Relating linear connections with principal connections

DEFINITION 2.4.10. Given a connection ∇on the tangent bundle T M, the torsion tensorof∇is the mapT :Γ(T M)×Γ(T M)→Γ(T M)defined by:

T(X, Y) =∇_XY − ∇_YX−[X, Y],

for allX, Y ∈Γ(T M). More generally, if∇is a connection on an arbitrary vector bundleπ : E → M and ifι : T M → E is a vector bundle morphism then the ι-torsion tensorof∇is the mapT^ι:Γ(T M)×Γ(T M)→Γ(E)defined by:

T^ι(X, Y) =∇_X ι(Y)

− ∇_Y ι(X)

−ι [X, Y] ,

for allX, Y ∈Γ(T M). A connection∇onT M whose torsion tensorT is identi-cally zero is said to besymmetric.

Clearly, ifE =T Mandι:T M →T Mis the identity thenT^ι =T. It is easy to check that theι-torsion tensorT^ιisC^∞(M)-bilinear and thus, for eachx∈M, it defines a bilinear mapT_x^ι :T_xM×T_xM →E_x(see Exercise 1.63). Obviously theι-torsion tensor is anti-symmetric.

2.5. RELATING LINEAR CONNECTIONS WITH PRINCIPAL CONNECTIONS 123

Now lets:U →FRE0(E)be a smooth localE0-frame of the vector bundleE and let˜:U →E₀denote the representation of a smooth local section:U →E ofEwith respect tos; then:

(C^E)⁻¹◦=q◦(s,˜),

where q : FRE0(E) ×E0 → FRE0(E)×_∼ E0 denotes the quotient map. The representation of(C^E)⁻¹◦with respect tosis also equal to˜(see Example 1.5.10).

Using equality (2.3.7) we obtain:

(2.5.3) ∇_v (C^E)⁻¹◦

= [s(x),d˜x(v) + ¯ωx(v)·˜(x)],

for allx ∈ U and allv ∈ T_xM, whereω¯ denotes the representation with respect tosof the connection formωcorresponding toHor FRE0(E)

. From (2.5.2) and (2.5.3) we get:

(2.5.4) ∇_v=s(x)

d˜_x(v) + ¯ω_x(v)·˜(x) , for all∈Γ(E|_U), allx∈U and allv∈TxM. If we set:

(2.5.5) Γx(v) =I_s(x) ω¯x(v)

=s(x)◦ω¯x(v)◦s(x)⁻¹ ∈gl(Ex), for allx∈U,v∈T_xM, then formula (2.5.4) becomes (recall (2.4.5)):

∇_v= dI^s_v+ Γx(v)·(x).

If follows that ∇ is indeed a connection on the vector bundle E and that the Christoffel tensor Γ of ∇with respect to the smooth local E₀-frame sis given by (2.5.5). We have proven:

PROPOSITION2.5.2. Letπ :E →Mbe a vector bundle over a differentiable manifoldM with typical fiberE₀and letHor FR_E0(E)

be a principal connec-tion on the frame bundle FR_E0(E); denote byHor(E) the induced generalized connection onE. The covariant derivative operator∇corresponding toHor(E) is a linear connection on the vector bundleE; moreover, ifs:U → FRE0(E)is a smooth localE₀-frame ofE then the Christoffel tensor of∇with respect to the smooth localE0-framesand the representation ω¯ = s^∗ωof the connection form ωofHor FR_E0(E)

with respect tosare related by equality(2.5.5).

As a converse to Proposition 2.5.2, we will now show that every linear connec-tion∇onEis induced by a unique principal connection on the principal bundle of frames ofE.

REMARK2.5.3. If U is an open subset of M andHor FRE0(E)

is a con-nection onFRE0(E)then we have a corresponding connectionHor FRE0(E)|_U onFR_E0(E)|_U = FR_E0(E|_U)(see Example 2.2.8). Clearly, if∇is the connec-tion onEassociated toHor FR_E0(E)

then the connection onE|_U associated to Hor FR_E0(E)|_U

is just∇^U (recall Lemma 2.4.4).

PROPOSITION 2.5.4. Let π : E → M be a vector bundle with typical fiber E₀. For every linear connection∇on the vector bundleE there exists a unique principal connectionHor FR_E0(E)

on the principal bundle of framesFR_E0(E)

such that∇is the covariant derivative operator corresponding to the induced gen-eralized connectionHor(E)onE.

PROOF. Using the results of Exercises 2.6, 2.15 and Remark 2.5.3, it is easy to see that it suffices to prove the proposition in the case where the frame bundle FRE0(E) admits a globally defined smooth local sections : M → FRE0(E).

Let us therefore assume that such globally defined smooth local sectionsexists.

Let Γ denote the Christoffel tensor of ∇with respect to s. Given a connection Hor FR_E0(E)

onFR_E0(E)with connection formω, let us denote byω¯the rep-resentation ofωwith respect tos. Then∇is associated withHor FR_E0(E)

if and only if (2.5.5) holds, for allx ∈ M. But (2.5.5) defines a unique smoothgl(E0 )-valued1-form onM and Lemma 2.2.10 says that there exists a unique connection formωonFR_E0(E)withω¯ =s^∗ω. The conclusion follows.

COROLLARY 2.5.5. Let π : E → M be a vector bundle with typical fiber E0. Given a linear connection ∇ on E then there exists a unique generalized connectionHor(E)onEwhose covariant derivative operator is∇.

PROOF. The existence follows from Proposition 2.5.4 and the uniqueness fol-lows from Corollary 2.1.6, keeping in mind the fact that the submersionπ has the

global extension property (see Exercise 1.62).

COROLLARY2.5.6. Letπ :E →M be a vector bundle with typical fiberE₀. IfHor(E)is a generalized connection onE whose covariant derivative operator

∇ is a linear connection on E then there exists a unique principal connection Hor FRE0(E)

on the principal bundle of framesFRE0(E)such thatHor(E)is induced byHor FRE0(E)

.

The result of Propositions 2.5.2, 2.5.4 and of Corollaries 2.5.5 and 2.5.6 can be summarized as follows: the set of linear connections on a vector bundleEis in one-to-one correspondence with a subset of the set of all generalized connections onE. Such subset of the set of generalized connections onEis precisely the set of generalized connections that are induced by principal connections onFR_E0(E).

Moreover, there is also a one-to-one correspondence between the set of principal connections onFRE0(E)and the set of generalized connections onEwhose co-variant derivative operator is a linear connection; in particular, there is a one-to-one correspondence between the set of principal connections onFRE0(E)and the set of linear connections onE.From now on, we use such one-to-one correspondence to identify the set of linear connections onEwith a subset of the set of generalized connections onE.

EXAMPLE 2.5.7. Let M be a differentiable manifold, E0 be a real finite-dimensional vector space and consider the trivial vector bundle E = M ×E₀. Its principal bundle ofE0-frames is the trivial principal bundle:

P =M×GL(E0).

We claim that the canonical connection dI of E (see Example 2.4.5) is induced by the canonical connection of P (see Example 2.2.7). To prove the claim, let

2.5. RELATING LINEAR CONNECTIONS WITH PRINCIPAL CONNECTIONS 125

s:M →P be the smooth section defined bys(x) = (x,Id), whereId∈GL(E0) denotes the identity map ofE₀. Obviously the connectiondI^s(Example 2.4.6) on E is the canonical connection of the trivial bundleE. If∇is the connection on E induced by the trivial connection on P then ∇ = dI^s + Γ, where Γ denotes the Christoffel tensor of ∇with respect to s. We have to check thatΓ = 0. If ω is the connection form of the trivial connection ofP then it is easy to see that

¯

ω=s^∗ω = 0and henceΓ = 0, by formula (2.5.5).

LEMMA2.5.8. Letπ : E → M be a vector bundle with typical fiberE₀,∇,

∇⁰ be connections onE andω, ω⁰ respectively be the connections forms of the corresponding connections on the principal bundleFR_E0(E). Sett =∇ − ∇⁰. If s:U →FRE0(E)is a smooth local section then the diagram:

gl(Ex)

TxM

tx

::v

vv vv vv vv

(s^∗(ω−ωHHH⁰))HHxHHHH$$

gl(E₀)

I_s(x)

OO

commutes, for allx∈U, wheret_x :T_xM →gl(E_x)denotes the mapv7→t_x(v,·) andI_s(x)denotes conjugation by the linear isomorphisms(x) :E0→Ex.

PROOF. LetΓ,Γ⁰denote the Christoffel tensors with respect tosof∇and∇⁰, respectively. Clearly,t = Γ−Γ⁰. The conclusion is obtained immediately using

(2.5.5).

EXAMPLE 2.5.9 (linear connection induced on an associated vector bundle).

LetΠ : P → M be a G-principal bundle,E₀ be a real finite-dimensional vec-tor space and ρ : G → GL(E₀) be a smooth representation of G on E₀, so that the associated bundle P ×_G E0 is a vector bundle (recall Example 1.5.5).

Given a principal connectionHor(P)on the principal bundleP, we obtain a prin-cipal connectionHor FRE0(P×_GE0)

onFRE0(P ×_GE0)by taking the push-forward (recall Definition 2.2.14) of Hor(P) by the morphism of principal bun-dles H : P → FRE0(P ×_G E0) defined in (1.5.3). The principal connection Hor FR_E0(P×_GE₀)

therefore induces a linear connection∇on the vector bun-dleP ×_GE0.

Notice that the principal connectionHor(P)ofPinduces an associated gener-alized connectionHor(P ×_GE0)on the associated bundleP×_GE0as explained in Section 2.3. We claim that ∇ is precisely the covariant derivative operator of such generalized connection. To prove the claim, we have to check that if FR_E0(P ×_G E0) ×_∼ E0 is endowed with the generalized connection associated toHor FRE0(P ×_GE0)

and if P ×_GE0 is endowed with the generalized con-nection associated toHor(P)then the contraction map (1.5.4) is connection pre-serving. But this follows from the observation that the contraction map (1.5.4) is

the inverse of the induced mapHb(recall Example 1.5.5) and from the fact thatHbis connection preserving (recall Corollary 2.3.4).

2.5.1. Connection preserving morphisms of vector bundles. Since linear connections on vector bundles are particular cases of generalized connections then it makes sense to talk about connection preserving maps between vector bundles endowed with linear connections (recall Definition 2.1.3).

LEMMA2.5.10. LetE,E⁰be vector bundles endowed with linear connections

∇and∇⁰, respectively. LetL : E → E⁰ be a morphism of vector bundles. The following conditions are equivalent:

(a) Lis connection preserving;

(b) ∇⁰_v(L◦) =L(∇_v), for allv∈T M and all∈Γ(E).

Moreover, ifE andE⁰ have the same typical fiberE₀ and ifLis an isomorphism of vector bundles then (a), (b) are also equivalent to:

(c) the morphism of principal bundlesL∗ : FRE0(E) → FRE0(E⁰)is con-nection preserving, whereFR_E0(E)andFR_E0(E⁰)are endowed with the unique principal connections that induces the linear connections∇and

∇⁰, respectively.

PROOF. The equivalence between (a) and (b) follows from the equivalence between (a) and (d) in Lemma 2.1.5, by observing that the projection of a vector bundle has the global extension property (see Exercise 1.62). To prove the equiv-alence between (a) and (c), consider the commutative diagram (1.5.6). Since the contraction maps C^E andC^E0 are connection preserving diffeomorphisms, it fol-lows thatLis connection preserving if and only ifLc∗is. Finally, since the action ofGL(E₀)onE₀is effective, it follows from Corollary 2.3.4 thatLc∗is connection

preserving if and only ifL∗ is.

EXAMPLE2.5.11. Letπ : E → M be a vector bundle with typical fiberE₀ ands:U →FRE0(E)be a smooth localE0-frame ofE. The corresponding trivi-alizationsˇ:U×E₀→E|_Uis a vector bundle isomorphism (see Example 1.5.13);

if the trivial vector bundleU×E0is endowed with its canonical connectiondI(see Example 2.4.5) andE|_U is endowed with the connectiondI^s (see Example 2.4.6) thensˇis connection preserving.

EXAMPLE2.5.12. Letπ : E → M be a vector bundle with typical fiberE₀ endowed with a connection∇and lets : U → FRE0(E)be a smooth localE0 -frame of E. Denote by ω the connection form of the connection on FR_E0(E) associated to∇ and setω¯ = s^∗ω. Then ω¯ is a smoothLin(E0)-valued covari-ant 1-tensor field on U that can be identified with a C^∞(U)-bilinear map from Γ(T M|_U)×Γ(U×E0)toΓ(U×E₀)(recall Examples 1.6.31 and 1.6.33). IfE|_U is endowed with the connection∇^Uand the trivial vector bundleU×E₀is endowed with the connectiondI + ¯ωthen it follows from (2.5.4) that the local trivialization ˇ

s:U ×E₀ →E|_U is a connection preserving vector bundle isomorphism.

EXAMPLE 2.5.13. Let π : E → M be a vector bundle with typical fiber E₀ endowed with a linear connection∇; denote byHor FR_E0(E)

the principal