The inner torsion of a G-structure

whereT^∗ :Ex→Exdenotes thetransposeofT with respect togx, i.e., the unique linear endomorphism ofE_xsuch that:

g_x T(e), e⁰

=g_x e, T^∗(e⁰) ,

for alle, e⁰ ∈ E_x. Thus, the inner torsionI^P_x is identified with a linear map from TxM tosym(Ex). Lets: U → P be a smooth local section withx ∈ U and let e, e⁰∈E_xbe fixed; consider the local sections, ⁰ :U →Edefined by:

(y) = s(y)◦s(x)⁻¹

·e, (2.11.2)

⁰(y) = s(y)◦s(x)⁻¹

·e⁰,

for ally ∈U. Since the representations ofand⁰ with respect tosare constant, we have:

∇_v= dI^s_v+ Γ_x(v)·(x) = Γ_x(v)·(x), (2.11.3)

∇_v⁰ = dI^s_v⁰+ Γ_x(v)·⁰(x) = Γ_x(v)·⁰(x), for allv∈TxM. Sincesis a local section ofFR^o_E0(E), it follows that:

g_y (y), ⁰(y)

=hs(x)⁻¹·e, s(x)⁻¹·e⁰i_E0, for ally∈U, so that the real-valued mapg(, ⁰)is constant. Thus:

0 =v g(, ⁰)

= (∇_vg)(e, e⁰) +g_x(∇_v, e⁰) +g_x(e,∇_v⁰)

= (∇_vg)(e, e⁰) +g_x Γ_x(v)·e, e⁰

+g_x e,Γ_x(v)·e⁰ , for allv∈T_xM. Then:

g_x

Γ_x(v) + Γ_x(v)^∗

·e, e⁰

=−(∇_vg)(e, e⁰) and (Lemma 2.11.1 and (2.11.1)):

gx I^P_x(v),·

= ¹₂gx

Γx(v) + Γx(v)^∗ ,·

=−¹₂∇_vg,

for all x ∈ M, v ∈ T_xM. Identifying ∇_vg : E_x ×E_x → R with a linear endomorphism ofEx, we obtain:

I^P_x(v) =−¹₂∇_vg.

Thus, the inner torsion of P is essentially the covariant derivative of the semi-Riemannian structureg. In particular, I^P = 0if and only if∇g = 0, i.e.,∇is compatible with the semi-Riemannian structureg.

EXAMPLE 2.11.4. Let π : E → M be a vector bundle with typical fiber E0 and F be a vector subbundle of E. If F0 is a subspace of E0 such that dim(F₀) = dim(F_x) for all x ∈ M then the set P = FR_E0(E;F₀, F) of all E₀-frames of Eadapted to (F₀, F)is aG-structure onE withG= GL(E₀;F₀) (Example 1.8.5). Let∇be a connection onEand let us compute the inner torsion I^P. Letx∈ M be fixed. ClearlyG_x = GL(E_x;F_x)andg_xis the Lie algebra of linear endomorphismsT :Ex → Ex withT(Fx)⊂ Fx. We identify the quotient gl(E_x)/g_xwith the spaceLin(F_x, E_x/F_x)via the map:

gl(E_x)/g_x3T+g_x7−→q◦T|_Fx ∈Lin(F_x, E_x/F_x),

2.11. THE INNER TORSION OF AG-STRUCTURE 159

where q : Ex → Ex/Fx denotes the quotient map. Thus, the inner torsionI^P_x is identified with a linear map fromT_xM toLin(F_x, E_x/F_x). Let s : U → P be a smooth local section with x ∈ U. Given e ∈ Fx, we define a local sec-tion : U → E as in (2.11.2). Then the representation ofwith respect tosis constant and (2.11.3) holds, for allv ∈ T_xM. Moreover, sincestakes values in FRE0(E;F0, F), we have(U)⊂F. Thus:

∇_v+F_x=α^F_x(v, e)∈E_x/F_x,

whereα^F denotes the second fundamental form of the vector subbundleF (Exer-cise 2.20). Then:

Γ_x(v)·e+F_x =α^F_x(v, e) and:

I^P_x(v) =α^F_x(v,·)∈Lin(F_x, E_x/F_x),

for allx ∈M,v ∈TxM. In particular,I^P = 0if and only ifα^F = 0, i.e., if and only if the covariant derivative of any smooth section ofF is a smooth section of F.

EXAMPLE 2.11.5. Let π : E → M, F, E0, F0 be as in Example 2.11.4.

Letg be a semi-Riemannian structure onE, h·,·i_E0 be an indefinite inner prod-uct on E0 and assume thatFR^o_E0(Ex;F0, Fx) 6= ∅, for all x ∈ M. Then P = FR^o_E0(E;F₀, F)is aG-structure onEwithG= O(E₀;F₀)(Example 1.8.5). For simplicity, we assume that the restriction ofgx toFx ×Fx is nondegenerate, for all x ∈ M; thus, E = F ⊕F^⊥. Denote byq : E → F^⊥ the projection. Let

∇ be a connection on E and let x ∈ M be fixed. We compute I^P_x. We have Gx= O(Ex;Fx)andg_xis the Lie algebra of linear endomorphismsT :Ex→Ex

that are anti-symmetric (with respect tog_x) and satisfyT(F_x)⊂ F_x. We have an isomorphism:

gl(E_x)/g_x−→sym(E_x)⊕Lin(F_x, F_x^⊥)

T+g_x7−→ ¹₂(T+T^∗),¹₂q_x◦(T−T^∗)|_Fx ,

so that we identify I^P_x with a linear map from T_xM to the space sym(E_x) ⊕ Lin(F_x, F_x^⊥). Consider the component:

α∈Γ Lin(T M, F;F^⊥)

of∇with respect to the decompositionE=F⊕F^⊥. Lets:U →Pbe a smooth local section withx∈U. As in Example 2.11.3, we have:

1

2 Γ_x(v) + Γ_x(v)^∗

=−¹₂∇_vg,

for allv∈TxM. Moreover, arguing as in Example 2.11.4, we obtain:

q Γ_x(v)·e

=α_x(v, e), for allv∈TxM,e∈Ex. Then:

(2.11.4) ¹₂ Γ_x(v)−Γ_x(v)^∗

= Γ_x(v)−¹₂ Γ_x(v) + Γ_x(v)^∗

= Γ_x(v) +¹₂∇_vg, and:

I^P_x(v) = −¹₂∇_vg, α_x(v,·) +¹₂q◦ ∇_vg|_Fx ,

for allx ∈M,v ∈TxM, where∇_vgis identified with a linear endomorphism of E_x. In particular,I^P = 0if and only if∇g = 0andα = 0, i.e., if and only if∇ is compatible withg and the covariant derivative of any smooth section ofF is a smooth section ofF.

EXAMPLE 2.11.6. Let π : E → M be a vector bundle with typical fiber E0 and ∈ Γ(E) be a smooth section ofE with(x) 6= 0, for allx ∈ M. If e₀ ∈E₀ is a nonzero vector thenP = FR_E0(E;e₀, )is aG-structure onE with G= GL(E0;e0)(Example 1.8.6). Let∇be a connection onEand let us compute I^P. Letx ∈ M be fixed. ThenG_x = GL E_x;(x)

andg_x is the Lie algebra of linear endomorphismsT :E_x → E_x such thatT (x)

= 0. We identify the quotientgl(Ex)/gxwithExvia the map:

gl(Ex)/gx 3T +g_x 7−→T (x)

∈Ex.

Lets:U →Pbe a smooth local section withx∈U. ThenI^P_x is identified with a linear map fromTxM toEx. Sincestakes values inFRE0(E;e0, ), we have:

(y) =s(y)·e0,

so that the representation ofwith respect tosis constant and:

(2.11.5) ∇_v= Γ_x(v)·(x),

for allv∈TxM. Then:

I^P_x(v) =∇_v, for allv∈T_xMand:

I^P_x = (∇)(x),

for all x ∈ M. In particular, I^P = 0 if and only if the section is parallel.

Assume now thatg is a semi-Riemannian structure onE,h·,·i_E0 is an indefinite inner product on E0 and that FR^o_E0 Ex;e0, (x)

6= ∅, for all x ∈ M. Then P = FR^o_E0(E;e0, )is aG-structure onEwithG= O(E0;e0). Let us compute I^P. Letx ∈ M be fixed. ThenG_x = O E_x;(x)

andg_x is the Lie algebra of anti-symmetric linear endomorphismsT ofE_x such thatT (x)

= 0. We have the following linear isomorphism:

gl(E_x)/g_x3T+g_x7−→ ¹₂(T+T^∗),¹₂(T−T^∗)·(x)

∈sym(E_x)⊕(x)^⊥ where(x)^⊥ denotes the kernel ofgx (x),·

. Lets:U → P be a smooth local section withx∈U. As in Example 2.11.3, we have:

1

2 Γ_x(v) + Γ_x(v)^∗

=−¹₂∇_vg, and, as in (2.11.4):

1

2 Γx(v)−Γx(v)^∗

= Γx(v) +¹₂∇_vg, for allv∈T_xM. Moreover, (2.11.5) holds. Then:

1

2 Γ_x(v)−Γ_x(v)^∗

·(x) =∇_v+¹₂(∇_vg) (x) . Hence:

I^P_x(v) =

−¹₂∇_vg,∇_v+¹₂(∇_vg) (x) ,

2.11. THE INNER TORSION OF AG-STRUCTURE 161

for allx∈M,v∈TxM. In particular,I^P = 0if and only if∇is compatible with gandis parallel.

EXAMPLE2.11.7. Letπ :E → M be a vector bundle with typical fiberE0, J be an almost complex structure onEandJ₀be a complex structure onE₀. The setP = FR^c_E0(E)is aG-structure onEwithG= GL(E0, J0)(Example 1.8.7).

Let∇be a connection onE and let us computeI^P. Letx ∈ M be fixed. Then Gx = GL(Ex, Jx)andg_x is the Lie algebra of linear endomorphismsT :Ex → E_xsuch thatT◦J_x =J_x◦T. We have an isomorphism:

gl(Ex)/gx3T +g_x 7−→[T, Jx]∈Lin(Ex, Jx),

where[T, Jx] =T◦Jx−Jx◦TandLin(Ex, Jx)denotes the space of linear maps T :E_x → E_x such thatT ◦J_x+J_x◦T = 0. Lets:U →P be a smooth local section withx∈Uand lete∈E_xbe fixed. We define a local section:U →Eas in (2.11.2). Then(x) =eand the representation ofwith respect tosis constant;

moreover, sincestakes values inFR^c_E0(E), also the representation ofJ() with respect tosis constant. Then:

∇_v= Γ_x(v, e), ∇_v J()

= Γ_x v, J_x(e) , and:

∇_v J()

= (∇_vJ)(e) +J_x(∇_v), for allv∈TxM. We therefore obtain:

Γ_x(v)◦J_x =∇_vJ+J_x◦Γ_x(v).

Hence:

I^P_x(v) =∇_vJ,

for allx∈M and allv∈T_xM. In particular,I^P = 0if and only ifJ is parallel.

EXAMPLE2.11.8. Letπ :E → M be a vector bundle with typical fiberE₀, J be an almost complex structure onE,g be a semi-Riemannian structure onE, J₀ be a complex structure onE₀andh·,·i_E0 be an indefinite inner product onE₀. Assume thatJxis anti-symmetric with respect togxfor allx∈M, thatJ0is anti-symmetric with respect toh·,·i_E0 and that gx has the same index as h·,·i_E0, for allx ∈ M. Then the setP = FR^u_E

0(E)is aG-structure onE withG = U(E₀) (Example 1.8.7). Let∇be a connection onEand let us computeI^P. Letx∈Mbe fixed. We haveG_x= U(E_x)andg_xis the Lie algebra of linear mapsT :E_x→E_x such thatT ◦Jx = Jx◦T and such thatT is anti-symmetric with respect togx. We have a linear isomorphism:

gl(Ex)/gx −→sym(Ex)⊕Lina(Ex, Jx) T +g_x 7−→ ¹₂(T +T^∗),¹₂[T −T^∗, Jx]

,

whereLin_a(E_x, J_x)denotes the space of linear mapsT :E_x → E_xthat are anti-symmetric with respect togxand such thatT ◦Jx+Jx◦T = 0. Lets:U →P be a smooth local section withx∈U. As in Example 2.11.3, we have:

1

2 Γ_x(v) + Γ_x(v)^∗

=−¹₂∇_vg,

for allv∈TxM. Moreover, as in Example 2.11.7, we have:

[Γ_x(v), J_x] =∇_vJ, for allv∈T_xM. Then:

1

2 Γx(v)−Γx(v)^∗

= Γx(v)− ∇_vg, and:

1

2[Γx(v)−Γx(v)^∗, Jx] =∇_vJ−[∇_vg, Jx].

Hence:

I^P_x(v) = −¹₂∇_vg,∇_vJ−[∇_vg, J_x] ,

for all x ∈ M and all v ∈ TxM. In particular, I^P = 0 if and only if ∇ is compatible withgandJ is parallel.

REMARK2.11.9. LetMbe ann-dimensional differentiable manifold endowed with a connection∇, Gbe a Lie subgroup ofGL(Rⁿ)andP ⊂ FR(T M)be a G-structure onT M. We denote byHor FR(T M)

the connection onFR(T M) associated to ∇ and by ω the corresponding connection form. Given x ∈ M, p∈Px, we have a linear isomorphism:

(2.11.6) (dΠp, ωp) :TpFR(T M)−→TxM⊕gl(Rⁿ)

as in (2.10.1). We have seen in Remark 2.10.5 that the image ofT_pP under the isomorphism (2.11.6) is equal to:

(v, X)∈TxM⊕gl(Rⁿ) : (Adp)⁻¹◦I^P_x

(v) =X+g .

Sincepis an isomorphism fromRⁿtoT_xM, by composing (2.11.6) withp⁻¹⊕Id we obtain another linear isomorphism (recall (2.9.11)):

(2.11.7) (θ_p, ω_p) :T_pFR(T M)−−→^∼= Rⁿ⊕gl(Rⁿ).

The image ofT_pP under (2.11.7) is obviously equal to:

(u, X)∈Rⁿ⊕gl(Rⁿ) : (Adp)⁻¹◦I^P_x ◦p

(u) =X+g .

IfI^P = 0, i.e., if∇is compatible withP and ifp :I → FR(E)is a smooth horizontal curve such thatp(t₀) ∈P for somet₀ ∈I thenp(t) ∈P for allt∈I. We now generalize this property to the case whereI^P is not necessarily zero.

PROPOSITION 2.11.10. LetE be a vector bundle of rank kover a manifold M,∇be a connection onE, Gbe a Lie subgroup of GL(R^k)andP ⊂ FR(E) be a G-structure onE. Let p : I → E be a smooth curve and set γ = Π◦p, whereΠ : FR(E)→M denotes the projection. Assume thatp(I)∩P 6=∅. Then p(I)⊂P if and only if:

(2.11.8) I^P_γ(t) γ⁰(t)

= (∇₁p)(t)◦p(t)⁻¹+g_γ(t), for allt∈I.

EXERCISES 163

PROOF. SinceT P is invariant by the action ofGinT FR(E)

, there exists a GL(R^k)-invariant smooth distributionDon the manifoldFR(E)such thatD_p = TpP, for all p ∈ P. Such distribution is integrable becauseP ·g is an integral submanifold ofD, for allg ∈GL(R^k). For allx ∈M and allp ∈FR(E_x), we defineL_p :T_xM → gl(R^k)/gby settingL_p = Ad_p⁻¹◦I^P_x and we defineV_p by setting:

(2.11.9) V_p =

(v, X)∈T_xM⊕gl(R^k) :L_p(v) =X+g . Clearly:

L_p◦g= Ad_g⁻¹ ◦ L_p, and therefore:

(2.11.10) (Id⊕Ad_g⁻¹)(V_p) =V_p◦g, for allp∈FR(E)and allg∈GL(R^k).

We claim that(dΠp, ωp)(D_p) =V_p, for allp∈F R(E). Namely, by the defini-tion of inner torsion, such equality holds forp∈P. The fact that the equality holds for anyp∈FR(E)follows from (2.11.10) and from the fact that the diagram:

Tp◦gFR(E) ^(dΠp◦g_∼^,ωp◦g)

= //TxM⊕gl(R^k) T_pFR(E)

g

OO

(dΠp,ωp)

∼= //TxM⊕gl(R^k)

Id⊕Ad_g−1

OO

commutes, for allx∈M,p∈FR(Ex)and allg∈GL(R^k). Now:

(dΠ_p(t), ω_p(t)) p⁰(t)

= γ⁰(t), p(t)⁻¹◦(∇₁p)(t) ,

for all t ∈ I and therefore p is tangent to D if and only if (2.11.8) holds. If p(I)⊂P then, sinceP is an integral submanifold ofD, it follows thatpis tangent to D and thus (2.11.8) holds. Conversely, assume that (2.11.8) holds and that p(I)∩P 6= ∅. Since for all g ∈ GL(R^k), P ·g is an integral submanifold of D, it follows that the set p⁻¹(P ·g) is open inI; thusp⁻¹(P) is both open and

closed inIand the conclusion follows.

Exercises The general concept of connection.

EXERCISE 2.1. LetV,W be vector spaces and letT : V → W be a linear map. Given a subspaceZofV, show thatV =Z⊕Ker(T)if and only if the map T|_Z :Z →T(V)is an isomorphism.

EXERCISE2.2. LetV1,V2,V⁰be vector spaces and assume that we are given linear mapsT₁:V₁→V⁰,T₂:V₂→V⁰,L:V₁ →V₂ such that the diagram:

V₁ ^{L //}

TA1AAAAAA

A V₂

T2

~~}}}}}}}}

V⁰

commutes and such thatT₁ andT₂ have the same image (this is the case, for in-stance, if bothT1 andT2 are surjective or ifLis surjective). LetZ be a subspace ofV1 withV1 =Z⊕Ker(T1). Show that the restriction ofLtoZis injective and thatV₂ =L(Z)⊕Ker(T₂).

EXERCISE2.3. LetW be a vector space andW₁,W₂,W₂⁰ be subspaces ofW such thatW =W1⊕W2andW1∩W₂⁰ ={0}. Show thatW2 ⊂W₂⁰ if and only ifW₂ = W₂⁰. Conclude that, under the hypotheses and notations of Exercise 2.2, ifZ⁰ is a subspace ofV2 withV2 =Z⁰ ⊕Ker(T2)thenL(Z) ⊂Z⁰ if and only if L(Z) =Z⁰.

EXERCISE2.4. LetV,W be vector spaces,T :V →W be a linear map and V0 ⊂V,W0 ⊂W be subspaces such thatT|_V0 :V0 → W0 is an isomorphism. If His a subspace ofW withW =H⊕W₀, show that:

V =T⁻¹(H)⊕V₀.

EXERCISE2.5. LetE,Mbe differentiable manifolds,π :E →Mbe a smooth submersion and:U →Ebe a smooth local section ofπ. Show that for allx∈U, the image ofd(x)is a horizontal subspace ofT_(x)E.

Connections on principal fiber bundles.

EXERCISE2.6. LetΠ :P →Mbe aG-principal bundle and letM =S

i∈IU_i be an open cover of M. Assume that for every i ∈ I it is given a connection Hor(P|_Ui)on the principal bundle P|_Ui and assume that for alli, j ∈ I and all x∈Ui∩Uj we haveHorx(P|_Ui) = Horx(P|_Uj). Show that there exists a unique connectionHor(P)on P such thatHorx(P) = Horx(P|_Ui), for alli ∈I and all x∈Ui.

EXERCISE 2.7. LetΠ : P → M be aG-principal bundle,V be a real finite-dimensional vector space and letρ :G → GL(V)be a smooth representation of GonV. Show that aV-valued differential formλonP isρ-pseudoG-invariant if and only if for everyx∈M, there exists a pointp∈P_xsuch that:

(γ_g^∗λ)p=ρ(g)⁻¹◦λp, for allg∈G.

EXERCISE 2.8. Let P be a G-principal bundle endowed with a connection Hor(P)and denote byp_ver :T P → Ver(P),p_hor :T P → Hor(P)respectively the vertical and the horizontal projections determined by the horizontal distribution Hor(P). Giveng∈G,p∈P,ζ ∈T_pP, show that:

p_ver(ζ·g) =p_ver(ζ)·g, p_hor(ζ·g) =p_hor(ζ)·g.

EXERCISES 165

EXERCISE 2.9. LetΠ : P → M be aG-principal bundle and letωbe a Ad-pseudo G-invariantg-valued1-form on P. Show that if for every x ∈ M there existsp ∈Px such that condition (2.2.5) holds then condition (2.2.5) holds for all p∈P.

EXERCISE2.10. LetΠ :P →M be aG-principal bundle,V be a real finite-dimensional vector space andρ:G→GL(V)be a smooth representation ofGon V. Letλ¹,λ²beV-valuedρ-pseudoG-invariantk-forms onP and assume that:

λ¹_p(ζ1, . . . , ζ_k) =λ²_p(ζ1, . . . , ζ_k),

for allp ∈ P,ζ1, . . . , ζk ∈TpP, provided that at least one of the vectorsζiis in Verp(P). Given a smooth local sections:U →P ofP, show that ifs^∗λ¹=s^∗λ² thenλ¹andλ²are equal onP|_U.

EXERCISE 2.11. LetP,Qbe principal bundles over the same differentiable manifoldM, with structural groups GandH, respectively. Letφ : P → Q be a morphism of principal bundles whose subjacent Lie group homomorphism is φ0 : G → H. Denote byg, hthe Lie algebras ofGandH respectively and by φ¯₀ : g → hthe differential of φ₀ at the identity. Forp ∈ P, q ∈ Q, denote by β_p^P : G → P, β_q^Q : H → Qthe maps given by action at p and by action atq, respectively; consider the linear isomorphisms:

dβ_p^P(1) :g−→Ver_p(P), dβ_q^Q(1) :h−→Ver_q(Q).

Letωbe anh-valued1-form onQsuch that:

ω_q|_Verq(Q) = dβ_q^Q(1)−1

, for allq∈Q. Show that:

(φ^∗ω)_p|_Verp(P)= ¯φ₀◦ dβ_p^P(1)−1

, for allp∈P.

EXERCISE 2.12. LetP,Qbe principal bundles over the same differentiable manifoldM, with structural groups GandH, respectively. Letφ : P → Q be a morphism of principal bundles and letφ0 : G → H denote its subjacent Lie group homomorphism. Let V be a real finite-dimensional vector space and let ρ : H → GL(V) be a smooth representation ofH onV. Ifλis aρ-pseudo H-invariant differential form onQ, show thatφ^∗λis a(ρ◦φ₀)-pseudoG-invariant differential form onP.

Connections on vector bundles.

EXERCISE2.13. LetV,W be vector spaces andT :V →W be a linear map.

Given subspacesZ,Z⁰ofV, show thatT(Z) =T(Z⁰)if and only ifZ+Ker(T) = Z⁰+ Ker(T).

EXERCISE 2.14. Letπ : E → M be a vector bundle and∇be a connection onE. Given a smooth section∈Γ(E)ofEthat vanishes on an open subsetUof M, show that∇_valso vanishes, for allv∈T M|_U. Conclude that, if, ⁰∈Γ(E)

are equal on an open subset U of M then ∇_vand ∇_v⁰ are also equal, for all v∈T M|_U.

EXERCISE 2.15. Letπ : E → M be a vector bundle and∇be a connection onE.

• Given open subsetsU, V ofM with V ⊂ U, consider the connection

∇^U induced by∇onE|_U and the connection(∇^U)^V induced by∇^U on (E|_U)|_V =E|_V. Show that(∇^U)^V is the same as∇^V.

• Let∇⁰ be another connection on E. If every point of M has an open neighborhoodU inMsuch that∇^U =∇^0U, show that∇=∇⁰.

EXERCISE 2.16. Let∇,∇⁰ be connections on a vector bundleπ : E → M. Assume that for all x ∈ M and alle ∈ E_x there exists a smooth local section :U →EofEdefined in an open neighborhoodU ofxinM such that(x) =e and:

∇_v=∇⁰_v, for allv∈T_xM. Show that∇=∇⁰.

EXERCISE 2.17. Let π : E → M be a vector bundle with typical fiber E₀ endowed with a connection∇and letE1be a real vector space isomorphic toE0. As we have seen in Exercise 1.61,π :E → M can be regarded also as a vector bundle with typical fiberE₁. Since the differential structure ofEdoes not depend on the typical fiber, the space Γ(E) also doesn’t depend on the typical fiber and hence∇is also a connection on the vector bundleπ :E → M with typical fiber E1. The connection ∇is associated to connections on both principal bundles of framesFR_E0(E)andFR_E1(E). Show that:

• the horizontal distribution on E defined by ∇does not depend on the typical fiber;

• for any linear isomorphismi : E1 → E0, the isomorphism of principal bundlesγ_idefined in Exercise 1.61 is connection preserving.

Pull-back of connections on vector bundles.

EXERCISE2.18. Assume that we are given a commutative diagram of sets and maps:

A ^{f //}

φ

B

ψ

C _g ^//D Given a subsetSofB, show thatφ f⁻¹(S)

⊂g⁻¹ ψ(S) .

Functorial constructions with connections on vector bundles.

EXERCISE 2.19. Letn ≥ 1 be fixed and letF : Vecⁿ → Vecbe a smooth functor. LetE¹, . . . ,Eⁿbe vector bundles over a differentiable manifoldM with typical fibers E₀¹, . . . ,E₀ⁿ, respectively. For eachi = 1, . . . , n, letE₀ⁱ be a real

EXERCISES 167

vector space isomorphic toE₀ⁱ. As we have seen in Exercise 1.61, the vector bun-dleEⁱcan also be regarded as a vector bundle with typical fiberE₀ⁱ; denote such vector bundle with changed typical fiber byEⁱ. As we have seen in Exercise 1.68, the vector bundlesF(E¹, . . . , Eⁿ)andF(E¹, . . . , Eⁿ)differ only by their typical fibers. Fori= 1, . . . , n, let∇ⁱbe a connection onEⁱ; then∇ⁱis also a connection on Eⁱ (recall Exercise 2.17). The reader should observe that the construction of the connection F(∇¹, . . . ,∇ⁿ) depends in principle not only on the connections

∇ⁱbut also on the typical fibers of the vector bundles. Show that, in fact, the con-nectionF(∇¹, . . . ,∇ⁿ)does not depend on the typical fibers of the vector bundles involved.

The components of a linear connection.

EXERCISE2.20. Letπ :E →M be a vector bundle endowed with a connec-tion∇andF be a vector subbundle ofE. Denote byq :E → E/F the quotient map. Show that the map:

Γ(T M)×Γ(F)3(X, )7−→q◦ ∇_X∈Γ(E/F)

isC^∞(M)-bilinear. Conclude that there exists a smooth sectionα^F ofLin(T M, F;E/F) such that:

∇_v+Fx =α^F_x v, (x)

∈Ex/Fx,

for allx∈M,v∈T_xM. We callα^F thesecond fundamental formof the subbun-dleF.

EXERCISE 2.21. Letπ : E → M be a vector bundle endowed with a semi-Riemannian structure gand a connection∇compatible withg. IfR denotes the curvature tensor of∇, show that for allx ∈M,v, w∈ TxM, the linear operator R_x(v, w) :E_x3e7→R_x(v, w)e∈E_xis anti-symmetric with respect tog_x, i.e.:

g_x R_x(v, w)e, e⁰

=−g_x e, R_x(v, w)e⁰ , for alle, e⁰ ∈E_x.

EXERCISE2.22. Let(M, g)be a semi-Riemannian manifold. Show that there exists a unique connection∇onM which is both symmetric and compatible with the semi-Riemannian metricg; such connection is defined by the equality:

(2.11) g(∇_XY, Z) = ¹₂

X g(Y, Z)

+Y g(Z, X)

−Z g(X, Y)

−g X,[Y, Z]

+g Y,[Z, X]

+g Z,[X, Y] , and is called theLevi-Civita connectionof the semi-Riemannian manifold(M, g).

Formula (2.11) is known asKoszul formula.

Relating connections with principal subbundles.

EXERCISE 2.23. LetΠ : P → M be aG-principal bundle endowed with a connectionHor(P)and letQbe anH-principal subbundle ofP; denote byωthe connection form ofHor(P). Show that the following conditions are equivalent:

• Horp(P)⊂TpQ, for allp∈Q;

• T_pQ= Hor_p(P)⊕Ver_p(Q), for allp∈Q;

• the1-formω|_Qtakes values inh, i.e.,ωp(TpQ)⊂h, for allp∈Q;

• there exists a connection on the principal bundleQsuch that the inclusion mapQ→P is connection preserving;

• the isomorphism (2.10.1) carriesTpQontoTxM⊕h, for allx∈M and allp∈Q_x.

The inner torsion of aG-structure.

EXERCISE 2.24. Let π : E → M be a vector bundle with typical fiber E0

endowed with a connection ∇, E1 be a real vector space andi : E1 → E0 be a linear isomorphism. LetGbe a Lie subgroup ofGL(E₀)andP ⊂ FR_E0(E) be a G-structure on E. Thenγi(P) ⊂ FRE1(E) is a I_i⁻¹(G)-structure on E (see Exercises 1.61 and 1.47). Show that the inner torsion ofP is equal to the inner torsion ofγ_i(P).

No documento The theory of connections and G-structures. Applications to affine and isometric immersions Paolo Piccione Daniel V. Tausk (páginas 165-177)