁, . . . , ζ_k)

=ρ(g)⁻¹·λp(ζ1, . . . , ζk), for allg∈G; therefore:

(2.9.10) dλ_p(ζ₀, . . . , ζ_k) = d(f ◦β_p)(1)·A₀ =−¯ρ(A₀)·λ_p(ζ₁, . . . , ζ_k).

Now let us compute the sum on the righthand side of (2.9.9); clearly, all the terms of that sum vanish, except for those with σ(0) = 0. Such terms are all equal to

¯

ρ(A₀)·λ_p(ζ₁, . . . , ζ_k)and therefore their sum is equal tok! ¯ρ(A₀)·λ_p(ζ₁, . . . , ζ_k).

Using (2.9.10), we conclude that the righthand side of (2.9.9) vanishes. Obviously also the lefthand side of (2.9.9) vanishes and the proof is complete.

REMARK2.9.13. Letπ : E → M be a vector bundle with typical fiber E₀ endowed with a connection∇and letHor FRE0(E)

be the corresponding con-nection on the principal bundle of framesFR_E0(E). Letρ: GL(E₀)→ GL(E₀) be the identity map. A horizontal ρ-pseudo GL(E0)-invariant differential form λonFR_E0(E)is associated to a unique differential form`on M with values in FR<sub>E0</sub>(E)×<sub>∼</sub>E<sub>0</sub>. By composing`with the contraction mapC^E, we obtain a differ-ential formC^E◦`onM with values onE. In this situation, we will also say that λandC<sup>E</sup>◦`are associated.

More generally, letP be aG-principal bundle over a differentiable manifold M, let E₀ be a real finite-dimensional vector space, let ρ : G → GL(E₀) be a smooth representation and letφ:P →FRE0(E)be a morphism of principal bun-dles whose subjacent Lie group homomorphism is the representationρ. We have then an isomorphism of vector bundles (recall Definition 1.5.17)C^φfromP×_GE₀ toE. A horizontalρ-pseudo invariant differential formλonP is associated to a unique differential form`onMwith values inP×<sub>G</sub>E<sub>0</sub>. By composing`with the φ-contraction mapC^φ, we obtain a differential formC^φ◦`onM with values onE.

In this situation, we will also say thatλandC^φ◦`are associated. A few particular situations where this occurs are presented in Remarks 1.6.1 and 1.6.9.

DEFINITION 2.9.14. Let M be a differentiable manifold and consider the GL(Rⁿ)-principal bundleFR(T M) of frames ofT M. The identity map ofT M can be identified with aT M-valued smooth1-form onM; thecanonical formθ ofFR(T M) is theRⁿ-valued smooth1-form on FR(T M) that is associated to the identity map ofT M. More generally, let π : E → M be a vector bundle with typical fiberE0 and letι :T M → E be a morphism of vector bundles. We can identifyιwith a smoothE-valued1-form on M. Theι-canonical formθ^ι of FRE0(E)is theE0-valued smooth1-form onFRE0(E)that is associated toι.

More explicitly, we have:

(2.9.11) θ_p(ζ) =p⁻¹ dΠ_p(ζ)

∈Rⁿ,

for allp∈FR(T M),ζ ∈TpFR(T M)and, more generally:

θ_p^ι(ζ) =p⁻¹ ιx·dΠp(ζ)

∈E0,

for allx∈M,p∈FRE0(Ex),ζ ∈TpFRE0(E). Notice that ifs:U →FRE0(E) is a smooth local section then:

(2.9.12) (s^∗θ^ι)x=s(x)⁻¹◦ιx :TxM −→E0,

for all x ∈ U; namely, if Π : FR_E0(E) → M denotes the projection then the compositiondΠ_s(x)◦dsxis the identity map ofTxM.

DEFINITION 2.9.15. Let M be a differentiable manifold and consider the GL(Rⁿ)-principal bundle FR(T M) of frames of T M. The torsion form Θ of FR(T M)is defined by:

Θ = Dθ.

More generally, letπ : E → M be a vector bundle with typical fiberE0 and let ι:T M →Ebe a morphism of vector bundles. Theι-torsion formΘ^ιofFR_E0(E) is defined by:

Θ^ι= Dθ^ι. Observe that by (2.9.8) we have:

(2.9.13) Θ^ι = dθ^ι+ω∧θ^ι,

where the wedge product is taken with respect to the obvious bilinear pairing of gl(E0)andE0.

The curvature tensorRof a connection∇on a vector bundleπ :E→M can be identified with a smoothgl(E)-valued2-form onM. We have the following:

LEMMA 2.9.16. Let π : E → M be a vector bundle with typical fiber E0

endowed with a connection ∇. The curvature formΩcorresponding to the con-nection on the principal bundle of framesFR_E0(E)is associated to the curvature tensorR; more explicitly:

(2.9.14) p◦Ω_p(ζ₁, ζ₂)◦p⁻¹ =R_x dΠ_p(ζ₁),dΠ_p(ζ₂)

∈Lin(E_x), for allx∈M,p∈FR_E0(E_x),ζ₁, ζ₂∈T_pFR_E0(E), where

Π : FRE0(E)−→M denotes the projection.

PROOF. Lets : U → FR_E0(E) be a smooth local section and setω¯ =s^∗ω, Ω = s^∗Ω. Keeping in mind equality (2.9.5) and Remark 2.9.13, we see that the proof will be concluded once we show that:

s(x)◦ Ω_x(v, w)

◦s(x)⁻¹=R_x(v, w),

for allx ∈ U and allv, w ∈ T_xM. Let X, Y ∈ Γ(T M|_U) and ∈ Γ(E|_U) be fixed; denote by˜:U →E₀ the representation ofwith respect tos. We have to show that:

s(x)

Ω_x X(x), Y(x)

·˜(x)

=R_x X(x), Y(x) (x),

2.9. DIFFERENTIAL FORMS IN A PRINCIPAL BUNDLE 151

for allx∈U. We computeR(X, Y)using (2.5.4) as follows; the representation of∇_Ywith respect tosis given by:

Y(˜) + ¯ω(Y)·˜.

Therefore, the representation of∇_X∇_Ywith respect tosis equal to:

(2.9.15) X Y ˜)

+X ω(Y¯ )

·˜+ ¯ω(Y)·X(˜)+ ¯ω(X)·Y(˜)+ ¯ω(X)◦ω(Y¯ )

·˜. Similarly the representation of∇_Y∇_Xwith respect tosis equal to:

(2.9.16) Y X ˜)

+Y ω(X)¯

·˜+ ¯ω(X)·Y(˜)+ ¯ω(Y)·X(˜)+ ¯ω(Y)◦ω(X)¯

·˜, and the representation of∇_[X,Y]with respect tosis equal to:

(2.9.17) [X, Y](˜) + ¯ω [X, Y]

·˜.

Hence, using (2.9.15), (2.9.16), (2.9.17) and Cartan’s formula for exterior differ-entiation (A.3.3), we obtain that the representation ofR(X, Y)with respect tos is equal to:

d¯ω(X, Y)·˜+ [¯ω(X),ω(Y¯ )]˜= Ω(X, Y)·˜.

The conclusion follows.

PROPOSITION2.9.17. LetΠ :P →Mbe aG-principal bundle endowed with a connection,E₀ be a real finite-dimensional vector space andρ :G→ GL(E₀) be a smooth representation ofGonE0. Assume that the vector bundleP ×_GE0

is endowed with connection defined in Example 2.5.9. If`is a smoothP ×<sub>G</sub>E<sub>0</sub> -valued k-form on M and λ is the associated E0-valued k-form on P then the covariant exterior differentialDλis associated to the covariant exterior differential D`.

PROOF. Lets : U → P be a smooth local section. Then, equality (2.9.5) holds. We have to prove that (see Remark 2.9.11):

(2.9.18) [s(x),(s^∗Dλ)x(v1, . . . , v_k+1)] = D`x(v1, . . . , v_k+1),

for allx ∈ U and allv1, . . . , vk+1 ∈ TxM. DefineH : P → FRE0(P ×_GE0) as in (1.5.3) and sets1 =H◦s, so thatsˆ= ˇs1(see (1.5.5)). Letωdenote the g-valued connection form of the connection ofP and letω⁰denote thegl(E₀)-valued connection form of the connection of FRE0(P ×_GE0). Since His connection preserving, we have:

H^∗ω⁰ = ¯ρ◦ω,

whereρ¯= dρ(1) :g→gl(E₀)(see (c) of Lemma 2.2.11). Settingω¯ =s^∗ωthen:

s^∗₁ω⁰ =s^∗H^∗ω⁰ =s^∗( ¯ρ◦ω) = ¯ρ◦ω.¯ By Example 2.5.12, the vector bundle isomorphism:

ˇ

s₁ :U ×E₀ −→(P|_U)×_GE₀

is connection preserving if the trivial vector bundleU ×E0 is endowed with the connectiondI +s^∗₁ω⁰ = dI + ¯ρ◦ω, where the¯ gl(E₀)-valued1-formρ¯◦ω¯onU is identified with theC^∞(U)-bilinear map:

Γ(T M|_U)×Γ(U ×E₀)3(X, )7−→ ρ¯◦ω(X)¯

()∈Γ(U ×E₀).

Set `˜ = ˇs<sup>−1</sup><sub>1</sub> ◦` = ˆs⁻¹ ◦`; by (2.9.5), we have `˜ = s^∗λ. Denote by D˜` the exterior covariant derivative of theE<sub>0</sub>-valuedk-form`˜associated to the connection dI + ¯ρ◦ ω¯ on the trivial bundle U ×E0; since sˇ1 is connection preserving, by Proposition 2.7.20, we have:

(2.9.19) sˆ◦D˜`= ˇs<sub>1</sub>◦D˜`= D`.

Moreover, by Corollary 2.7.21 and Example 2.7.19:

D˜`= d˜`+ ( ¯ρ◦ω)¯ ∧`,˜

where the wedge product is taken with respect to the obvious bilinear pairing of gl(E0)withE0. If we consider the bilinear pairing ofgwithE0given by:

g×E0 3(A, e)7−→ρ(A)¯ ·e∈E0

then( ¯ρ◦ω)¯ ∧`˜= ¯ω∧`, so that:˜

D˜`= d˜`+ ¯ω∧`.˜

Taking the pull-back byson both sides of (2.9.8) and using thats^∗λ= ˜`we obtain:

s^∗Dλ= d˜`+ ¯ω∧`,˜ so that:

s^∗Dλ= D˜`, and, by (2.9.19):

ˆ

s◦s^∗Dλ= D`,

proving (2.9.18). This concludes the proof.

COROLLARY2.9.18. Theι-torsion formΘ^ι is associated to theι-torsion ten-sorT^ι; more explicitly:

(2.9.20) p Θ^ι_p(ζ1, ζ2)

=Tx dΠp(ζ1),dΠp(ζ2)

∈Ex, for allx∈M,p∈FR_E0(E_x),ζ₁, ζ₂∈T_pFR_E0(E), where

Π : FRE0(E)−→M denotes the projection.

PROOF. Follows immediately from Proposition 2.9.17 and from Example 2.7.22.

No documento The theory of connections and G-structures. Applications to affine and isometric immersions Paolo Piccione Daniel V. Tausk (páginas 152-160)