Chapter 2. The theory of connections
2.1. The general concept of connection
Letπ :E →Mbe a vector bundle with typical fiberE0and let∈Γ(E)be a smooth section ofE. IfE =M×E0is the trivial vector bundle overM thenis of the form(x) = x,˜(x)
, where˜:M →E0 is a smooth map; let us identity the smooth sectionofE =M ×E0 with the smooth map˜: M → E0. Given a point x ∈ M and a tangent vector v ∈ TxM, we can consider the directional derivatived˜(x)·v of˜at the point x, in the direction ofv. In general, ifE is an arbitrary vector bundle, what sense can be made of the directional derivative of a smooth section ∈ Γ(E) at a point x ∈ M, in the direction of a vector v ∈ TxM? Let us first approach the problem by considering a smooth localE0 -frames:U → FRE0(E)withx∈U. Let˜:U →E0 denote the representation of|U with respect tos. The directional derivatived˜(x)·vis an element of the typical fiber E0 and it corresponds via the isomorphisms(x) : E0 → Ex to a vector of the fiberEx; an apparently reasonable attempt at defining the directional derivative ofat the pointxin the direction ofvis:
directional derivative ofat the pointxin the direction ofv
=s(x) d˜(x)·v . Of course, in order to check that such definition makes sense, one has to look at what happens when another smooth local E0-frame s0 : V → FRE0(E) with x ∈V is chosen. Letg :U ∩V → GL(E0)denote the transition map froms0to s, so thats(y) =s0(y)◦g(y), for ally∈U∩V; then:
(y) =s(y)·(y) =˜ s0(y)· g(y)·˜(y) ,
for ally∈U ∩V, so that the representation˜0of|V with respect tos0satisfies:
˜
0(y) =g(y)·˜(y), for ally∈U ∩V. Then:
d˜0(x)·v= dg(x)·v
·˜(x) +g(x)· d˜(x)·v , and:
s0(x) d˜0(x)·v
=s(x)
g(x)−1
dg(x)·v
·(x)˜
+s(x) d˜(x)·v . The presence of the first term in the righthand side of the equality above shows that our plan for defining the directional derivative of a smooth section didn’t work. Let us look at the problem from a different angle. If : M → E is a
96
2.1. THE GENERAL CONCEPT OF CONNECTION 97
smooth section ofEthen for everyx∈Mwe can consider the differentiald(x), which is a linear map fromTxMtoT(x)E; for everyv∈TxM, we have therefore d(x)·v∈T(x)E. In the case thatE=M×E0is the trivial bundle overM then is of the form(x) = x,˜(x)
and:
d(x)·v= v,d˜(x)·v
∈T(x)E=TxM⊕E0.
Hence, in the case of the trivial bundle, the object that we wish to call the direc-tional derivative ofat the pointxin the direction ofvis the second coordinate of the vectord(x)·v. Ifπ :E →M is a general vector bundle thend(x)·vis just an element ofT(x)Eand it makes no sense to talk about the “second coordinate”
ofd(x)·v. Notice that, sinceπ◦is the identity map ofM, we have:
dπ(x) d(x)·v
=v,
so that, just in the case of the trivial bundle, the vectord(x)·vcontainsvas one of its components. The difficulty here is that there is no canonical way of extracting the “other component” from d(x) ·v. More precisely, the difficulty is that we don’t have a direct sum decomposition TxM ⊕E0 ofT(x)E just like we had in the case of the trivial bundleM ×E0. We have a canonical subspaceVer(x)Eof T(x)E(recall Definition 1.5.6) but such subspace has no canonical complement in the case of a general vector bundleE.
The problems we have encountered in the attempts to define a notion of di-rectional derivative for sections of an arbitrary vector bundle indicate that indeed no canonical notion of directional derivative for sections of general vector bundles exists. In order to define such a notion, the vector bundleEhas to be endowed with some additional structure. The additional structure onEthat will allow us to define a notion of directional derivative for smooth sections ofE is what we shall call a connectiononE. In order to make this definition precise, we start by considering the problem of lack of a natural complement for the vertical spaceVere(E)in the tangent space to the total spaceTeE. Let us give some definitions.
DEFINITION2.1.1. LetE,M be differentiable manifolds and letπ :E →M be a smooth submersion. Givene ∈ E then the spaceKer dπ(e)
is called the vertical subspaceofTeEat the pointewith respect to the submersionπ; assuming that the submersionπ is fixed by the context, we denote the vertical subspace by Vere(E). A subspace H of TeE is called horizontal with respect to π if it is a complement ofVere(E)inTeE, i.e., if:
TeE =H⊕Vere(E).
A distributionHon the manifoldEis calledhorizontalwith respect toπifHeis a horizontal subspace ofTeEfor everye∈ E. A smooth horizontal distribution onE will also be called ageneralized connectiononE (with respect toπ).
Notice that for allx∈M,π−1(x)is a smooth submanifold ofEand for every e∈π−1(x)we have:
Vere(E) =Te π−1(x) .
We set:
Ver(E) = [
e∈E
Vere(E)⊂TE.
The result of Exercise 1.64 says thatVer(E)is a smooth distribution onE. We call it thevertical distributiononEor also thevertical bundleofEdetermined byπ.
Notice that a subspaceHofTeEis horizontal with respect toπif and only if the restriction ofdπ(e)toHis an isomorphism ontoTπ(e)M(see Exercise 2.1).
When a horizontal distribution on E is fixed by the context we will usually denote it byHor(E); then:
(2.1.1) TeE= Hore(E)⊕Vere(E),
for alle∈ E. We denote bypver :TE →Ver(E)(resp.,phor :TE →Hor(E)) the map whose restriction toTeEis equal to the projection onto the second coordinate (resp., the first coordinate) corresponding to the direct sum decomposition (2.1.1), for alle∈ E. We callpver(resp.,phor) thevertical projection(resp., thehorizontal projection) determined by the horizontal distributionHor(E). Notice that ifHor(E) is a smooth distribution then the projectionspverandphorare morphisms of vector bundles; in this case, we also callHor(E)thehorizontal bundleofE.
DEFINITION2.1.2. LetE,M be differentiable manifolds and letπ :E →M be a smooth submersion. By alocal sectionofπwe mean a map:U → Edefined on an open subset U ofM such thatπ ◦is the inclusion map ofU inM. Let Hor(E)be a generalized connection onE. If:U → E is a smooth local section ofπ then, givenx ∈U,v ∈TxM, thecovariant derivativeofat the pointxin the direction ofvwith respect to the generalized connectionHor(E)is denoted by
∇vand it is defined by:
(2.1.2) ∇v=pver d(x)·v
∈Ver(x)(E);
we call∇thecovariant derivative operatorassociated to the generalized connec-tionHor(E). Givenx∈U, if∇v= 0, for allv∈TxM then the local sectionis said to beparallel atxwith respect toHor(E); ifis parallel at everyx ∈U we say simply thatisparallelwith respect toHor(E).
Clearly the covariant derivative∇vis linear inv. Moreover,is parallel atx with respect toHor(E)if and only if:
dx(TxM) = Hor(x)E.
DEFINITION2.1.3. Letπ :E →M,π0 :E0 → M be smooth submersions; a mapφ:E → E0 is said to befiber preservingif:
π0◦φ=π.
Let Hor(E), Hor(E0) be generalized connections on E and E0 respectively. A smooth mapφ:E → E0is said to beconnection preservingif it is fiber preserving and:
(2.1.3) dφe Hore(E)
= Horφ(e)(E0), for alle∈ E.
2.1. THE GENERAL CONCEPT OF CONNECTION 99
Clearly the composition of fiber preserving (resp., connection preserving) maps is also fiber preserving (resp., connection preserving). Moreover, the inverse of a bijective fiber preserving map (resp., of a smooth connection preserving diffeomor-phism) is also fiber preserving (resp., connection preserving).
Observe that ifφ : E → E0 is fiber preserving and if : U → E is a local section ofπthenφ◦:U → E0is a local section ofπ0. Ifφ:E → E0is a smooth fiber preserving map then for allx∈Mand alle∈π−1(x)the following diagram commutes:
(2.1.4)
TeE dφe //
dπEEeEEEEEE""
E Tφ(e)E0
dπφ(e)0
zzvvvvvvvvv
TxM In particular, we have:
(2.1.5) dφe Vere(E)
⊂Verφ(e)(E0).
DEFINITION 2.1.4. A smooth submersion π : E → M is said to have the global extension property if for every smooth local section : U → E ofπ and everyx∈U there exists a smooth global section¯:M → E such thatand¯are equal on some neighborhood ofxcontained inU.
The result of Exercise 1.62 shows that the projection of a vector bundle has the global extension property.
LEMMA2.1.5. Letπ :E → M,π0 :E0 →M be smooth submersions and let Hor(E),Hor(E0)be generalized connections onE andE0 respectively. Denote by
∇and∇0respectively the covariant derivative operators corresponding toHor(E) andHor(E0). Given a smooth fiber preserving mapφ:E → E0then the following conditions are equivalent:
(a) φis connection preserving;
(b) dφe Hore(E)
⊂Horφ(e)(E0), for alle∈ E; (c) for any smooth local section:U → Eofπ, it is:
(2.1.6) ∇0v(φ◦) = dφ(x)(∇v), for allx∈U and allv∈TxM.
Ifπ : E → M has the global extension property then conditions (a), (b) and (c) are also equivalent to:
(d) for any smooth global section:M → Eofπ, equality(2.1.6)holds, for allx∈M and allv∈TxM.
PROOF. The equivalence between (a) and (b) follows from the commutativity of diagram (2.1.4), applying the results of Exercises 2.2 and 2.3. Now assume (a) and let us prove (c). Denote bypverandp0verthe vertical projections determined by Hor(E)and byHor(E0), respectively. From (2.1.3) and (2.1.5) we get that:
p0ver dφe(ζ)
= dφe pver(ζ) ,
for alle∈ E and allζ ∈TeE. Thus, given a smooth local section:U → Eofπ, we have:
∇0v(φ◦) =p0ver
dφ(x) dx(v)
= dφ(x)
pver dx(v)
= dφ(x)(∇v), for allx ∈ U and allv ∈ TxM. This proves (c). Conversely, assume (c) and let us prove (a). Let e ∈ E be fixed and set π(e) = x ∈ M. Choose an arbitrary submanifoldSofEwithe∈SandTeS= Hore(E). Since:
d(π|S)e= dπe|TeS :TeS −→TxM
is an isomorphism then, possibly taking a smallerS, we may assume thatπ|Sis a smooth diffeomorphism onto an open neighborhoodU ofxinM. Then:
= (π|S)−1:U −→ E
is a smooth local section of π, (x) = e and is parallel at x with respect to Hor(E). Now (2.1.6) implies thatφ◦is parallel atxwith respect toHor(E0)and hence:
dφe Hore(E)
= (dφe◦dx)(TxM) = d(φ◦)x(TxM) = Horφ(e)(E0), proving (a). Finally, assume thatπ : E → M has the global extension property and let us prove that (a), (b) and (c) are all equivalent to (d). It is obvious that (c) implies (d). The proof of the fact that (d) implies (a) can be done by repeating the same steps of our proof that (c) implies (a), keeping in mind that the smooth local section : U → E of π constructed in that proof can be replaced by a smooth
global section¯:M → E.
COROLLARY2.1.6. Letπ : E → M be a smooth submersion endowed with generalized connectionsHor(E) andHor0(E); denote by∇ and∇0 respectively the covariant derivative operators corresponding toHor(E)andHor0(E). If:
(2.1.7) ∇v=∇0v,
for every smooth local section : U → E of π and for every v ∈ T M|U then Hor(E) = Hor0(E). Moreover, ifπhas the global extension property and if(2.1.7) holds for every smooth global section:M → Eofπand for everyv∈T M then Hor(E) = Hor0(E).
PROOF. Apply Lemma 2.1.5 withφthe identity map ofE.
Let us go back to our discussion about directional derivatives of smooth sec-tions of a vector bundleπ : E → M. The projectionπ of the vector bundle is a smooth submersion and the notions of vertical space and local section given in Definitions 2.1.1 and 2.1.2 are consistent with the ones given in Section 1.5. If Hor(E)is a generalized connection onEthen for every smooth section∈Γ(E), every pointx ∈ M and every vectorv ∈ TxM, the covariant derivative ∇vis an element of the vertical space Ver(x)E, which is identified with the fiber Ex. Although the covariant derivative∇vis linear inv, it doesn’t have in general the
2.1. THE GENERAL CONCEPT OF CONNECTION 101
other “nice” properties that one would expect from a notion of directional deriva-tive; for instance, the covariant derivative∇vis not in general linear in. It turns out that for some generalized connectionsHor(E), the corresponding notion of co-variant derivative of smooth sections ofEsatisfies all the desirable properties. The difficulty is that it is not so easy to give a direct description of the properties that the generalized connectionHor(E)should satisfy in order that the corresponding covariant derivative∇satisfies all the desirable properties.
Our plan for developing the theory of connections is the following: we first study the notion of connection on principal bundles. A principal connection on a principal bundle is just a generalized connection on the total space that is invariant under the action of the structural group. We show how a principal connection on a principal bundle induces a generalized connection in any of its associated bundles.
In particular, ifE is a vector bundle, a principal connection on the principal bun-dle of framesFRE0(E)induces a generalized connectionHor(E)onE(recall the isomorphism given by the contraction map (1.5.1)). Looking at the situation from a different perspective, we will define the notion of linear connection on a vector bundleE simply by stating that a linear connection onEis the same as a covari-ant derivative operator ∇satisfying some natural properties. It will be seen that the covariant derivative operator determined by a generalized connectionHor(E) induced from a principal connection onFRE0(E)is indeed a linear connection on E; moreover, there is a one to one correspondence between the principal connec-tions on the principal bundleFRE0(E)and the linear connections∇on the vector bundleE.
2.1.1. Pull-back of generalized connections and submersions.
DEFINITION2.1.7. LetE,M,M0be differentiable manifolds,π :E →M be a smooth submersion and letf :M0 → M be a smooth map. By alocal section ofπ alongf we mean a mapε:U0 → E withπ◦ε=f|U0, whereU0 is an open subset ofM0. IfHor(E)is a generalized connection onE with respect toπand if ε:U0 → Eis a smooth local section ofπalongf then we set:
∇vε=pver dε(y)·v
∈Verε(y)(E),
for ally∈U0,v ∈TyM0and we call∇vεthecovariant derivativeofεat the point yin the direction ofv. Giveny ∈U0, if∇vε= 0, for allv ∈TyM0then the local sectionεis said to beparallel atywith respect toHor(E); ifεis parallel at every y∈U0we say simply thatεisparallelwith respect toHor(E).
Clearly the covariant derivative∇vεis linear inv. Moreover,εis parallel aty with respect toHor(E)if and only if:
dεy(TyM0)⊂Horε(y)E.
Ifπ : E → M,π0 : E0 → M are smooth submersions,φ : E → E0 is fiber preserving and ifε:U0 → Eis a local section ofπalong a mapf :M0→M then obviouslyφ◦ε:U → E0 is a local section ofπ0 alongf. We have the following analogue of (2.1.6):
LEMMA2.1.8. Letπ:E →M,π0 :E0 →Mbe smooth submersions endowed with generalized connectionsHor(E),Hor(E0), respectively,M0be a differentiable manifold, f : M0 → M be a smooth map,φ : E → E0 be a smooth connection preserving map andε:U0→ Ebe a local section ofπalongf. Then:
∇0v(φ◦ε) = dφε(y)(∇vε),
for ally∈U0,v∈TyM0, where∇,∇0denote respectively the covariant derivative operators with respect toHor(E)andHor(E0).
PROOF. It is analogous to the proof of (2.1.6) in Lemma 2.1.5.
LetE,M,M0be differentiable manifolds,π :E →Mbe a smooth submersion and letf :M0 →M be a smooth map. We set:
f∗E =
(y, e)∈M0× E :f(y) =π(e)
and we denote byπ1:f∗E →M0,f¯:f∗E → Erespectively the restriction tof∗E of the first and of the second projection of the cartesian productM0× E. Sinceπis a submersion, the result of Exercise 1.55 says thatf∗Eis a smooth submanifold of M0× E and that the triple(f∗E, π1,f¯)is the pull-back of(f, π, M, M0,E)in the category of differentiable manifolds and smooth maps. Sinceπ is a submersion, it follows easily from (1.19) that alsoπ1 :f∗E →M0 is a submersion. We call the submersionπ1 :f∗E →M0thepull-backof the submersionπ :E →M byfand we callf¯:f∗E → E thecanonical mapof the pull-backf∗E.
REMARK2.1.9. Giveny∈M0then:
π1−1(y) ={y} ×π−1 f(y)
; we thus identifyπ−11 (y) withπ−1 f(y)
in the obvious way. Under such identi-fication, the restriction to π−11 (y) of the canonical mapf¯is the identity map of π−1 f(y)
. In particular, for alle∈f∗E, we identify the vertical spaceVere(f∗E) with the vertical spaceVerf(e)¯ (E)and the restriction toVere(f∗E)of the differen-tiald ¯fewith the identity map ofVerf(e)¯ (E).
Clearly the composition on the left withf¯of a (smooth) local section of the submersionπ1 : f∗E →M0 is a (smooth) local section ofπ :E → M alongf. Conversely, using the property of pull-backs described in diagram (1.17), we see that ifε : U0 → E is a (smooth) local section ofπ : E → M alongf then there exists a unique (smooth) local section←−ε :U0 →f∗E ofπ1 :f∗E →M0such that f¯◦ ←−ε =ε. The situation is illustrated by the following commutative diagram:
f∗E f¯ //E
π
U0
←−
OO
=={
{{ {{ {{ {{
f //M