Chapter 9 Integration on Manifolds

Home , Metric connection, Orientability, Riemann integral, Volume form

Chapter 9

Integration on Manifolds

9.1 Integration in Rn As we said in Section 8.1, one of the raison d’être for differential forms is that they are the objects that can be integrated on manifolds. We will be integrating differential forms that are at least continuous (in most cases, smooth) and with compact support. In the case of n n forms, ω,onR ,thismeansthattheclosureoftheset, x R ωx =0,iscompact. { ∈ | } Similarly, for a form, ω ∗(M), where M is a manifod, the support, supp (ω), of ω is the ∈A M closure of the set, p M ωp =0 .Welet c∗(M)denotethesetofdifferentialformswith compact support on{ M∈ .If| M is a} smooth manifoldA of dimension n,ourultimategoalisto define a linear operator, n : c (M) R, A −→ M which generalizes in a natural way the usual integral on Rn. In this section, we assume that M = Rn,orsomeopensubsetofRn. Now, every n-form (with compact support) on Rn is given by ω = f(x) dx dx , x 1 ∧···∧ n where f is a smooth function with compact support. Thus, we set

ω = f(x)dx1 dxn, n n ··· R R where the expression on the right-hand side is the usual Riemann integral of f on Rn. Actually, we will need to integrate smooth forms, ω n(U), with compact support deﬁned ∈Ac on some open subset, U Rn (with supp(ω) U). However, this is no problem since we still have ⊆ ⊆ ω = f(x) dx dx , x 1 ∧···∧ n where f : U R is a smooth function with compact support contained in U and f can be → smoothly extended to Rn by setting it to 0 on Rn supp(f). We write ω for this integral. − V 309 310 CHAPTER 9. INTEGRATION ON MANIFOLDS

It is crucial for the generalization of the integral to manifolds to see what the change of variable formula looks like in terms of diﬀerential forms.

Proposition 9.1 Let ϕ: U V be a diﬀeomorphism between two open subsets of Rn. If the → n Jacobian determinant, J(ϕ)(x), has a constant sign, δ = 1 on U, then for every ω c (V ), we have ± ∈A

ϕ∗ω = δ ω. U V Proof .Weknowthatω can be written as ω = f(x) dx dx ,xV, x 1 ∧···∧ n ∈ where f : V R has compact support. From the example before Proposition 8.6, we have → (ϕ∗ω) = f(ϕ(y))J(ϕ) dy dy y y 1 ∧···∧ n = δf(ϕ(y)) J(ϕ) dy dy . | y| 1 ∧···∧ n On the other hand, the change of variable formula (using ϕ)is

f(x) dx dx = f(ϕ(y)) J(ϕ) dy dy , 1 ··· n | y| 1 ··· n ϕ(U) U so the formula follows. We will promote the integral on open subsets of Rn to manifolds using partitions of unity.

9.2 Integration on Manifolds

Intuitively, for any n-form, ω n(M), on a smooth n-dimensional oriented manifold, M, ∈Ac the integral, M ω,iscomputedbypatchingtogethertheintegralsonsmall-enoughopen subsets covering M using a partition of unity. If (U,ϕ) is a chart such that supp(ω) U, 1 n 1 ⊆ then the form (ϕ− )∗ω is an n-form on R and the integral, ϕ(U)(ϕ− )∗ω,makessense.The orientability of M is needed to ensure that the above integrals have a consistent value on overlapping charts.

Remark: It is also possible to deﬁne integration on non-orientable manifolds using densities but we have no need for this extra generality. Proposition 9.2 Let M be a smooth oriented manifold of dimension n. Then, there exists a unique linear operator, n : c (M) R, M A −→ n with the following property: For any ω c (M), if supp(ω) U, where (U,ϕ) is a positively oriented chart, then ∈A ⊆ 1 ω = (ϕ− )∗ω. ( ) † M ϕ(U) 9.2. INTEGRATION ON MANIFOLDS 311

Proof . First, assume that supp(ω) U,where(U,ϕ)isapositivelyorientedchart.Then, we wish to set ⊆ 1 ω = (ϕ− )∗ω. M ϕ(U) However, we need to prove that the above expression does not depend on the choice of the 1 chart. Let (V,ψ)beanotherchartsuchthatsupp(ω) V .Themap,θ = ψ ϕ− ,isa ⊆ ◦ diﬀeomorphism from W = ϕ(U V )toW = ψ(U V ) and, by hypothesis, its Jacobian determinant is positive on W .Since∩ ∩

1 1 supp ((ϕ− )∗ω) W, supp ((ψ− )∗ω) W , ϕ(U) ⊆ ψ(V ) ⊆ 1 1 1 1 and θ∗ (ψ− )∗ω =(ϕ− )∗ ψ∗ (ψ− )∗ω =(ϕ− )∗ω, Proposition 9.1 yields ◦ ◦ ◦ 1 1 (ϕ− )∗ω = (ψ− )∗ω, ϕ(U) ψ(V ) as claimed. In the general case, using Theorem 3.26, for every open cover of M by positively oriented charts, (Ui,ϕi), we have a partition of unity, (ρi)i I , subordinate to this cover. Recall that ∈ supp(ρ ) U ,iI. i ⊆ i ∈

Thus, ρiω is an n-form whose support is a subset of Ui.Furthermore,as i ρi =1,

ω = ρiω. i Deﬁne

I(ω)= ρiω, i Ui where each term in the sum is deﬁned by

1 ρiω = (ϕi− )∗ρiω, Ui ϕi(Ui) where (U ,ϕ)isthechartassociatedwithi I.ItremainstoshowthatI(ω)doesnot i i ∈ depend on the choice of open cover and on the choice of partition of unity. Let (Vj,ψj) be another open cover by positively oriented charts and let (θj)j J be a partition of unity ∈ subordinate to the open cover, (Vj). Note that

ρiθjω = ρiθjω, Ui Vj since supp(ρ θ ω) U V ,andas ρ =1and θ =1,wehave i j ⊆ i ∩ j i i j j ρiω = ρiθjω = ρiθjω = θjω, i Ui i,j Ui i,j Vj j Vj 312 CHAPTER 9. INTEGRATION ON MANIFOLDS

proving that I(ω)isindeedindependentoftheopencoverandofthepartitionofunity.The uniqueness assertion is easily proved using a partition of unity. The integral of Deﬁnition 9.2 has the following properties:

Proposition 9.3 Let M be an oriented manifold of dimension n. The following properties hold:

n (1) If M is connected, then for every n-form, ω c (M), the sign of M ω changes when the orientation of M is reversed. ∈A (2) For every n-form, ω n(M), if supp(ω) W , for some open subset, W , of M, then ∈Ac ⊆ ω = ω, M W where W is given the orientation induced by M.

n (3) If ϕ: M N is an orientation-preserving diﬀeomorphism, then for every ω c (N), we have → ∈A

ω = ϕ∗ω. N M Proof . Use a partition of unity to reduce to the case where supp(ω)iscontainedinthe domain of a chart and then use Proposition 9.1 and ( )fromProposition9.2. † The theory or integration developed so far deals with domains that are not general enough. Indeed, for many applications, we need to integrate over domains with boundaries.

9.3 Integration on Regular Domains and Stokes’ Theorem

Given a manifold, M,wedeﬁneaclassofsubsetswithboundariesthatcanbeintegratedon and for which Stokes’ Theorem holds. In Warner [147] (Chapter 4), such subsets are called regular domains and in Madsen and Tornehave [100] (Chapter 10) they are called domains with smooth boundary.

Deﬁnition 9.1 Let M be a smooth manifold of dimension n.Asubset,N M,iscalleda domain with smooth boundary (or codimension zero submanifold with boundary⊆ )iﬀforevery p M,thereisachart,(U,ϕ), with p U, such that ∈ ∈ ϕ(U N)=ϕ(U) Hn, ( ) ∩ ∩ ∗ where Hn is the closed upper-half space,

n n H = (x1,...,xn) R xn 0 . { ∈ | ≥ } 9.3. INTEGRATION ON REGULAR DOMAINS AND STOKES’ THEOREM 313

Note that ( ) is automatically satisﬁed when p is an interior or an exterior point of N, ∗ since we can pick a chart such that ϕ(U)iscontainedinanopenhalfspaceofRn deﬁned by either xn > 0orxn < 0. If p is a boundary point of N,thenϕ(p)hasitslastcoordinate equal to 0. If M is orientable, then any orientation of M induces an orientation of ∂N,the boundary of N.Thisfollowsfromthefollowingproposition:

Proposition 9.4 Let ϕ: Hn Hn be a diffeomorphism with everywhere positive Jacobian → determinant. Then, ϕ induces a diffeomorphism, Φ: ∂Hn ∂Hn, which, viewed as a diffeo- n 1 → morphism of R − also has everywhere positive Jacobian determinant.

Proof .Bytheinversefunctiontheorem,everyinteriorpointofHn is the image of an interior point, so ϕ maps the boundary to itself. If ϕ =(ϕ1,...,ϕn), then

Φ=(ϕ1(x1,...,xn 1, 0),...,ϕn 1(x1,...,xn 1, 0)), − − −

∂ϕn since ϕn(x1,...,xn 1, 0) = 0. It follows that (x1,...,xn 1, 0) = 0, for i =1,...,n 1, − ∂xi − − and as ϕ maps Hn to itself, ∂ϕn (x1,...,xn 1, 0) > 0. ∂xn − Now, the Jacobian matrix of ϕ at q = ϕ(p) ∂Hn is of the form ∈

∗ . dΦq . dϕq =  

 ∂ϕn∗  0 0 (q)  ··· ∂xn    and since ∂ϕn (q) > 0andbyhypothesisdet(dϕ ) > 0, we have det(dΦ ) > 0, as claimed. ∂xn q q n In order to make Stokes’ formula sign free, if H has the orientation given by dx1 dxn, n n ∧···∧ then H is given the orientation given by ( 1) dx1 dxn 1 if n 2and 1forn =1. − This choice of orientation can be explained in− terms of∧··· the∧ notion of outward≥ directed− tangent vector.

Deﬁnition 9.2 Given any domain with smooth boundary, N M,atangentvector, ⊆ w TpM, at a boundary point, p ∂N,isoutward directed iﬀthere is a chart, (U,ϕ), with ∈ ∈ n th p U and ϕ(U N)=ϕ(U) H and such that dϕp(w) has a negative n coordinate ∈ ∩ ∩ prn(dϕp(w)).

Let (V,ψ)beanotherchartwithp V .Then,thetransitionmap, ∈ 1 θ = ψ ϕ− : ϕ(U V ) ψ(U V ) ◦ ∩ → ∩ induces a map ϕ(U V ) Hn ψ(U V ) Hn ∩ ∩ −→ ∩ ∩ 314 CHAPTER 9. INTEGRATION ON MANIFOLDS

which restricts to a diﬀeomorphism

Θ: ϕ(U V ) ∂Hn ψ(U V ) ∂Hn. ∩ ∩ → ∩ ∩ n The proof of Proposition 9.4 shows that the Jacobian matrix of dθq at q = ϕ(p) ∂H is of the form ∈ ∗ . dΘq . dθq =  

 ∂θn∗  0 0 (q)  ··· ∂xn  ∂θn   1 with θ =(θ1,...,θn)andthat (q) > 0. As dψp = d(ψ ϕ− )q dϕp,weseethatforany ∂xn ◦ ◦ w TpM with prn(dϕp(w)) < 0, we also have prn(dψp(w)) < 0. Therefore, the negativity condition∈ of Deﬁnition does not depend on the chart at p.Thefollowingpropositionisthen easy to show:

Proposition 9.5 Let N M be a domain with smooth boundary where M is a smooth manifold of dimension n. ⊆ (1) The boundary, ∂N, of N is a smooth manifold of dimension n 1. − (2) Assume M is oriented. If n 2, there is an induced orientation on ∂N determined as follows: For every p ∂N,≥ if v T M is an outward directed tangent vector then a ∈ 1 ∈ p basis, (v2,...,vn) for Tp∂N is positively oriented iﬀthe basis (v1,v2,...,vn) for TpM is positively oriented. When n =1, every p ∂N has the orientation +1 iﬀfor every outward directed tangent vector, v T M,∈ the vector v is a positively oriented basis 1 ∈ p 1 of TpM.

n If M is oriented, then for every n-form, ω c (M), the integral N ω is well-deﬁned. More precisely, Proposition 9.2 can be generalized∈A to domains with a smooth boundary. This can be shown in various ways. In Warner, this is shown by covering N with special kinds of open subsets arising from regular simplices (see Warner [147], Chapter 4). In Madsen and Tornehave [100], it is argued that integration theory goes through for continuous n-forms with compact support. If σ is a volume form on M,thenforeverycontinuousfunctionwith compact support, f,themap f I (f)= fσ → σ M is a linear positive operator. By Riesz’ representation theorem, Iσ determines a positive Borel measure, µσ,whichsatisﬁes

fdµσ = fσ. M M Then, we can set

ω = 1N ω, N M 9.4. INTEGRATION ON RIEMANNIAN MANIFOLDS AND LIE GROUPS 315

where 1N is the function with value 1 on N and 0 outside N. Another way to proceed is to prove an extension of Proposition 9.1 using a slight generalization of the change of variable formula:

Proposition 9.6 Let ϕ: U V be a diﬀeomorphism between two open subsets of Rn and → assume that ϕ maps U Hn to V Hn. Then, for every smooth function, f : V R, with compact support, ∩ ∩ →

f(x)dx1 dxn = f(ϕ(y)) J(ϕ)y dy1 dyn. V Hn ··· U Hn | | ··· ∩ ∩ We now have all the ingredient to state Stokes’s formula. We omit the proof as it can be found in many places (for example, Warner [147], Chapter 4, Bott and Tu [19], Chapter 1, and Madsen and Tornehave [100], Chapter 10). The proof is fairly easy and it is not particularly illuminating, although one has to be very careful about matters of orientation. Theorem 9.7 (Stokes’ Theorem) Let N M be a domain with smooth boundary where M is a smooth oriented manifold of dimension⊆ n, give ∂N the orientation induced by M and let i: ∂N M be the inclusion map. For every diﬀerential form with compact support, n 1 → ω − (M), we have ∈Ac i∗ω = dω. ∂N N In particular, if N = M is a smooth oriented manifold with boundary, then

i∗ω = dω ∂M M and if M is a smooth oriented manifold without boundary, then

dω =0. M

Of course, i∗ω is the restriction of ω to ∂N and for simplicity of notation, i∗ω is usually written ω and Stokes’ formula is written

ω = dω. ∂N N 9.4 Integration on Riemannian Manifolds and Lie Groups

We saw in Section 8.6 that every orientable Riemannian manifold has a uniquely deﬁned volume form, VolM (see Proposition 8.26). given any smooth function, f,withcompact support on M, we deﬁne the integral of f over M by

f = f VolM . M M 316 CHAPTER 9. INTEGRATION ON MANIFOLDS

Actually, it is possible to deﬁne the integral, M f, even if M is not orientable but we do not need this extra generality. If M is compact, then 1 = Vol is the volume of M M M M M (where 1 is the constant function with value 1). M If M and N are Riemannian manifolds, then we have the following version of Proposition 9.3 (3):

Proposition 9.8 If M and N are oriented Riemannian manifolds and if ϕ: M N is an → orientation preserving diﬀeomorphism, then for every function, f C∞(M), with compact support, we have ∈ f Vol = f ϕ det(dϕ) Vol , N ◦ | | M N M where f ϕ det(dϕ) denotes the function, p f(ϕ(p)) det(dϕp) , with dϕp : TpM Tϕ(p)N. In particular,◦ | if ϕ is| an orientation preserving→ isometry| (see Deﬁnition| 7.11), then→

f Vol = f ϕ Vol . N ◦ M N M

We often denote M f VolM by M f(t)dt. If G is a Lie group, we know from Section 8.6 that G is always orientable and that n G possesses left-invariant volume forms. Since dim( g∗)=1ifdim(G)=n and since every left-invariant volume form is determined by its value at the identity, the space of left- invariant volume forms on G has dimension 1. If we pick some left-invariant volume form, ω,definingtheorientationofG,theneveryotherleft-invariantvolumeformisproportional to ω.Givenanysmoothfunction,f,withcompactsupportonG,wedefinetheintegral of f over G (w.r.t. ω) by f = fω. G G This integral depends on ω but since ω is defined up to some positive constant, so is the integral. When G is compact, we usually pick ω so that

ω =1. G

For every g G,asω is left-invariant, Lg∗ω = ω,soLg∗ is an orientation-preserving diﬀeomorphism and∈ by Proposition 9.3 (3),

fω = Lg∗(fω), G G so we get

f = fω = L∗(fω)= L∗fL∗ω = (f L )ω = f L . g g g ◦ g ◦ g G G G G G G 9.4. INTEGRATION ON RIEMANNIAN MANIFOLDS AND LIE GROUPS 317

The property f = f L ◦ g G G is called left-invariance. It is then natural to ask when our integral is right-invariant, that is, when

f = f R . ◦ g G G

Observe that Rg∗ω is left-invariant, since

Lh∗ Rg∗ω = Rg∗Lh∗ ω = Rg∗ω.

It follows that Rg∗ω is some constant multiple of ω,andso,thereisafunction,∆: G R such that →

Rg∗ω = ∆(g)ω. One can check that ∆issmoothandwelet ∆(g)= ∆(g) . | | Clearly, ∆(gh)=∆(g)∆(h), so ∆is a homorphism of G into R+. The function ∆is called the modular function of G. Now, by Proposition 9.3 (3), as Rg∗ is an orientation-preserving diﬀeomorphism,

fω = R∗(fω)= R∗f R∗ω = (f R )∆(g)ω g g ◦ g ◦ g G G G G or, equivalently, 1 fω =∆(g− ) (f R )ω. ◦ g G G It follows that if ωl is any left-invariant volume form on G and if ωr is any right-invariant volume form in G,then 1 ωr(g)=c∆(g− )ωl(g), 1 for some constant c =0.Indeed,ifletω(g)=∆(g− )ω (g), then l 1 Rh∗ ω = ∆((gh)− )Rh∗ ωl 1 1 = ∆(h)− ∆(g− )∆(h)ωl 1 = ∆(g− )ωl,

1 which shows that ω is right-invariant and thus, ωr(g)=c∆(g− )ωl(g), as claimed (since 1 1 ∆(g− )= ∆(g− )). Actually, it is not diﬃcult to prove that ± 1 ∆(g)= det(Ad(g− )) . | | 318 CHAPTER 9. INTEGRATION ON MANIFOLDS

n 1 For this, recall that Ad(g)=d(Lg Rg− )1.Foranyleft-invariantn-form, ω g∗,we claim that ◦ ∈ 1 (Rg∗ω)h = det(Ad(g− ))ωh, 1 which shows that ∆(g)= det(Ad(g− )) .Indeed,forallv ,...,v T G, we have | | 1 n ∈ h

(Rg∗ω)h(v1,...,vn)

= ωhg(d(Rg)h(v1),...,d(Rg)h(vn))

= ω (d(L L 1 R L L 1 ) (v ),...,d(L L 1 R L L 1 ) (v )) hg g ◦ g− ◦ g ◦ h ◦ h− h 1 g ◦ g− ◦ g ◦ h ◦ h− h n = ω (d(L L L 1 R L 1 ) (v ),...,d(L L L 1 R L 1 ) (v )) hg h ◦ g ◦ g− ◦ g ◦ h− h 1 h ◦ g ◦ g− ◦ g ◦ h− h n 1 1 1 1 = ωhg(d(Lhg Lg− Rg Lh− )h(v1),...,d(Lhg Lg− Rg Lh− )h(vn)) ◦ ◦ 1 ◦ ◦ ◦ 1◦ 1 1 = ωhg d(Lhg)1(Ad(g− )(d(Lh− )h(v1))),...,d(Lhg)1(Ad(g− )(d(Lh− )h(vn))) 1 1 =(L∗ ω) Ad(g− )(d(L 1 ) (v )),...,Ad(g− )(d(L 1 ) (v )) hg 1 h− h 1 h− h n 1 1 = ω Ad(g− )(d(L 1 ) (v )),...,Ad(g− )(d(L 1 ) (v )) 1 h− h 1 h− h n 1 = det(Ad(g− )) ω d(L 1 ) (v ),...,d(L 1 ) (v ) 1 h− h 1 h− h n 1 = det(Ad(g− )) (L∗ 1 ω)h(v1,...,vn) h− 1 = det(Ad(g− )) ωh(v1,...,vn),

where we used the left-invariance of ω twice. Consequently, our integral is right-invariant iﬀ ∆ 1onG.Thus,ourintegralisnot always right-invariant. When it is, i.e. when ∆ 1on≡ G,wesaythatG is unimodular. This happens in particular when G is compact, since≡ in this case,

1= ω = 1Gω = ∆(g)ω =∆(g) ω =∆(g), G G G G for all g G.Therefore,foracompact Lie group, G,ourintegralisbothleftandright invariant.∈ We say that our integral is bi-invariant.

As a matter of notation, the integral G f = G fω is often written G f(g)dg.Then, left-invariance can be expressed as

f(g)dg = f(hg)dg G G and right-invariance as f(g)dg = f(gh)dg, G G for all h G.Ifω is left-invariant, then it can be shown that ∈

1 1 f(g− )∆(g− )dg = f(g)dg. G G 9.4. INTEGRATION ON RIEMANNIAN MANIFOLDS AND LIE GROUPS 319

Consequently, if G is unimodular, then

1 f(g− )dg = f(g)dg. G G

In general, if G is not unimodular, then ωl = ωr.Asimpleexampleisthegroup,G,of aﬃne transformations of the real line, which can be viewed as the group of matrices of the form ab A = , a, b, R,a=0. 01 ∈ Then, it it is easy to see that the left-invariant volume form and the right-invariant volume form on G are given by dadb dadb ω = ,ω= , l a2 r a 1 and so, ∆(A)= a− . | |

Remark: By the Riesz’ representation theorem, ω deﬁnes a positive measure, µω,which satisﬁes

fdµω = fω. G G Using what we have shown, this measure is left-invariant. Such measures are called left Haar measures and similarly, we have right Haar measures.Itcanbeshownthateverytwoleft Haar measures on a Lie group are proportional (see Knapp, [89], Chapter VIII). Given a left Haar measure, µ, the function, ∆, such that

µ(Rgh)=∆(g)µ(h)

for all g, h G is the modular function of G. However, beware that some authors, including ∈ 1 Knapp, use ∆(g− )insteadof∆(g). As above, we have

1 ∆(g)= det(Ad(g− )) . | |

1 Beware that authors who use ∆(g− )insteadof∆(g), give a formula where Ad(g)appears 1 instead of Ad(g− ). Again, G is unimodular iﬀ ∆ 1. It can be shown that compact, semisimple, reductive and nilpotent Lie groups are≡ unimodular (for instance, see Knapp, [89], Chapter VIII). On such groups, left Haar measures are also right Haar measures (and vice versa). In this case, we can speak of Haar measures on G. For more details on Haar measures on locally compact groups and Lie groups, we refer the reader to Folland [54] (Chapter 2), Helgason [72] (Chapter 1) and Dieudonn´e[47] (Chapter XIV). 320 CHAPTER 9. INTEGRATION ON MANIFOLDS Chapter 10

Distributions and the Frobenius Theorem

10.1 Tangential Distributions, Involutive Distributions

Given any smooth manifold, M,(ofdimensionn)foranysmoothvectorfield,X,onM, we know from Section 3.5 that for every point, p M,thereisauniquemaximalintegral curve through p.Furthermore,anytwodistinctintegralcurvesdonotintersecteachother∈ and the union of all the integral curves is M itself. A nonvanishing vector field, X,canbe viewed as the smooth assignment of a one-dimensional vector space to every point of M, namely, p RXp TpM,whereRXp denotes the line spanned by Xp. Thus, it is natural to consider→ the more⊆ general situation where we fix some integer, r,with1 r n and we ≤ ≤ have an assignment, p D(p) TpM,whereD(p)issomer-dimensional subspace of TpM such that D(p)“variessmoothly”with→ ⊆ p M.Isthereanotionofintegralmanifoldfor such assignments? Do they always exist? ∈ It is indeed possible to generalize the notion of integral curve and to define integral manifolds but, unlike the situation for vector fields (r =1),noteveryassignment,D,as above, possess an integral manifold. However, there is a necessary and sufficient condition for the existence of integral manifolds given by the Frobenius Theorem. This theorem has several equivalent formulations. First, we will present a formulation in terms of vector fields. Then, we will show that there are advantages in reformulating the notion of involutivity in terms of differential ideals and we will state a differential form version of the Frobenius Theorem. The above versions of the Frobenius Theorem are “local”. We will briefly discuss the notion of foliation and state a global version of the Frobenius Theorem. Since Frobenius’ Theorem is a standard result of differential geometry, we will omit most proofs and instead refer the reader to the literature. A complete treatment of Frobenius’ Theorem can be found in Warner [147], Morita [114] and Lee [98]. Our first task is to define precisely what we mean by a smooth assignment, p D(p) → ⊆ TpM,whereD(p)isanr-dimensional subspace.

321 322 CHAPTER 10. DISTRIBUTIONS AND THE FROBENIUS THEOREM

Definition 10.1 Let M be a smooth manifold of dimension n.Foranyintegerr,with 1 r n,anr-dimensional tangential distribution (for short, a distribution)isamap, D ≤: M ≤ TM, such that → (a) D(p) T M is an r-dimensional subspace for all p M. ⊆ p ∈ (b) For every p M,thereissomeopensubset,U,withp U, and r smooth vector fields, X ,...,X ,definedon∈ U,suchthat(X (q),...,X (q))∈ is a basis of D(q)forallq U. 1 r 1 r ∈ We say that D is locally spanned by X1,...,Xr.

An immersed submanifold, N,ofM is an integral manifold of D iﬀ D(p)=TpN,forall p N.WesaythatD is completely integrable iﬀthere exists an integral manifold of D through∈ every point of M.

We also write Dp for D(p).

Remarks: (1) An r-dimensional distribution, D,isjustasmoothsubbundleofTM.

(2) An integral manifold is only an immersed submanifold, not necessarily an embedded submanifold.

(3) Some authors (such as Lee) reserve the locution “completely integrable” to a seemingly strongly condition (See Lee [98], Chapter 19, page 500). This condition is in fact equivalent to “our” deﬁnition (which seems the most commonly adopted).

(4) Morita [114] uses a stronger notion of integral manifold, namely, an integral manifold is actually an embedded manifold. Most of the results, including Frobenius Theorem still hold but maximal integral manifolds are immersed but not embedded manifolds and this is why most authors prefer to use the weaker deﬁnition (immersed manifolds).

Here is an example of a distribution which does not have any integral manifolds: This is the two-dimensional distribution in R3 spanned by the vector ﬁelds ∂ ∂ ∂ X = + y ,X= . ∂x ∂z ∂y We leave it as an exercise to the reader to show that the above distribution is not integrable. The key to integrability is an involutivity condition. Here is the deﬁnition.

Definition 10.2 Let M be a smooth manifold of dimension n and let D be an r-dimensional distribution on M.Foranysmoothvectorfield,X,wesaythatX belongs to D (or lies in D)iffXp Dp,forallp M.WesaythatD is involutive ifffor any two smooth vector fields, X, Y∈,onM,ifX and∈ Y belong to D,then[X, Y ]alsobelongstoD. 10.2. FROBENIUS THEOREM 323

Proposition 10.1 Let M be a smooth manifold of dimension n. If an r-dimensional distribution, D, is completely integrable, then D is involutive.

Proof .AproofcanbefoundininWarner[147](Chapter1),andLee[98](Proposition19.3). These proofs use Proposition 3.14. Another proof is given in Morita [114] (Section 2.3) but beware that Morita deﬁnes an integral manifold to be an embedded manifold. In the example before Deﬁnition 10.1, we have ∂ [X, Y ]= , −∂z so this distribution is not involutive. Therefore, by Proposition 10.1, this distribution is not completely integrable.

10.2 Frobenius Theorem

Frobenius’ Theorem asserts that the converse of Proposition 10.1 holds. Although we do not intend to prove it in full, we would like to explain the main idea of the proof of Frobenius’ Theorem. It turns out that the involutivity condition of two vector fields is equivalent to the commutativity of their corresponding flows and this is the crucial fact used in the proof. Given a manifold, M,wesathattwovectorfields,X and Y are mutually commutative 2 ∂ ∂ ∂ iff[X, Y ]=0.Forexample,onR ,thevectorfields ∂x and ∂y are commutative but ∂x and ∂ x ∂y are not. Recall from Definition 3.23 that we denote by ΦX the (global) flow of the vector field, X X.Foreveryp M, the map, t Φ (t, p)=γp(t)isthemaximalintegralcurvethrough ∈ X → p.WealsowriteΦt(p)forΦ (t, p)(droppingX). Recall that the map, p Φt(p), is a diffeomorphism on its domain (an open subset of M). For the next proposition,→ given two vector fields, X and Y ,wewillwriteΦfortheflowassociatedwithX and Ψfor the flow associated with Y .

Proposition 10.2 Given a manifold, M, for any two smooth vector ﬁelds, X and Y , the following conditions are equivalent: (1) X and Y are mutually commutative (i.e. [X, Y ]=0).

(2) Y is invariant under Φt, that is, (Φt) Y = Y , whenever the left-hand side is deﬁned. ∗

(3) X is invariant under Ψs, that is, (Ψs) X = X, whenever the left-hand side is deﬁned. ∗

(4) The maps Φt and Ψt are mutually commutative. This means that

Φ Ψ =Ψ Φ , t ◦ s s ◦ t for all s, t such that both sides are deﬁned. 324 CHAPTER 10. DISTRIBUTIONS AND THE FROBENIUS THEOREM

(5) Y =0. LX (6) X =0. LY (In (5) Y is the Lie derivative and similarly in (6).) LX Proof .AproofcanbefoundinLee[98](Chapter18,Proposition18.5)andinMorita[114] (Chapter 2, Proposition 2.18). For example, to prove the implication (2) = (4), we observe that if ϕ is a diﬀeomorphism on some open subset, U,ofM,thentheintegralcurvesof⇒ ϕ Y ∗ through a point p M are of the form ϕ γ,whereγ is the integral curve of Y through 1 ∈ ◦ 1 ϕ− (p). Consequently, the local one-parameter group generated by ϕ Y is ϕ Ψs ϕ− .If 1 ∗ { ◦ ◦ } we apply this to ϕ =Φt,as(Φt) Y = Y ,wegetΦt Ψs Φt− =Ψs and hence, Φt Ψs =Ψs Φt. ∗ ◦ ◦ ◦ ◦

In order to state our ﬁrst version of the Frobenius Theorem we make the following deﬁnition:

Definition 10.3 Let M be a smooth manifold of dimension n.Givenanysmoothr- dimensional distribution, D,onM,achart,(U,ϕ), is flat for D iff

r n r ϕ(U) = U U R R − , ∼ × ⊆ × where U and U are connected open subsets such that for every p U,thedistributionD is spanned by the vector ﬁelds ∈ ∂ ∂ ,..., . ∂x1 ∂xr

If (U,ϕ)isﬂatforD,thenitisclearthateachsliceof(U,ϕ),

S = q U x = c ,...,x = c , c { ∈ | r+1 r+1 n n} th is an integral manifold of D,wherexi = pri ϕ is the i -coordinate function on U and n r ◦ c =(cr+1,...,cn) R − is a fixed vector. ∈ Theorem 10.3 (Frobenius) Let M be a smooth manifold of dimension n. A smooth r- dimensional distribution, D, on M is completely integrable iffit is involutive. Furthermore, for every p U, there is flat chart, (U,ϕ), for D with p U, so that every slice of (U,ϕ) is an integral manifold∈ of D. ∈

Proof . A proof of Theorem 10.3 can be found in Warner [147] (Chapter 1, Theorem 1.60), Lee [98] (Chapter 19, Theorem 19.10) and Morita [114] (Chapter 2, Theorem 2.17). Since we already have Proposition 10.1, it is only necessary to prove that if a distribution is involutive then it is completely integrable. Here is a sketch of the proof, following Morita. 10.2. FROBENIUS THEOREM 325

Pick any p M. As D is a smooth distribution, we can ﬁnd some chart, (U,ϕ), with ∈ p U,andsomevectorﬁelds,Y1,...,Yr,sothatY1(q),...,Yr(q)arelinearlyindependent and∈ span D for all q U.Locally,wecanwrite q ∈ n ∂ Y = a ,i=1,...,r. i ij ∂x j=1 j

Since the Yi are linearly independent, by renumbering the coordinates if necessary, we may assume that the r r matrices × A(q)=(a (q)) q U ij ∈ 1 are invertible. Then, the inverse matrices, B(q)=A− (q)deﬁner r functions, bij(q)and let × r

Xi = bijYj,j=1,...,r. j=1 Now, in matrix form, ∂ , Y1 ∂x1 . .  .  = AR .  ∂ Yr ,    ∂xn      for some r (n r)matrix,R and × −

X1 Y1 . .  .  = B  .  , Xr Yr         so we get ∂ X1 ∂x1 . .  .  = IBR .  , ∂ Xr    ∂xn      that is, ∂ n ∂ X = + c ,i=1,...,r, ( ) i ∂x ij ∂x ∗ i j=r+1 j where the cij are functions deﬁned on U.Obviously,X1,...,Xr are linearly independent and they span Dq for all q U.SinceD is involutive, there are some functions, fk,deﬁned on U,sothat ∈ r

[Xi,Xj]= fkXk. k=1 326 CHAPTER 10. DISTRIBUTIONS AND THE FROBENIUS THEOREM

∂ ∂ On the other hand, by ( ), each [Xi,Xj]isalinearcombinationof ,..., .Therefore, ∗ ∂xr+1 ∂xn fk =0,fork =1,...,r,whichshowsthat

[X ,X ]=0, 1 i, j r, i j ≤ ≤ that is, the vector ﬁelds X1,...,Xr are mutually commutative. i Let Φt be the local one-parameter group associated with Xi.ByProposition10.2(4), i the Φt commute, that is,

Φi Φj =Φj Φi 1 i, j r, t ◦ s s ◦ t ≤ ≤ whenever both sides are defined. We can pick a sufficiently open subset, V ,inRr containing the origin and define the map, Φ: V U by → Φ(t ,...,t )=Φ1 Φr (p). 1 r t1 ◦···◦ tr j Clearly, Φis smooth and using the fact that each Xi is invariant under each Φs,forj = i, and ∂ dΦi = X (p), p ∂t i i we get ∂ dΦ = X (p). p ∂t i i r As X1,...,Xr are linearly independent, we deduce that dΦp : T0R TpM is an injection and thus, we may assume by shrinking V if necessary that our map,→ Φ: V M,isan embedding. But then, N =Φ(V )isasubmanifoldofM and it only remains to→ prove that N is an integral manifold of D through p.

Obviously, TpN = Dp,sowejusthavetoprovethatTqN = DqN for all q N. Now, for every q N,wecanwrite ∈ ∈ q =Φ(t ,...,t )=Φ1 Φr (p), 1 r t1 ◦···◦ tr for some (t ,...,t ) V .SincetheΦi commute, for any i,with1 i r,wecanwrite 1 r ∈ t ≤ ≤ i 1 i 1 i+1 r − q =Φti Φt1 Φti 1 Φti+1 Φtr (p). ◦ ◦···◦ − ◦ ◦···◦

If we ﬁx all the tj but ti and vary ti by a small amount, we obtain a curve in N through q i and this is an orbit of Φt.Therefore,thiscurveisanintegralcurveofXi through q whose velocity vector at q is equal to X (q)andso,X (q) T N.Sincetheabovereasoningholds i i ∈ q for all i,wegetTqN = Dq,asclaimed.Therefore,N is an integral manifold of D through p. In preparation for a global version of Frobenius Theorem in terms of foliations, we state the following Proposition proved in Lee [98] (Chapter 19, Proposition 19.12): 10.3. DIFFERENTIAL IDEALS AND FROBENIUS THEOREM 327

Proposition 10.4 Let M be a smooth manifold of dimension n and let D be an involutive r-dimensional distribution on M. For every ﬂat chart, (U,ϕ), for D, for every integral manifold, N, of D, the set N U is a countable disjoint union of open parallel k-dimensional slices of U, each of which is open∩ in N and embedded in M.

We now describe an alternative method for describing involutivity in terms of diﬀerential forms.

10.3 Diﬀerential Ideals and Frobenius Theorem

First, we give a smoothness criterion for distributions in terms of one-forms.

Proposition 10.5 Let M be a smooth manifold of dimension n and let D be an assignment, p Dp TpM, of some r-dimensional subspace of TpM, for all p M. Then, D is a smooth distribution→ ⊆ iﬀfor every p U, there is some open subset, U, with∈p U, and some linearly ∈ ∈ independent one-forms, ω1,...,ωn r, deﬁned on U, so that −

Dq = u TqM (ω1)q(u)= =(ωn r)q(u)=0 , for all q U. { ∈ | ··· − } ∈ Proof . Proposition 10.5 is proved in Lee [98] (Chapter 19, Lemma 19.5). The idea is to either extend a set of linearly independent differential one-forms to a coframe and then consider the dual frame or to extend some linearly independent vector fields to a frame and then take the dual basis. Proposition 10.5 suggests the following definition:

Definition 10.4 Let M be a smooth manifold of dimension n and let D be an r-dimensional distibution on M. Some linearly independent one-forms, ω1,...,ωn r,definedsomeopen − subset, U M,arecalledlocal defining one-forms for D if ⊆

Dq = u TqM (ω1)q(u)= =(ωn r)q(u)=0 , for all q U. { ∈ | ··· − } ∈ We say that a k-form, ω k(M), annihilates D iﬀ ∈A

ωq(X1(q),...,Xr(q)) = 0, for all q M and for all vector ﬁelds, X ,...,X , belonging to D.Wewrite ∈ 1 r Ik(D)= ω k(M) ω (X (q),...,X (q)) = 0 , { ∈A | q 1 r } for all q M and for all vector ﬁelds, X ,...,X , belonging to D and we let ∈ 1 r n I(D)= Ik(D). k=1 328 CHAPTER 10. DISTRIBUTIONS AND THE FROBENIUS THEOREM

Thus, I(D)isthecollectionofdifferentialformsthat“vanishonD.” In the classical terminology, a system of local defining one-forms as above is called a system of Pfaffian equations. It turns out that I(D)isnotonlyavectorspacebutalsoanidealof •(M). A Asubspace,I,of •(M)isanideal ifffor every ω I,wehaveθ ω I for every A ∈ ∧ ∈ θ •(M). ∈A Proposition 10.6 Let M be a smooth n-dimensional manifold and D be an r-dimensional distribution. If I(D) is the space of forms annihilating D then the following hold:

(a) I(D) is an ideal in •(M). A (b) I(D) is locally generated by n r linearly independent one-forms, which means: For every p U, there is some open− subset, U M, with p U and a set of linearly ∈ ⊆ ∈ independent one-forms, ω1,...,ωn r, deﬁned on U, so that − k (i) If ω I (D), then ω U belongs to the ideal in •(U) generated by ω1,...,ωn r, − that∈ is, A n r − ω = θ ω , on U, i ∧ i i=1 k 1 for some (k 1)-forms, θ − (U). − i ∈A (ii) If ω k(M) and if there is an open cover by subsets U (as above) such that for ∈A every U in the cover, ω U belongs to the ideal generated by ω1,...,ωn r, then − ω I(D). ∈ (c) If I •(M) is an ideal locally generated by n r linearly independent one-forms, ⊆A − then there exists a unique smooth r-dimensional distribution, D, for which I = I(D).

Proof . Proposition 10.6 is proved in Warner (Chapter 2, Proposition 2.28). See also Morita [114] (Chapter 2, Lemma 2.19) and Lee [98] (Chapter 19, page 498-500). In order to characterize involutive distributions, we need the notion of diﬀerential ideal.

Definition 10.5 Let M be a smooth manifold of dimension n. An ideal, I •(M), is a differential ideal iffit is closed under exterior differentiation, that is ⊆A

dω I whenever ω I, ∈ ∈ which we also express by dI I. ⊆ Here is the diﬀerential ideal criterion for involutivity.

Proposition 10.7 Let M be a smooth manifold of dimension n. A smooth r-dimensional distribution, D, is involutive iﬀthe ideal, I(D), is a diﬀerential ideal. 10.3. DIFFERENTIAL IDEALS AND FROBENIUS THEOREM 329

Proof .Proposition10.7isprovedinWarner[147](Chapter2,Proposition2.30),Morita [114] (Chapter 2, Proposition 2.20) and Lee [98] (Chapter 19, Proposition 19.19). Here is one direction of the proof. Assume I(D)isadifferentialideal.Weknowthatforany one-form, ω, dω(X, Y )=X(ω(Y )) Y (ω(X)) ω([X, Y ]), − − for any vector fields, X, Y . Now, if ω1,...,ωn r are linearly independent one-forms that − define D locally on U,usingabumpfunction,wecanextendω1,...,ωn r to M and then − using the above equation, for any vector fields X, Y belonging to D,weget

ω ([X, Y ]) = X(ω (Y )) Y (ω (X)) dω (X, Y )=0, i i − i − i

and since ωi(X)=ωi(Y )=dωi(X, Y )=0,wegetωi([X, Y ]) = 0 for i =1,...,n r,which means that [X, Y ]belongstoD. − Using Proposition 10.6, we can give a more concrete criterion: D is involutive iﬀfor every local deﬁning one-forms, ω1,...,ωn r,forD (on some open subset, U), there are some − one-forms, ω 1(U), so that ij ∈A n r − dω = ω ω (i =1,...,n r). i ij ∧ j − j=1 The above conditions are often called the integrability conditions.

Deﬁnition 10.6 Let M be a smooth manifold of dimension n.GivenanyidealI •(M), ⊆A an immersed manifold, (M,ψ), of M is an integral manifold of I iﬀ

ψ∗ω =0, for all ω I. ∈ A connected integral manifold of the ideal I is maximal iﬀits image is not a proper subset of the image of any other connected integral manifold of I.

Finally, here is the diﬀerential form version of the Frobenius Theorem.

Theorem 10.8 (Frobenius Theorem, Diﬀerential Ideal Version) Let M be a smooth manifold of dimension n.IfI •(M) is a diﬀerential ideal locally generated by n r linearly independent one-forms, then⊆A for every p M, there exists a unique maximal,− connected, ∈ integral manifold of I through p and this integral manifold has dimension r.

Proof .Theorem10.8isprovedinWarner[147].Thistheoremfollowsimmediatelyfrom Theorem 1.64 in Warner [147]. Another version of the Frobenius Theorem goes as follows: 330 CHAPTER 10. DISTRIBUTIONS AND THE FROBENIUS THEOREM

Theorem 10.9 (Frobenius Theorem, Integrability Conditions Version) Let M be a smooth manifold of dimension n.Anr-dimensional distribution, D, on M is completely integrable iﬀfor every local deﬁning one-forms, ω1,...,ωn r, for D (on some open subset, U), there − are some one-forms, ω 1(U), so that we have the integrability conditions ij ∈A n r − dω = ω ω (i =1,...,n r). i ij ∧ j − j=1

There are applications of Frobenius Theorem (in its various forms) to systems of partial diﬀerential equations but we will not deal with this subject. The reader is advised to consult Lee [98], Chapter 19, and the references there.

10.4 A Glimpse at Foliations and a Global Version of Frobenius Theorem

All the maximal integral manifolds of an r-dimensional involutive distribution on a manifold, M,yieldadecompositionofM with some nice properties, those of a foliation.

Definition 10.7 Let M be a smooth manifold of dimension n.Afamily, = α α,of subsets of M is a k-dimensional foliation iffit is a family of pairwise disjoint,F connected,{F } immersed k-dimensional submanifolds of M,calledtheleaves of the foliation, whose union is M and such that, for every p M,thereisachart,(U,ϕ), with p U,calledaflat chart for the foliation and the following∈ property holds: ∈

r n r ϕ(U) = U U R R − , ∼ × ⊆ ×

where U and U are some connected open subsets and for every leaf, α, of the foliation, if U = ,then U is a countable union of k-dimensional slicesF given by Fα ∩ ∅ Fα ∩

xr+1 = cr+1,...,xn = cn,

for some constants, cr+1,...,cn R. ∈

The structure of a foliation can be very complicated. For instance, the leaves can be dense in M. For example, there are spirals on a torus that form the leaves of a foliation (see Lee [98], Example 19.9). Foliations are in one-to-one correspondence with involutive distributions.

Proposition 10.10 Let M be a smooth manifold of dimension n. For any foliation, , on M, the family of tangent spaces to the leaves of forms an involutive distribution onFM. F 10.4. A GLIMPSE AT FOLIATIONS 331

The converse to the above proposition may be viewed as a global version of Frobenius Theorem.

Theorem 10.11 Let M be a smooth manifold of dimension n. For every r-dimensional smooth, involutive distribution, D, on M, the family of all maximal, connected, integral manifolds of D forms a foliation of M.

Proof .TheproofofTheorem10.11canbefoundinLee[98](Theorem19.21). 332 CHAPTER 10. DISTRIBUTIONS AND THE FROBENIUS THEOREM Chapter 11

Connections and Curvature in Vector Bundles

11.1 Connections and Connection Forms in Vector Bundles and Riemannian Manifolds

Given a manifold, M,ingeneral,foranytwopoints,p, q M, there is no “natural” isomor- ∈ phism between the tangent spaces TpM and TqM.Moregenerally,givenanyvectorbundle, ξ =(E,π,B,V ), for any two points, p, q B,thereisno“natural”isomorphismbetween 1 1 ∈ the fibres, E = π− (p)andE = π− (q). Given a curve, c:[0, 1] M,onM (resp. a curve, p q → c:[0, 1] E,onB), as c(t)movesonM (resp. on B), how does the tangent space, Tc(t)M → 1 (resp. the fibre Ec(t) = π− (c(t))) change as c(t)moves? n n n If M = R ,thenthespacesTc(t)R are canonically isomorphic to R and any vector, n n v Tc(0)R = R ,issimplymovedalongc by parallel transport,thatit,atc(t), the tangent ∈ ∼ n vector, v,alsobelongstoTc(t)R . However, if M is curved, for example, a sphere, then it is not obvious how to “parallel transport” a tangent vector at c(0) along a curve c.Away to achieve this is to define the notion of parallel vector field along a curve and this, in turn, can be defined in terms of the notion of covariant derivative of a vector field (or covariant derivative of a section, in the case of vector bundles). Assume for simplicity that M is a surface in R3.Givenanytwovectorfields,X and Y 3 defined on some open subset, U R ,foreveryp U,thedirectional derivative, DX Y (p), of Y with respect to X is defined⊆ by ∈ Y (p + tX(p)) Y (p) DX Y (p)=lim − . t 0 → t If f : U R is a differentiable function on U,foreveryp U,thedirectional derivative, X[f](p) →(or X(f)(p)), of f with respect to X is defined by ∈ f(p + tX(p)) f(p) X[f](p)=lim − . t 0 → t 333 334 CHAPTER 11. CONNECTIONS AND CURVATURE IN VECTOR BUNDLES

We know that X[f](p)=df p(X(p)).

It is easily shown that DX Y (p)isR-bilinear in X and Y ,isC∞(U)-linear in X and satisﬁes the Leibnitz derivation rule with respect to Y ,thatis:

Proposition 11.1 The directional derivative of vector ﬁelds satisﬁes the following properties:

DX1+X2 Y (p)=DX1 Y (p)+DX2 Y (p)

DfXY (p)=fDX Y (p)

DX (Y1 + Y2)(p)=DX Y1(p)+DX Y2(p)

DX (fY)(p)=X[f](p)Y (p)+f(p)DX Y (p), for all X, X ,X ,Y,Y ,Y X(U) and all f C∞(U). 1 2 1 2 ∈ ∈

Now, if p U where U M is an open subset of M,foranyvectorﬁeld,Y ,deﬁned ∈ ⊆ on U (Y (p) TpM,forallp U), for every X TpM,thedirectionalderivative,DX Y (p), makes sense∈ and it has an orthogonal∈ decomposition,∈

D Y (p)= Y (p)+(D ) Y (p), X ∇X n X where its horizontal (or tangential) component is X Y (p) TpM and its normal component is (D ) Y (p). The component, Y (p), is the ∇covariant∈ derivative of Y with respect to n X ∇X X T M and it allows us to define the covariant derivative of a vector field, Y X(U), ∈ p ∈ with respect to a vector field, X X(M), on M.Weeasilycheckthat X Y satisfies the four equations of Proposition 11.1.∈ ∇ In particular, Y ,maybeavectorfieldassociatedwithacurve,c:[0, 1] M.Avector → field along a curve, c, is a vector field, Y ,suchthatY (c(t)) Tc(t)M,forallt [0, 1]. We also write Y (t)forY (c(t)). Then, we say that Y is parallel along∈ c iff Y =0along∈ c. ∇∂/∂t The notion of parallel transport on a surface can be defined using parallel vector fields along curves. Let p, q be any two points on the surface M and assume there is a curve, c:[0, 1] M, joining p = c(0) to q = c(1). Then, using the uniqueness and existence theorem→ for ordinary differential equations, it can be shown that for any initial tangent vector, Y T M, there is a unique parallel vector field, Y ,alongc, with Y (0) = Y .If 0 ∈ p 0 we set Y1 = Y (1), we obtain a linear map, Y0 Y1,fromTpM to TqM which is also an isometry. → As a summary, given a surface, M,ifwecandefineanotionofcovariantderivative, : X(M) X(M) X(M), satisfying the properties of Proposition 11.1, then we can define the∇ notion× of parallel→ vector field along a curve and the notion of parallel transport, which yields a natural way of relating two tangent spaces, TpM and TqM,usingcurvesjoiningp and q. This can be generalized to manifolds and even to vector bundles using the notion of connection. We will see that the notion of connection induces the notion of curvature. 11.1. CONNECTIONS IN VECTOR BUNDLES AND RIEMANNIAN MANIFOLDS 335

Moreover, if M has a Riemannian metric, we will see that this metric induces a unique connection with two extra properties (the Levi-Civita connection). Given a manifold, M,asX(M)=Γ(M,TM)=Γ(TM), the set of smooth sections of the tangent bundle, TM,itisnaturalthatforavectorbundle,ξ =(E,π,B,V ), a connection on ξ should be some kind of bilinear map, X(B) Γ(ξ) Γ(ξ), × −→ that tells us how to take the covariant derivative of sections. Technically, it turns out that it is cleaner to deﬁne a connection on a vector bundle, ξ, as an R-linear map, :Γ(ξ) 1(B) Γ(ξ), ( ) ∇ →A ⊗C∞(B) ∗ that satisﬁes the “Leibnitz rule” (fs)=df s + f s, ∇ ⊗ ∇ 1 with s Γ(ξ)andf C∞(B), where Γ(ξ)and (B)aretreatedasC∞(B)-modules. Since 1 ∈ ∈ A (B)=Γ(B,T∗B)=Γ(T ∗B)and,byProposition7.12, A 1 (B) Γ(ξ)=Γ(T ∗B) Γ(ξ) A ⊗C∞(B) ⊗C∞(B) = Γ(T ∗B ξ) ∼ ⊗ = Γ( om(TB,ξ)) ∼ H = Hom (Γ(TB), Γ(ξ)) ∼ C∞(B)

= HomC∞(B)(X(B), Γ(ξ)), the range of can be viewed as a space of Γ(ξ)-valued diﬀerential forms on B.Milnorand Stasheﬀ[110]∇ (Appendix C) use the version where

:Γ(ξ) Γ(T ∗B ξ) ∇ → ⊗ and Madsen and Tornehave [100] (Chapter 17) use the equivalent version stated in ( ). A thorough presentation of connections on vector bundles and the various ways to define∗ them can be found in Postnikov [125] which also constitutes one of the most extensive references on differential geometry. Set 1(ξ)= 1(B; ξ)= 1(B) Γ(ξ) A A A ⊗C∞(B) and, more generally, for any i 0, set ≥ i i i i (ξ)= (B; ξ)= (B) Γ(ξ) = Γ T ∗B ξ . A A A ⊗C∞(B) ∼ ⊗ 0 0 Obviously, (ξ)=Γ(ξ)(andrecallthat (B)=C∞(B)). The space of differential forms, i(B; ξ), withA values in Γ(ξ)isageneralizationofthespace,A i(M,F), of differential forms withA values in F encountered in Section 8.4. A 336 CHAPTER 11. CONNECTIONS AND CURVATURE IN VECTOR BUNDLES

If we use the isomorphism

1(B) Γ(ξ) = Hom (X(B), Γ(ξ)), A ⊗C∞(B) ∼ C∞(B) then a connection is an R-linear map, :Γ(ξ) Hom (X(B), Γ(ξ)), ∇ −→ C∞(B) satisfying a Leibnitz-type rule or equivalently, an R-bilinear map, : X(B) Γ(ξ) Γ(ξ), ∇ × −→ such that, for any X X(B)ands Γ(ξ), if we write s instead of (X, s), then the ∈ ∈ ∇X ∇ following properties hold for all f C∞(B): ∈ s = f s ∇fX ∇X (fs)=X[f]s + f s. ∇X ∇X This second version may be considered simpler than the ﬁrst since it does not involve a tensor product. Since

1 (B)=Γ(T ∗B) = Hom (X(B),C∞(B)) = X(B)∗, A ∼ C∞(B) using Proposition 22.36, the isomorphism

α: 1(B) Γ(ξ) = Hom (X(B), Γ(ξ)) A ⊗C∞(B) ∼ C∞(B) can be described in terms of the evaluation map,

Ev : 1(B) Γ(ξ) Γ(ξ), X A ⊗C∞(B) → given by Ev (ω s)=ω(X)s, X X(B),ω 1(B),s Γ(ξ). X ⊗ ∈ ∈A ∈ Namely, for any θ 1(B) Γ(ξ), ∈A ⊗C∞(B)

α(θ)(X)=EvX (θ).

In particular, the reader should check that

Ev (df s)=X[f]s. X ⊗ Then, it is easy to see that we pass from the ﬁrst version of ,where ∇ :Γ(ξ) 1(B) Γ(ξ)() ∇ →A ⊗C∞(B) ∗ with the Leibnitz rule (fs)=df s + f s, ∇ ⊗ ∇ 11.1. CONNECTIONS IN VECTOR BUNDLES AND RIEMANNIAN MANIFOLDS 337

to the second version of ,denoted ,where ∇ ∇

: X(B) Γ(ξ) Γ(ξ), ( ) ∇ × → ∗∗ is R-bilinear and where the two conditions

s = f s ∇fX ∇X (fs)=X[f]s + f s ∇X ∇X hold, via the equation =Ev . ∇X X ◦∇ From now on, we will simply write X s instead of X s, unless confusion arise. As summary of the above discussion, we make the∇ following deﬁnition:∇

Deﬁnition 11.1 Let ξ =(E,π,B,V )beasmoothrealvectorbundle.Aconnection on ξ is an R-linear map, :Γ(ξ) 1(B) Γ(ξ)() ∇ →A ⊗C∞(B) ∗ such that the Leibnitz rule (fs)=df s + f s ∇ ⊗ ∇ holds, for all s Γ(ξ)andallf C∞(B). For every X X(B), we let ∈ ∈ ∈ =Ev ∇X X ◦∇ and for every s Γ(ξ), we call s the covariant derivative of s relative to X.Then,the ∈ ∇X family, ( X ), induces a R-bilinear map also denoted , ∇ ∇ : X(B) Γ(ξ) Γ(ξ), ( ) ∇ × → ∗∗ such that the following two conditions hold:

s = f s ∇fX ∇X (fs)=X[f]s + f s, ∇X ∇X for all s Γ(ξ), all X X(B)andallf C∞(B). We refer to ( )astheﬁrst version of a connection∈ and to ( )asthe∈ second version∈ of a connection. ∗ ∗∗

Observe that in terms of the i(ξ)’s, a connection is a linear map, A : 0(ξ) 1(ξ), ∇ A →A satisfying the Leibnitz rule. When ξ = TB,aconnection(secondversion)iswhatisknown as an aﬃne connection on a manifold, B. 338 CHAPTER 11. CONNECTIONS AND CURVATURE IN VECTOR BUNDLES

Remark: Given two connections, 1 and 2,wehave ∇ ∇ 1(fs) 2(fs)=df s + f 1s df s f 2s = f( 1s 2s), ∇ −∇ ⊗ ∇ − ⊗ − ∇ ∇ −∇ 1 2 1 which shows that is a C∞(B)-linear map from Γ(ξ)to (B) Γ(ξ). However ∇ −∇ A ⊗C∞(B) 0 i i HomC∞(B)( (ξ), (ξ)) = HomC∞(B)(Γ(ξ), (B) C∞(B) Γ(ξ)) A A iA ⊗ = Γ(ξ)∗ ( (B) Γ(ξ)) ∼ C∞(B) C∞(B) i ⊗ A ⊗ = (B) (Γ(ξ)∗ Γ(ξ)) ∼ A ⊗C∞(B) ⊗C∞(B) = i(B) Hom (Γ(ξ), Γ(ξ)) ∼ A ⊗C∞(B) C∞(B) = i(B) Γ( om(ξ,ξ)) ∼ A ⊗C∞(B) H = i( om(ξ,ξ)). A H Therefore, 1 2 1( om(ξ,ξ)), that is, it is a one-form with values in Γ( om(ξ,ξ)). But then, the∇ −∇ vector∈ space,A H Γ( om(ξ,ξ)), acts on the space of connections (byH addition) and makes the space of connectionsH into an aﬃne space. Given any connection, and any one-form, ω Γ( om(ξ,ξ)), the expression + ω is also a connection. Equivalently,∇ any aﬃne combination∈ H of connections is also a connection.∇ Abasicpropertyof is that it is a local operator. ∇ Proposition 11.2 Let ξ =(E,π,B,V ) be a smooth real vector bundle and let be a connection on ξ. For every open subset, U B, for every section, s Γ(ξ), if s 0∇on U, then s 0 on U, that is, is a local operator.⊆ ∈ ≡ ∇ ≡ ∇ Proof .ByProposition3.24appliedtotheconstantfunctionwithvalue1,foreveryp U, ∈ there is some open subset, V U,containingp and a smooth function, f : B R,such that supp f U and f 1on⊆V .Consequently,fs is a smooth section which is→ identically zero. By applying⊆ the Leibnitz≡ rule, we get

0= (fs)=df s + f s, ∇ ⊗ ∇ which, evaluated at p yields ( s)(p)=0,sincef(p)=1anddf 0onV . ∇ ≡ As an immediate consequence of Proposition 11.2, if s1 and s2 are two sections in Γ(ξ) that agree on U,thens1 s2 is zero on U,so (s1 s2)= s1 s2 is zero on U,thatis, s and s agree on U.− ∇ − ∇ −∇ ∇ 1 ∇ 2 Proposition 11.2 also implies that a connection, ,onξ,restrictstoaconnection, U ∇ ∇ on the vector bundle, ξ U,foreveryopensubset,U B.Indeed,lets be a section of ξ over U.Pickanyb U and deﬁne ( s)(b) as follows: Using⊆ Proposition 3.24, there is some ∈ ∇ open subset, V1 U,containingb and a smooth function, f1 : B R, such that supp f1 U and f 1onV⊆so, let s = f s,aglobalsectionofξ.Clearly,→s = s on V ,andset ⊆ 1 ≡ 1 1 1 1 1 ( s)(b)=( s )(b). ∇ ∇ 1 11.1. CONNECTIONS IN VECTOR BUNDLES AND RIEMANNIAN MANIFOLDS 339

This deﬁnition does not depend on (V1,f1), because if we had used another pair, (V2,f2), as above, since b V V ,wehave ∈ 1 ∩ 2 s = f s = s = f s = s on V V 1 1 2 2 1 ∩ 2 so, by Proposition 11.2, ( s )(b)=( s )(b). ∇ 1 ∇ 2 It should also be noted that ( s)(b) only depends on X(b), that is, for any two vector ∇X ﬁelds, X, Y X(B), if X(b)=Y (b)forsomeb B,then ∈ ∈ ( s)(b)=( s)(b), for every s Γ(ξ). ∇X ∇Y ∈

As above, by linearity, it it enough to prove that if X(b)=0,then( X s)(b)=0.Toprove this, pick any local trivialization, (U,ϕ), with b U.Then,wecanwrite∇ ∈ d ∂ X U = X . i ∂x i=1 i However, as before, we can find a pair, (V,f), with b V U,suppf U and f =1onV , so that f ∂ is a smooth vector field on B and f ∂ ∈agrees⊆ with ∂ on⊆V ,fori =1,...,n. ∂xi ∂xi ∂xi 2 Clearly, fX C∞(B)andfX agrees with X on V so if we write X = f X,then i ∈ i i d ∂ X = f 2X = fX f i ∂x i=1 i and we have d 2 f X s = X s = fXi f ∂ s. ∇ ∇ ∇ ∂xi i=1 Since X (b)=0andf(b)=1,weget( s)(b)=0,asclaimed. i ∇X Using the above property, for any point, p B, we can define the covariant derivative, ( s)(p), of a section, s Γ(ξ), with respect to∈ a tangent vector, u T B.Indeed,pickany ∇u ∈ ∈ p vector field, X X(B), such that X(p)=u (such a vector field exists locally over the domain ∈ of a chart and then extend it using a bump function) and set ( us)(p)=( X s)(p). By the above property, if X(p)=Y (p), then ( s)(p)=( s)(p)so(∇ s)(p)iswell-defined.∇ ∇X ∇Y ∇u Since is a local operator, ( us)(p) is also well defined for any tangent vector, u TpB, and any∇ local section, s Γ(U,∇ξ), defined in some open subset, U,withp U.Fromnow∈ on, we will use this property∈ without any further justification. ∈

Since ξ is locally trivial, it is interesting to see what U looks like when (U,ϕ)isa local trivialization of ξ. ∇ 340 CHAPTER 11. CONNECTIONS AND CURVATURE IN VECTOR BUNDLES

Fix once and for all some basis, (v1,...,vn), of the typical ﬁbre, V (n =dim(V )). To 1 every local trivialization, ϕ: π− (U) U V ,ofξ (for some open subset, U B), we → × ⊆ associate the frame, (s1,...,sn), over U given by 1 s (b)=ϕ− (b, v ),bU. i i ∈ n Then, every section, s,overU, can be written uniquely as s = i=1 fisi,forsomefunctions fi C∞(U)andwehave ∈ n n s = (f s )= (df s + f s ). ∇ ∇ i i i ⊗ i i∇ i i=1 i=1 On the other hand, each s can be written as ∇ i n s = ω s , ∇ i ij ⊗ j j=1 for some n n matrix, ω =(ω ), of one-forms, ω 1(U), so we get × ij ij ∈A n n n n n n s = df s + f s = df s + f ω s = (df + f ω ) s . ∇ i ⊗ i i∇ i i ⊗ i i ij ⊗ j j i ij ⊗ j i=1 i=1 i=1 i,j=1 j=1 i=1 With respect to the frame, (s ,...,s ), the connection has the matrix form 1 n ∇ (f ,...,f )=(df ,...,df )+(f ,...,f )ω ∇ 1 n 1 n 1 n 1 and the matrix, ω =(ωij), of one-forms, ωij (U), is called the connection form or ∈1 A connection matrix of with respect to ϕ: π− (U) U V .Theabovecomputation also shows that on U,anyconnectionisuniquelydeterminedbyamatrixofone-forms,∇ → × ω 1(U). In particular, the connection on U for which ij ∈A s =0,..., s =0, ∇ 1 ∇ n corresponding to the zero matrix is called the ﬂat connection on U (w.r.t. (s1,...,sn)). Some authors (such as Morita [114]) use a notation involving subscripts and superscripts, namely n s = ωj s . ∇ i i ⊗ j j=1 j But, beware, the expression ω =(ωi )denotesthen n-matrix whose rows are indexed by j and whose columns are indexed by i! Accordingly,× if θ = ωη,then

i i k θj = ωkηj . k i The matrix, (ωj)isthusthe transpose of our matrix (ωij). This has the eﬀects that some of the results diﬀer either by a sign (as in ω ω)orbyapermutationofmatrices(asinthe formula for a change of frame). ∧ 11.1. CONNECTIONS IN VECTOR BUNDLES AND RIEMANNIAN MANIFOLDS 341

1 Remark: If (θ1,...,θn)isthedualframeof(s1,...,sn), that is, θi (U), is the one-form deﬁned so that ∈A θ (b)(s (b)) = δ , for all b U, 1 i, j n, i j ij ∈ ≤ ≤ n k then we can write ωik = j=1 Γjiθj and so, n s = Γk (θ s ), ∇ i ji j ⊗ k j,k=1 k where the Γ C∞(U)aretheChristoﬀel symbols. ji ∈ Proposition 11.3 Every vector bundle, ξ, possesses a connection.

Proof .Sinceξ is locally trivial, we can find a locally finite open cover, (Uα)α,ofB such that 1 α π− (U )istrivial.If(f )isapartitionofunitysubordinatetothecover(U ) and if is α α α α ∇ any flat connection on ξ Uα,thenitisimmediatelyverifiedthat = f α ∇ α∇ α is a connection on ξ. 1 1 If ϕα : π− (Uα) Uα V and ϕβ : π− (Uβ) Uβ V are two overlapping trivializations, we know that for every→ b× U U ,wehave → × ∈ α ∩ β 1 ϕ ϕ− (b, u)=(b, g (b)u), α ◦ β αβ where g : U U GL(V ) is the transition function. As αβ α ∩ β → 1 1 ϕβ− (b, u)=ϕα− (b, gαβ(b)u),

if (s1,...,sn)istheframeoverUα associated with ϕα and (t1,...,tn)istheframeoverUβ associated with ϕβ,weseethat n

ti = gijsj, j=1 where gαβ =(gij).

Proposition 11.4 With the notations as above, the connection matrices, ωα and ωβ respectively over Uα and Uβ obey the tranformation rule 1 1 ωβ = gαβωαgαβ− +(dgαβ)gαβ− ,

where dgαβ =(dgij).

To prove the above proposition, apply to both side of the equations ∇ n

ti = gijsj j=1 and use ω and ω to express t and s .Thedetailsareleftasanexercise. α β ∇ i ∇ j 342 CHAPTER 11. CONNECTIONS AND CURVATURE IN VECTOR BUNDLES

In Morita [114] (Proposition 5.22), the order of the matrices in the equation of Proposi- tion 11.4 must be reversed.

If ξ = TM,thetangentbundleofsomesmoothmanifold,M,thenaconnectiononTM, also called a connection on M is a linear map,

: X(M) 1(M) X(M) = Hom (X(M), (X(M)), ∇ −→ A ⊗C∞(M) ∼ C∞(M)

since Γ(TM)=X(M). Then, for ﬁxed Y (M), the map Y is C∞(M)-linear, which implies that Y is a (1, 1) tensor. In a local∈ chart,X (U,ϕ), we have∇ ∇ n ∂ k ∂ ∂ = Γij , ∇ ∂xi ∂xj ∂xk k=1 k where the Γij are Christoﬀel symbols.

The covariant derivative, X , given by a connection, ,onTM,canbeextendedtoa covariant derivative, r,s,definedontensorfieldsin∇ Γ(M,T∇ r,s(M)), for all r, s 0, where ∇X ≥ r,s r s T (M)=T ⊗ M (T ∗M)⊗ . ⊗ 1,0 0,0 We already have X = X and it is natural to set X f = X[f]=df (X). Recall that there is an isomorphism∇ ∇ between the set of tensor fields,∇ Γ(M,Tr,s(M)), and the set of C∞(M)-multilinear maps,

1 1 Φ: (M) (M) X(M) X(M) C∞(M), A ×···×A × ×···× −→ r s

1 where (M)andX(M)areC∞(M)-modules. A The next proposition is left as an exercise. For help, see O’Neill [119], Chapter 2, Propo- sition 13 and Theorem 15.

Proposition 11.5 for every vector ﬁeld, X X(M), there is a unique family of R-linear map, r,s :Γ(M,Tr,s(M)) Γ(M,Tr,s(M)), with∈ r, s 0, such that ∇ → ≥ 0,0 1,0 (a) f = df (X), for all f C∞(M) and = , for all X X(M). ∇X ∈ ∇X ∇X ∈ (b) r1+r2,s1+s2 (S T )= r1,s1 (S) T + S r2,s2 (T ), for all S Γ(M,Tr1,s1 (M)) and ∇X ⊗ ∇X ⊗ ⊗∇X ∈ all T Γ(M,Tr2,s2 (M)). ∈ r 1,s 1 r,s r,s (c) X− − (cij(S)) = cij( X (S)), for all S Γ(M,T (M)) and all contractions, cij, of Γ∇(M,Tr,s(M)). ∇ ∈ 11.1. CONNECTIONS IN VECTOR BUNDLES AND RIEMANNIAN MANIFOLDS 343

Furthermore, ( 0,1θ)(Y )=X[θ(Y )] θ( Y ), ∇X − ∇X for all X, Y X(M) and all one-forms, θ 1(M) and for every S Γ(M,Tr,s(M)), with r + s 2, the∈ covariant derivative, r,s(S)∈, isA given by ∈ ≥ ∇X r,s ( X S)(θ1,...,θr,X1,...,Xs)=X[S(θ1,...,θr,X1,...,Xs)] ∇ r S(θ ,..., 0,1θ ,...,θ ,X ,...,X ) − 1 ∇X i r 1 s i=1 s S(θ ,...,...,θ ,X ,..., X ,...,X ), − 1 r 1 ∇X j s j=1 for all X ,...,X X(M) and all one-forms, θ ,...,θ 1(M). 1 s ∈ 1 r ∈A We define the covariant differential, r,sS,ofatensor,S Γ(M,Tr,s(M)), as the (r, s +1)-tensorgivenby ∇ ∈ ( r,sS)(θ ,...,θ ,X,X ,...,X )=( r,sS)(θ ,...,θ ,X ,...,X ), ∇ 1 r 1 s ∇X 1 r 1 s 1 for all X, Xj X(M)andallθi (M). For simplicity of notation we usually omit the superscripts r∈and s.Inparticular,for∈A S = g,theRiemannianmetriconM (a (0, 2) tensor), we get (g)(Y,Z)=d(g(Y,Z))(X) g( Y,Z) g(Y, Z), ∇X − ∇X − ∇X for all X, Y, Z X(M). We will see later on that a connection on M is compatible with a metric, g,iff ∈(g)=0. ∇X Everything we did in this section applies to complex vector bundles by considering complex vector spaces instead of real vector spaces, C-linear maps instead of R-linear map, and the space of smooth complex-valued functions, C∞(B; C) ∼= C∞(B) R C.Wealsousespaces of complex-valued differentials forms, ⊗

i i i 1 (B; C)= (B) C (B) C∞(B; C) = Γ T ∗B , A A ⊗ ∞ ∼ ⊗ C 1 i where is the trivial complex line bundle, B C,andwedefine (ξ)as C × A i i (ξ)= (B; C) C (B; ) Γ(ξ). A A ⊗ ∞ C AconnectionisaC-linear map, :Γ(ξ) 1(ξ), that satisfies the same Leibnitz-type rule ∇ →A as before. Obviously, every differential form in i(B; C)canbewrittenuniquelyasω + iη, with ω,η i(B). The exterior differential, A ∈A d: i(B; C) i+1(B; C) A →A is defined by d(ω + iη)=dω + idη.Weobtaincomplex-valueddeRhamcohomologygroups,

i i HDR(M; C)=HDR(M) C. ⊗R 344 CHAPTER 11. CONNECTIONS AND CURVATURE IN VECTOR BUNDLES 11.2 Curvature, Curvature Form and Curvature Ma- trix

If ξ = B V is the trivial bundle and is a ﬂat connection on ξ, we obviously have × ∇ = , ∇X ∇Y −∇Y ∇X ∇[X,Y ] where [X, Y ]istheLiebracketofthevectorﬁeldsX and Y . However, for general bundles and arbitrary connections, the above fails. The error term,

R(X, Y )= , ∇X ∇Y −∇Y ∇X −∇[X,Y ] measures what’s called the curvature of the connection. The curvature of a connection also turns up as the failure of a certain sequence involving the spaces i(ξ)tobeacochain complex. Recall that a connection on ξ is a linear map A

: 0(ξ) 1(ξ) ∇ A →A satisfying a Leibnitz-type rule. It is natural to ask whether can be extended to a family i i+1 ∇ of operators, d∇ : (ξ) (ξ), with properties analogous to d on ∗(B). A →A A This is indeed the case and we get a sequence of map,

d d 0 0(ξ) ∇ 1(ξ) ∇ 2(ξ) i(ξ) ∇ i+1(ξ) , −→ A −→ A −→ A −→ ···−→ A −→ A −→ ··· but in general, d∇ d∇ =0fails.Inparticular,d∇ =0generallyfails.Theterm ◦ ◦∇ K∇ = d∇ is the curvature form (or tensor) of the connection . As we will see it yields our previous◦∇ curvature, R,back. ∇

Our next goal is to define d∇.Forthis,wefirstdefineanC∞(B)-bilinear map

: i(ξ) j(η) i+j(ξ η) ∧ A ×A −→ A ⊗ as follows: (ω s) (τ t)=(ω τ) (s t), ⊗ ∧ ⊗ ∧ ⊗ ⊗ where ω i(B), τ j(B), s Γ(ξ), and t Γ(η), where we used the fact that ∈A ∈A ∈ ∈ Γ(ξ η)=Γ(ξ) Γ(η). ⊗ ⊗C∞(B) First, consider the case where ξ = 1 = B R,thetriviallinebundleoverB.Inthiscase, i(ξ)= i(B)andwehaveabilinearmap× A A : i(B) j(η) i+j(η) ∧ A ×A −→ A given by ω (τ t)=(ω τ) t. ∧ ⊗ ∧ ⊗ 11.2. CURVATURE AND CURVATURE FORM 345

For j =0,wehavethebilinearmap

: i(B) Γ(η) i(η) ∧ A × −→ A given by ω t = ω t. ∧ ⊗ It is clear that the bilinear map

: r(B) s(η) r+s(η) ∧ A ×A −→ A has the following properties:

(ω τ) θ = ω (τ θ) ∧ ∧ ∧ ∧ 1 θ = θ, ∧ for all ω i(B),τ j(B), θ k(ξ) and where 1 denotes the constant function in ∈A ∈A ∈A C∞(B)withvalue1.

Proposition 11.6 For every vector bundle, ξ, for all j 0, there is a unique R-linear map j j+1≥ (resp. C-linear if ξ is a complex VB), d∇ : (ξ) (ξ), such that A →A

(i) d∇ = for j =0. ∇ i i j (ii) d∇(ω t)=dω t +( 1) ω d∇t, for all ω (B) and all t (ξ). ∧ ∧ − ∧ ∈A ∈A

j j j j+1 Proof .Recallthat (ξ)= (B) Γ(ξ)anddeﬁned∇ : (B) Γ(ξ) (ξ)by A A ⊗C∞(B) A × →A j d∇(ω, s)=dω s +( 1) ω s, ⊗ − ∧∇ j j for all ω (B)andalls Γ(ξ). We claim that d∇ induces an R-linear map on (ξ)but ∈A ∈ A there is a complication as d∇ is not C∞(B)-bilinear. The way around this problem is to use Proposition 22.37. For this, we need to check that d∇ satisﬁes the condition of Proposition j 22.37, where the right action of C∞(B)on (B)isequaltotheleftaction,namelywedging: A 0 j f ω = ω ffC∞(B)= (B),ω (B). ∧ ∧ ∈ A ∈A i As is C∞(B)-bilinear and τ s = τ s for all τ (B)andalls Γ(ξ), we have ∧ ⊗ ∧ ∈A ∈ j d∇(ωf, s)=d(ωf) s +( 1) (ωf) s ⊗ − ∧∇ = d(ωf) s +( 1)jfω s ∧ − ∧∇ =((dω)f +( 1)jω df ) s +( 1)jfω s − ∧ ∧ − ∧∇ = fdω s +( 1)jω df s +( 1)jfω s ∧ − ∧ ∧ − ∧∇ 346 CHAPTER 11. CONNECTIONS AND CURVATURE IN VECTOR BUNDLES and

j d∇(ω, fs)=dω (fs)+( 1) ω (fs) ⊗ − ∧∇ = dω (fs)+( 1)jω (fs) ∧ − ∧∇ = fdω s +( 1)jω (df s + f s) ∧ − ∧ ⊗ ∇ = fdω s +( 1)jω (df s + f s) ∧ − ∧ ∧ ∇ = fdω s +( 1)jω df s +( 1)jfω s. ∧ − ∧ ∧ − ∧∇ j j+1 Thus, d∇(ωf, s)=d∇(ω, fs), and Proposition 22.37 shows that d∇ : (ξ) (ξ)isa A →A well-deﬁned R-linear map for all j 0. Furthermore, it is clear that d∇ = for j = 0. Now, for ω i(B)andt = τ s j(≥ξ)wehave ∇ ∈A ⊗ ∈A

d∇(ω (τ s)) = d∇((ω τ) s)) ∧ ⊗ ∧ ⊗ = d(ω τ) s +( 1)i+j(ω τ) s ∧ ⊗ − ∧ ∧∇ =(dω τ) s +( 1)i(ω dτ) s +( 1)i+j(ω τ) s ∧ ⊗ − ∧ ⊗ − ∧ ∧∇ = dω (τ s +( 1)iω (dτ s)+( 1)i+jω (τ s) ∧ ⊗ − i ∧ ⊗ − ∧ ∧∇ = dω (τ s)+( 1) ω d∇(τ s), ∧ ⊗ − i ∧ ∧ = dω (τ s)+( 1) ω d∇(τ s), ∧ ⊗ − ∧ ⊗ which proves (ii). As a consequence, we have the following sequence of linear maps:

d d 0 0(ξ) ∇ 1(ξ) ∇ 2(ξ) i(ξ) ∇ i+1(ξ) . −→ A −→ A −→ A −→ ···−→ A −→ A −→ ··· but in general, d∇ d∇ = 0 fails. Although generally d∇ =0fails,themapd∇ is ◦ ◦∇ ◦∇ C∞(B)-linear. Indeed,

(d∇ )(fs)=d∇(df s + f s) ◦∇ ⊗ ∇ = d∇(df s + f s) ∧ ∧∇ = ddf s df s + df s + f d∇( s) ∧ − ∧∇ ∧∇ ∧ ∇ = f(d∇ )(s)). ◦∇ 0 2 Therefore, d∇ : (ξ) (ξ)isaC∞(B)-linear map. However, recall that just before Proposition 11.2◦∇ weA showed→ thatA

Hom ( 0(ξ), i(ξ)) = i( om(ξ,ξ)), C∞(B) A A ∼ A H 2 therefore, d∇ ( om(ξ,ξ)), that is, d∇ is a two-form with values in Γ( om(ξ,ξ)). ◦∇ ∈ A H ◦∇ H Definition 11.2 For any vector bundle, ξ,andanyconnection, ,onξ,thevector-valued 2 ∇ two-form, R∇ = d∇ ( om(ξ,ξ)) is the curvature form (or curvature tensor)ofthe ◦∇∈A H connection .Wesaythat is a flat connection iff R∇ =0. ∇ ∇ 11.2. CURVATURE AND CURVATURE FORM 347

For simplicity of notation, we also write R for R∇.TheexpressionR∇ is also denoted F ∇ or K∇. As in the case of a connection, we can express R∇ locally in any local trivialization, 1 2 j ϕ: π− (U) U V ,ofξ.SinceR∇ = d∇ (ξ)= (B) C∞(B) Γ(ξ), if (s1,...,sn) is the frame→ associated× with (ϕ,U), then ◦∇∈A A ⊗ n R∇(s )= Ω s , i ij ⊗ j j=1 2 for some matrix, Ω= (Ωij), of two forms, Ωij (B). We call Ω= (Ωij)thecurvature matrix (or curvature form)associatedwiththelocaltrivialization.Therelationshipbetween∈A the connection form, ω,andthecurvatureform,Ω,issimple: Proposition 11.7 (Structure Equations) Let ξ be any vector bundle and let be any con- 1 ∇ nection on ξ. For every local trivialization, ϕ: π− (U) U V , the connection matrix, → × ω =(ωij), and the curvature matrix, Ω=(Ωij), associated with the local trivialization, (ϕ,U), are related by the structure equation: Ω=dω ω ω. − ∧ Proof .Bydeﬁnition, n (s )= ω s , ∇ i ij ⊗ j j=1 so if we apply d∇ and use property (ii) of Proposition 11.6 we get n d∇( (s )) = Ω s ∇ i ik ⊗ k k=1 n = d∇(ω s ) ij ⊗ j j=1 n n = dω s ω s ij ⊗ j − ij ∧∇ j j=1 j=1 n n n = dω s ω ω s ij ⊗ j − ij ∧ jk ⊗ k j=1 j=1 k=1 n n n = dω s ω ω s , ik ⊗ k − ij ∧ jk ⊗ k j=1 k=1 k=1 and so, n Ω = dω ω ω , ik ik − ij ∧ jk j=1 which, means that Ω=dω ω ω, − ∧ as claimed. 348 CHAPTER 11. CONNECTIONS AND CURVATURE IN VECTOR BUNDLES

Some other texts, including Morita [114] (Theorem 5.21) state the structure equations as Ω=dω + ω ω. ∧

Although this is far from obvious from Definition 11.2, the curvature form, R∇,isrelated to the curvature, R(X, Y ), defined at the beginning of Section 11.2. For this, we define the evaluation map

Ev : 2( om(ξ,ξ)) 0( om(ξ,ξ)) = Γ( om(ξ,ξ)), X,Y A H →A H H 2 2 as follows: For all X, Y X(B), all ω h ( om(ξ,ξ)) = (B) C∞(B) Γ( om(ξ,ξ)), set ∈ ⊗ ∈A H A ⊗ H Ev (ω h)=ω(X, Y )h. X,Y ⊗ 2 It is clear that this map is C∞(B)-linear and thus well-deﬁned on ( om(ξ,ξ)). (Recall that 0( om(ξ,ξ)) = Γ( om(ξ,ξ)) = Hom (Γ(ξ), Γ(ξ)).) We writeA H A H H C∞(B) R∇ =Ev (R∇) Hom (Γ(ξ), Γ(ξ)). X,Y X,Y ∈ C∞(B) Proposition 11.8 For any vector bundle, ξ, and any connection, , on ξ, for all X, Y ∇ ∈ X(B), if we let R(X, Y )= , ∇X ◦∇Y −∇Y ◦∇X −∇[X,Y ] then

R(X, Y )=RX,Y∇ .

Sketch of Proof . First, check that R(X, Y )isC∞(B)-linear and then work locally using the frame associated with a local trivialization using Proposition 11.7.

Remark: Proposition 11.8 implies that R(Y,X)= R(X, Y )andthatR(X, Y )(s)is − C∞(B)-linear in X, Y and s. 1 1 If ϕ : π− (U ) U V and ϕ : π− (U ) U V are two overlapping trivializa- α α → α × β β → β × tions, the relationship between the curvature matrices Ωα and Ωβ,isgivenbythefollowing proposition which is the counterpart of Proposition 11.4 for the curvature matrix:

1 1 Proposition 11.9 If ϕα : π− (Uα) Uα V and ϕβ : π− (Uβ) Uβ V are two overlapping trivializations of a vector bundle, ξ→, then× we have the following→ transformation× rule for the curvature matrices Ωα and Ωβ: 1 Ωβ = gαβΩαgαβ− , where g : U U GL(V ) is the transition function. αβ α ∩ β → Proof Sketch. Use the structure equations (Proposition 11.7) and apply d to the equations of Proposition 11.4. Proposition 11.7 also yields a formula for dΩ, know as Bianchi’s identity (in local form). 11.2. CURVATURE AND CURVATURE FORM 349

Proposition 11.10 (Bianchi’s Identity) For any vector bundle, ξ, any connection, , on ξ, if ω and Ω are respectively the connection matrix and the curvature matrix, in some∇ local trivialization, then dΩ=ω Ω Ω ω. ∧ − ∧ Proof .Ifweapplyd to the structure equation, Ω= dω ω ω,weget − ∧ dΩ=ddω dω ω + ω dω − ∧ ∧ = (Ω+ ω ω) ω + ω (Ω+ ω ω) − ∧ ∧ ∧ ∧ = Ω ω ω ω ω + ω Ω+ω ω ω − ∧ − ∧ ∧ ∧ ∧ ∧ = ω Ω Ω ω, ∧ − ∧ as claimed. i We conclude this section by giving a formula for d∇ d∇(t), for any t (ξ). Consider the special case of the bilinear map ◦ ∈A

: i(ξ) j(η) i+j(ξ η) ∧ A ×A −→ A ⊗ deﬁned just before Proposition 11.6 with j =2andη = om(ξ,ξ). This is the C∞-bilinear map H : i(ξ) 2( om(ξ,ξ)) i+2(ξ om(ξ,ξ)). ∧ A ×A H −→ A ⊗H We also have the evaluation map,

ev: j(ξ om(ξ,ξ)) = j(B) Γ(ξ) Hom (Γ(ξ), Γ(ξ)) A ⊗H ∼ A ⊗C∞(B) ⊗C∞(B) C∞(B) j(B) Γ(ξ)= j(ξ), −→ A ⊗C∞(B) A given by ev(ω s h)=ω h(s), ⊗ ⊗ ⊗ with ω j(B), s Γ(ξ)andh Hom (Γ(ξ), Γ(ξ)). Let ∈A ∈ ∈ C∞(B) : i(ξ) 2( om(ξ,ξ)) i+2(ξ) ∧ A ×A H −→ A be the composition

ev i(ξ) 2( om(ξ,ξ)) ∧ i+2(ξ om(ξ,ξ)) i+2(ξ). A ×A H −→ A ⊗H −→ A More explicitly, the above map is given (on generators) by

(ω s) H = ω H(s), ⊗ ∧ ∧ where ω i(B), s Γ(ξ)andH Hom (Γ(ξ), 2(ξ)) = 2( om(ξ,ξ)). ∈A ∈ ∈ C∞(B) A ∼ A H Proposition 11.11 For any vector bundle, ξ, and any connection, , on ξ the composition i i+2 i ∇ d∇ d∇ : (ξ) (ξ) maps t to t R∇, for any t (ξ). ◦ A →A ∧ ∈A 350 CHAPTER 11. CONNECTIONS AND CURVATURE IN VECTOR BUNDLES

i i Proof . Any t (ξ)issomelinearcombinationofelementsω s (B) C∞(B) Γ(ξ) and by Proposition∈A 11.6, we have ⊗ ∈A ⊗

i d∇ d∇(ω s)=d∇(dω s +( 1) ω s) ◦ ⊗ ⊗ −i+1 ∧∇ i i i = ddω s +( 1) dω s +( 1) dω s +( 1) ( 1) ω d∇ s ⊗ − ∧∇ − ∧∇ − − ∧ ◦∇ = ω d∇ s ∧ ◦∇ =(ω s) R∇, ⊗ ∧ as claimed.

Proposition 11.11 shows that d∇ d∇ =0iffR∇ = d∇ =0,thatis,ifftheconnection is flat. Thus, the sequence ◦ ◦∇ ∇ d d 0 0(ξ) ∇ 1(ξ) ∇ 2(ξ) i(ξ) ∇ i+1(ξ) , −→ A −→ A −→ A −→ ···−→ A −→ A −→ ··· is a cochain complex iff is flat. ∇ Again, everything we did in this section applies to complex vector bundles.

11.3 Parallel Transport

The notion of connection yields the notion of parallel transport in a vector bundle. First, we need to deﬁne the covariant derivative of a section along a curve.

Deﬁnition 11.3 Let ξ =(E,π,B,V ) be a vector bundle and let γ :[a, b] B be a smooth curve in B.Asmooth section along the curve γ is a smooth map, X :[a, b→] E,suchthat π(X(t)) = γ(t), for all t [a, b]. When ξ = TB,thetangentbundleofthemanifold,→ B,we use the terminology smooth∈ vector ﬁeld along γ.

Recall that the curve γ :[a, b] B is smooth iﬀ γ is the restriction to [a, b]ofasmooth curve on some open interval containing→ [a, b].

Proposition 11.12 Let ξ be a vector bundle, be a connection on ξ and γ :[a, b] B be ∇ → a smooth curve in B. There is a R-linear map, D/dt, deﬁned on the vector space of smooth sections, X, along γ, which satisﬁes the following conditions:

(1) For any smooth function, f :[a, b] R, → D(fX) df DX = X + f dt dt dt

(2) If X is induced by a global section, s Γ(ξ), that is, if X(t )=s(γ(t )) for all ∈ 0 0 t0 [a, b], then ∈ DX (t )=( s) . dt 0 ∇γ(t0) γ(t0) 11.3. PARALLEL TRANSPORT 351

Proof .Sinceγ([a, b]) is compact, it can be covered by a finite number of open subsets, Uα, such that (U ,ϕ )isalocaltrivialization.Thus,wemayassumethatγ :[a, b] U for some α α → local trivialization, (U,ϕ). As ϕ γ :[a, b] Rn,wecanwrite ◦ → ϕ γ(t)=(u (t),...,u (t)), ◦ 1 n where each u = pr ϕ γ is smooth. Now (see Definition 3.13), for every g C∞(B), as i i ◦ ◦ ∈ n d d d 1 dui ∂ dγt0 (g)= (g γ) = ((g ϕ− ) (ϕ γ)) = g, dt dt ◦ dt ◦ ◦ ◦ dt ∂xi t0 t0 t0 i=1 γ(t0) since by definition of γ(t0), d γ(t )=dγ 0 t0 dt t0 (see the end of Section 3.2), we have n dui ∂ γ(t0)= . dt ∂xi i=1 γ(t0) If (s1,...,sn)isaframeoverU,wecanwrite n

X(t)= Xi(t)si(γ(t)), i=1 for some smooth functions, Xi.Then,conditions(1)and(2)implythat DX n dX = j s (γ(t)) + X (t) (s (γ(t))) dt dt j j ∇γ(t) j j=1 and since n dui ∂ γ(t)= , dt ∂x i=1 i γ(t) k there exist some smooth functions, Γij,sothat n dui dui k ∂ γ(t)(sj(γ(t))) = (sj(γ(t))) = Γijsk(γ(t)). ∇ dt ∇ ∂xi dt i=1 i,k It follows that DX n dX du = k + Γk i X s (γ(t)). dt dt ij dt j k ij k=1 Conversely, the above expression deﬁnes a linear operator, D/dt,anditiseasytocheckthat it satisﬁes (1) and (2). The operator, D/dt is often called covariant derivative along γ and it is also denoted by or simply . ∇γ(t) ∇γ 352 CHAPTER 11. CONNECTIONS AND CURVATURE IN VECTOR BUNDLES

Deﬁnition 11.4 Let ξ be a vector bundle and let be a connection on ξ.Foreverycurve, γ :[a, b] B,inB,asection,X,alongγ is parallel∇ (along γ) iﬀ → DX =0. dt

If M was embedded in Rd (for some d), then to say that X is parallel along γ would

mean that the directional derivative, (Dγ X)(γ(t)), is normal to Tγ(t)M. The following proposition can be shown using the existence and uniqueness of solutions of ODE’s (in our case, linear ODE’s) and its proof is omitted:

Proposition 11.13 Let ξ be a vector bundle and let be a connection on ξ. For every C1 ∇ 1 curve, γ :[a, b] B, in B, for every t [a, b] and every v π− (γ(t)), there is a unique parallel section,→X, along γ such that X(∈t)=v. ∈

For the proof of Proposition 11.13 it is suﬃcient to consider the portions of the curve γ contained in some local trivialization. In such a trivialization, (U,ϕ), as in the proof of Proposition 11.12, using a local frame, (s1,...,sn), over U,wehave

DX n dX du = k + Γk i X s (γ(t)), dt dt ij dt j k ij k=1 with u = pr ϕ γ.Consequently,X is parallel along our portion of γ iﬀthe system of i i ◦ ◦ linear ODE’s in the unknowns, Xk, dX du k + Γk i X =0,k=1,...,n, dt ij dt j ij is satisﬁed.

Remark: Proposition 11.13 can be extended to piecewise C1 curves.

Definition 11.5 Let ξ be a vector bundle and let be a connection on ξ.Foreverycurve, γ :[a, b] B,inB,foreveryt [a, b], the parallel∇ transport from γ(a) to γ(t) along γ is → ∈ 1 1 the linear map from the fibre, π− (γ(a)), to the fibre, π− (γ(t)), which associates to any 1 1 v π− (γ(a)) the vector X (t) π− (γ(t)), where X is the unique parallel section along γ ∈ v ∈ v with Xv(a)=v.

The following proposition is an immediate consequence of properties of linear ODE’s:

Proposition 11.14 Let ξ =(E,π,B,V ) be a vector bundle and let be a connection on ξ. For every C1 curve, γ :[a, b] B, in B, the parallel transport along∇ γ deﬁnes for every → 1 1 1 t [a, b] a linear isomorphism, Pγ : π− (γ(a)) π− (γ(t)), between the ﬁbres π− (γ(a)) and ∈1 → π− (γ(t)). 11.4. CONNECTIONS COMPATIBLE WITH A METRIC 353

In particular, if γ is a closed curve, that is, if γ(a)=γ(b)=p,weobtainalinear 1 isomorphism, Pγ,oftheﬁbreEp = π− (p), called the holonomy of γ.Theholonomy group of based at p, denoted Holp( ), is the subgroup of GL(V,R)givenby ∇ ∇

Holp( )= Pγ GL(V,R) γ is a closed curve based at p . ∇ { ∈ | } If B is connected, then Hol ( )dependsonthebasepointp B up to conjugation and so p ∇ ∈ Holp( ) and Holq( )areisomorphicforallp, q B.Inthiscase,itmakessensetotalk about∇ the holonomy∇ group of .Ifξ = TB,thetangentbundleofamanifold,∈ B,byabuse of language, we call Hol ( )the∇ holonomy group of B. p ∇ 11.4 Connections Compatible with a Metric; Levi-Civita Connections

If a vector bundle (or a Riemannian manifold), ξ,hasametric,thenitisnaturaltodeﬁne when a connection, ,onξ is compatible with the metric. So, assume the vector bundle, ξ, has a metric, , ∇.Wecanusethismetrictodeﬁnepairings − − 1(ξ) 0(ξ) 1(B)and 0(ξ) 1(ξ) 1(B) A ×A −→ A A ×A −→ A as follows: Set (on generators)

ω s ,s = s ,ω s = ω s ,s , ⊗ 1 2 1 ⊗ 2 1 2 1 for all ω (B),s1,s2 Γ(ξ)andwhere s1,s2 is the function in C∞(B)givenby b s (b)∈,sA(b) ,forallb ∈B.Moregenerally,wedefineapairing → 1 2 ∈ i(ξ) j(ξ) i+j(B), A ×A −→ A by ω s ,η s = s ,s ω η, ⊗ 1 ⊗ 2 1 2 ∧ for all ω i(B),η j(B), s ,s Γ(ξ). ∈A ∈A 1 2 ∈ Definition 11.6 Given any metric, , ,onavectorbundle,ξ,aconnection, ,onξ is compatible with the metric,forshort,a−metric− connection iff ∇

d s ,s = s ,s + s , s , 1 2 ∇ 1 2 1 ∇ 2 for all s ,s Γ(ξ). 1 2 ∈

In terms of version-two of a connection, is a metric connection iﬀ ∇X X( s ,s )= s ,s + s , s , 1 2 ∇X 1 2 1 ∇X 2 354 CHAPTER 11. CONNECTIONS AND CURVATURE IN VECTOR BUNDLES

for every vector ﬁeld, X X(B). ∈ Deﬁnition 11.6 remains unchanged if ξ is a complex vector bundle. The condition of compatibility with a metric is nicely expressed in a local trivialization. Indeed, let (U,ϕ) be a local trivialization of the vector bundle, ξ (of rank n). Then, using the Gram-Schmidt procedure, we obtain an orthonormal frame, (s1,...,sn), over U. Proposition 11.15 Using the above notations, if ω =(ω ) is the connection matrix of ij ∇ w.r.t. (s1,...,sn), then ω is skew-symmetric. Proof .Since n e = ω s ∇ i ij ⊗ j j=1 and since si,sj = δij (as (s1,...,sn)isorthonormal),wehaved si,sj =0onU.Conse- quently 0=d s ,s i j = si,sj + si, sj ∇n ∇ n = ω s ,s + s , ω s ik ⊗ k j i jl ⊗ l k=1 l=1 n n = ω s ,s + ω s ,s ik k j jl i l k=1 l=1 = ωij + ωji, as claimed. In Proposition 11.15, if ξ is a complex vector bundle, then ω is skew-Hermitian. This means that ω = ω, − where ω is the conjugate matrix of ω,thatis,(ω)ij = ωij.Itisalsoeasytoprovethatmetric connections exist. Proposition 11.16 Let ξ be a rank n vector with a metric, , . Then, ξ, possesses metric connections. − −

Proof .Wecanpickalocallyﬁnitecover,(Uα)α,ofB such that (Uα,ϕα)isalocaltrivialization of ξ. Then, for each (Uα,ϕα), we use the Gram-Schmidt procedure to obtain α α α an orthonormal frame, (s1 ,...,sn), over Uα and we let be the trivial connection on 1 α ∇ π− (Uα). By construction, is compatible with the metric. We ﬁnish the argumemt by using a partition of unity, leaving∇ the details to the reader. If ξ is a complex vector bundle, then we use a Hermitian metric and we call a connection compatible with this metric a Hermitian connection. In any local trivialization, the connection matrix, ω, is skew-Hermitian. The existence of Hermitian connections is clear. If is a metric connection, then the curvature matrices are also skew-symmetric. ∇ 11.4. CONNECTIONS COMPATIBLE WITH A METRIC 355

Proposition 11.17 Let ξ be a rank n vector bundle with a metric, , . In any local − − trivialization of ξ, the curvature matrix, Ω=(Ωij) is skew-symmetric. If ξ is a complex vector bundle, then Ω=(Ωij) is skew-Hermitian.

Proof .Bythestructureequation(Proposition11.7),

Ω=dω ω ω, − ∧ that is, Ω = dω n ω ω ,so,usingtheskewsymetryofω and wedge, ij ij − k=1 ik ∧ kj ij n Ω = dω ω ω ji ji − jk ∧ ki k=1 n = dω ω ω − ij − kj ∧ ik k=1 n = dω + ω ω − ij ik ∧ kj k=1 = Ω , − ij as claimed. We now restrict our attention to a Riemannian manifold, that is, to the case where our bundle, ξ,isthetangentbundle,ξ = TM,ofsomeRiemannianmanifold,M.Weknow from Proposition 11.16 that metric connections on TM exist. However, there are many metric connections on TM and none of them seems more relevant than the others. If M is aRiemannianmanifold,themetric, , ,onM is often denoted g.Inthiscase,forevery − − chart, (U,ϕ), we let g C∞(M)bethefunctiondeﬁnedby ij ∈ ∂ ∂ gij(p)= , . ∂xi ∂xj p pp (Note the unfortunate clash of notation with the transitions functions!)

The notations g = ij gijdxi dxj or simply g = ij gijdxidxj are often used to denote the metric in local coordinates. We⊗ observed immediately after stating Proposition 11.5 that the covariant diﬀerential, g,oftheRiemannianmetric, g,onM is given by ∇ (g)(Y,Z)=d(g(Y,Z))(X) g( Y,Z) g(Y, Z), ∇X − ∇X − ∇X for all X, Y, Z X(M). Therefore, a connection, ,onaRiemannianmanifold,(M,g), is compatible with∈ the metric iﬀ ∇ g =0. ∇ It is remarkable that if we require a certain kind of symmetry on a metric connection, then it is uniquely determined. Such a connection is known as the Levi-Civita connection. 356 CHAPTER 11. CONNECTIONS AND CURVATURE IN VECTOR BUNDLES

The Levi-Civita connection can be characterized in several equivalent ways, a rather simple way involving the notion of torsion of a connection. Recall that one way to introduce the curvature is to view it as the “error term”

R(X, Y )= . ∇X ∇Y −∇Y ∇X −∇[X,Y ] Another natural error term is the torsion, T (X, Y ), of the connection, ,givenby ∇ T (X, Y )= Y X [X, Y ], ∇X −∇Y − which measures the failure of the connection to behave like the Lie bracket.

Another way to characterize the Levi-Civita connection uses the cotangent bundle, T ∗M. It turns out that a connection, ,onavectorbundle(metricornot),ξ,naturallyinduces ∇ aconnection, ∗, on the dual bundle, ξ∗. Now, if is a connection on TM,then ∗ is is a ∇ ∇ 1 ∇ connection on T ∗M,namely,alinearmap, ∗ :Γ(T ∗M) (M) Γ(T ∗M), that is ∇ →A ⊗C∞(B) 1 1 1 ∗ : (M) (M) (M) = Γ(T ∗M T ∗M), ∇ A →A ⊗C∞(B) A ∼ ⊗ 1 since Γ(T ∗M)= (M). If we compose this map with ,wegetthemap A ∧

1(M) ∇∗ 1(M) 1(M) ∧ 2(M). A −→ A ⊗C∞(B) A −→ A Then, miracle, a metric connection is the Levi-Civita connection iﬀ

d = ∗, ∧◦∇ where d: 1(M) 2(M) is exterior diﬀerentiation. There is also a nice local expression of the aboveA equation.→A First, we consider the deﬁnition involving the torsion.

Proposition 11.18 (Levi-Civita, Version 1) Let M be any Riemannian manifold. There is a unique, metric, torsion-free connection, , on M, that is, a connection satisfying the conditions ∇

X( Y,Z )= Y,Z + Y, Z ∇X ∇X Y X =[X, Y ], ∇X −∇Y for all vector ﬁelds, X, Y, Z X(M). This connection is called the Levi-Civita connection (or canonical connection∈) on M. Furthermore, this connection is determined by the Koszul formula

2 Y,Z = X( Y,Z )+Y ( X, Z ) Z( X, Y ) ∇X − Y,[X, Z] X, [Y,Z] Z, [Y,X] . − − − 11.4. CONNECTIONS COMPATIBLE WITH A METRIC 357

Proof . First, we prove uniqueness. Since our metric is a non-degenerate bilinear form, it suﬃces to prove the Koszul formula. As our connection is compatible with the metric, we have

X( Y,Z )= Y,Z + Y, Z ∇X ∇X Y ( X, Z )= X, Z + X, Z ∇Y ∇Y Z( X, Y )= X, Y X, Y − −∇Z − ∇Z and by adding up the above equations, we get

X( Y,Z )+Y ( X, Z) Z( X, Y )= Y, Z X − ∇X −∇Z + X, Z Y ∇Y −∇Z + Z, Y + X . ∇X ∇Y Then, using the fact that the torsion is zero, we get

X( Y,Z )+Y ( X, Z ) Z( X, Y )= Y,[X, Z] + X, [Y,Z] − + Z, [Y,X] +2 Z, Y ∇X which yields the Koszul formula. Next, we prove existence. We begin by checking that the right-hand side of the Koszul formula is C∞(M)-linear in Z,forX and Y fixed. But then, the linear map Z Y,Z → ∇X induces a one-form and X Y is the vector field corresponding to it via the non-degenerate pairing. It remains to check∇ that satisfies the properties of a connection, which it a bit tedious (for example, see Kuhnel [91],∇ Chapter 5, Section D).

Remark: In a chart, (U,ϕ), if we set

∂ ∂kgij = (gij) ∂xk then it can be shown that the Christoﬀel symbols are given by

1 n Γk = gkl(∂ g + ∂ g ∂ g ), ij 2 i jl j il − l ij l=1 kl where (g )istheinverseofthematrix(gkl). Let us now consider the second approach to torsion-freeness. For this, we have to explain how a connection, ,onavectorbundle,ξ =(E,π,B,V ), induces a connection, ∗,onthe ∇ 1 ∇ dual bundle, ξ∗. First, there is an evaluation map Γ(ξ ξ∗) Γ( )orequivalently, ⊗ −→

, :Γ(ξ) Hom (Γ(ξ),C∞(B)) C∞(B), − − ⊗C∞(B) C∞(B) −→ 358 CHAPTER 11. CONNECTIONS AND CURVATURE IN VECTOR BUNDLES

given by s ,s∗ = s∗(s ),sΓ(ξ),s∗ Hom (Γ(ξ),C∞(B)) 1 2 2 1 1 ∈ 2 ∈ C∞(B) and thus a map

k k id , k k (ξ ξ∗)= (B) Γ(ξ ξ∗) ⊗− − (B) C∞(B) = (B). A ⊗ A ⊗C∞(B) ⊗ −→ A ⊗C∞(B) ∼ A Using this map we obtain a pairing

i j i+j i+j ( , ): (ξ) (ξ∗) ∧ (ξ ξ∗) (B), − − A ⊗A −→ A ⊗ −→ A given by (ω s ,η s∗)=(ω η) s ,s∗ , ⊗ 1 ⊗ 2 ∧ ⊗ 1 2 i j where ω (B), η (B), s Γ(ξ), s∗ Γ(ξ∗). It is easy to check that this pairing is ∈A ∈A 1 ∈ 2 ∈ non-degenerate. Then, given a connection, , on a rank n vector bundle, ξ,wedeﬁne ∗ ∇ ∇ on ξ∗ by d s ,s∗ = (s ),s∗ + s , ∗(s∗) , 1 2 ∇ 1 2 1 ∇ 2 where s Γ(ξ)ands∗ Γ(ξ∗). Because the pairing ( , )isnon-degenerate, ∗ is well- 1 ∈ 2 ∈ − − ∇ deﬁned and it is immediately that it is a connection on ξ∗.Letusseehowitisexpressed locally. If (U,ϕ)isalocaltrivializationand(s1,...,sn)istheframeoverU associated with (U,ϕ), then let (θ1,...,θn)bethedualframe(calledacoframe). We have

s ,θ = θ (s )=δ , 1 i, j n. j i i j ij ≤ ≤ Recall that n s = ω s ∇ j jk ⊗ k k=1 and write n ∗θ = ω∗ θ . ∇ i ik ⊗ k k=1 Applying d to the equation s ,θ = δ and using the equation deﬁning ∗,weget j i ij ∇ 0=d s ,θ j i = (sj),θi + sj, ∗(θi) ∇n ∇ n = ω s ,θ + s , ω∗ θ jk ⊗ k i j il ⊗ l k=1 l=1 n n

= ωjk sk,θi + ωil∗ sj,θl k=1 l=1 = ωji + ωij∗ . 11.4. CONNECTIONS COMPATIBLE WITH A METRIC 359

Therefore, if we write ω∗ =(ωij∗ ), we have

ω∗ = ω. −

If is a metric connection, then ω is skew-symmetric, that is, ω = ω.Inthiscase, ∇ − ω∗ = ω = ω. − If ξ is a complex vector bundle, then there is a problem because if (s1,...,sn)isaframe over U,thentheθj(b)’s deﬁned by

s (b),θ (b) = δ i j ij are not linear, but instead conjugate-linear. (Recall that a linear form, θ,isconjugate linear (or semi-linear)iffθ(λu)=λθ(u), for all λ C.) Instead of ξ∗,weneedtoconsiderthe ∈ bundle ξ∗,whichisthebundlewhosefibreoverb B consist of all conjugate-linear forms 1 ∈ over π− (b). In this case, the evaluation pairing, s, θ is conjugate-linear in s and we find that ω∗ = ω,whereω∗ is the connection matrix of ξ∗ over U.Ifξ is a Hermitian bundle, as − ω is skew-Hermitian, we find that ω∗ = ω,whichmakessensesinceξ and ξ∗ are canonically isomorphic. However, this does not give any information on ξ∗.Forthis,weconsiderthe conjugate bundle, ξ.Thisisthebundleobtainedfromξ by redefining the vector space 1 structure on each fibre, π− (b), b B,sothat ∈ (x + iy)v =(x iy)v, − 1 for every v π− (b). If ω is the connection matrix of ξ over U,thenω is the connection ∈ matrix of ξ over U.Ifξ has a Hermitian metric, it is easy to prove that ξ∗ and ξ are canonically isomorphic (see Proposition 11.32). In fact, the Hermitian product, , , − − establishes a pairing between ξ and ξ∗ and, basically as above, we can show that if ω is the connection matrix of ξ over U,thenω∗ = ω is the the connection matrix of ξ∗ over U. − As ω is skew-Hermitian, ω∗ = ω.

Going back to a connection, ,onamanifold,M,theconnection, ∗,isalinearmap, ∇ ∇ 1 1 1 ∗ : (M) (M) (M) = (X(M))∗ (X(M))∗ = (X(M) X(M))∗. ∇ A −→ A ⊗A ∼ ⊗C∞(M) ∼ ⊗C∞(M)

Let us figure out what ∗ is using the above interpretation. By the definition of ∗, ∧◦∇ ∇ ∗(X, Y )=X(θ(Y )) θ( Y ), ∇θ − ∇X for every one-form, θ 1(M)andallvectorfields,X, Y X(M). Applying ,weget ∈A ∈ ∧ ∗(X, Y ) ∗(Y,X)=X(θ(Y )) θ( Y ) Y (θ(X)) + θ( X) ∇θ −∇θ − ∇X − ∇Y = X(θ(Y )) Y (θ(X)) θ( Y X). − − ∇X −∇Y However, recall that

dθ(X, Y )=X(θ(Y )) Y (θ(X)) θ([X, Y ]), − − 360 CHAPTER 11. CONNECTIONS AND CURVATURE IN VECTOR BUNDLES

so we get

( ∗)(θ)(X, Y )= ∗(X, Y ) ∗(Y,X) ∧◦∇ ∇θ −∇θ = dθ(X, Y ) θ( Y X [X, Y ]) − ∇X −∇Y − = dθ(X, Y ) θ(T (X, Y )). − 1 It follows that for every θ (M), we have ( ∗)θ = dθ iff θ(T (X, Y )) = 0 for all ∈A ∧◦∇ X, Y X(M), that is iff T (X, Y )=0,forallX, Y X(M). We record this as ∈ ∈ Proposition 11.19 Let ξ be a manifold with connection . Then, is torsion-free (i.e., ∇ ∇ T (X, Y )= Y X [X, Y ]=0, for all X, Y X(M)) iff ∇X −∇Y − ∈ ∗ = d, ∧◦∇ where d: 1(M) 2(M) is exterior differentiation. A →A Proposition 11.19 together with Proposition 11.18 yield a second version of the Levi- Civita Theorem:

Proposition 11.20 (Levi-Civita, Version 2) Let M be any Riemannian manifold. There is a unique, metric connection, , on M, such that ∇ ∗ = d, ∧◦∇ where d: 1(M) 2(M) is exterior diﬀerentiation. This connection is equal to the Levi- Civita connectionA → inA Proposition 11.18.

Remark: If is the Levi-Civita connection of some Riemannian manifold, M,forevery ∇ chart, (U,ϕ), we have ω∗ = ω,whereω is the connection matrix of over U and ω∗ is the ∇ connection matrix of the dual connection ∗.ThisimpliesthattheChristoﬀelsymbolsof ∇ and ∗ over U are identical. Furthermore, ∗ is a linear map ∇ ∇ ∇ 1 ∗ : (M) Γ(T ∗M T ∗M). ∇ A −→ ⊗ Thus, locally in a chart, (U,ϕ), if (as usual) we let x = pr ϕ,thenwecanwrite i i ◦ k ∗(dx )= Γ dx dx . ∇ k ij i ⊗ j ij

Now, if we want ∗ = d,wemusthave ∗(dx )=ddx =0,thatis ∧◦∇ ∧∇ k k k k Γij =Γji, for all i, j. Therefore, torsion-freeness can indeed be viewed as a symmetry condition on the Christoﬀel symbols of the connection . ∇ 11.4. CONNECTIONS COMPATIBLE WITH A METRIC 361

Our third version is a local version due to Elie´ Cartan. Recall that locally in a chart, (U,ϕ), the connection, ∗,isgivenbythematrix,ω∗,suchthatω∗ = ω where ω is the connection matrix of TM∇ over U.Thatis,wehave −

n ∗θ = ω θ , ∇ i − ji ⊗ j j=1 for some one-forms, ω 1(M). Then, ij ∈A n ∗θ = ω θ ∧◦∇ i − ji ∧ j j=1

so the requirement that d = ∗ is expressed locally by ∧◦∇ n dθ = ω θ . i − ji ∧ j j=1

In addition, since our connection is metric, ω is skew-symmetric and so, ω∗ = ω.Then,itis not too surprising that the following proposition holds:

Proposition 11.21 Let M be a Riemannian manifold with metric, , . For every chart, − − (U,ϕ), if (s1,...,sn) is the frame over U associated with (U,ϕ) and (θ1,...,θn) is the corresponding coframe (dual frame), then there is a unique matrix, ω =(ωij), of one-forms, ω 1(M), so that the following conditions hold: ij ∈A (i) ω = ω . ji − ij n (ii) dθ = ω θ or, in matrix form, dθ = ω θ. i ij ∧ j ∧ j=1 Proof .Thereisadirectproofusingacombinatorialtrick,forinstance,seeMorita[114], Chapter 5, Proposition 5.32 or Milnor and Stasheﬀ[110], Appendix C, Lemma 8. On the other hand, if we view ω =(ωij)asaconnectionmatrix,thenweobservedthat(i)assertsthat the connection is metric and (ii) that it is torsion-free. We conclude by applying Proposition 11.20. As an example, consider an orientable (compact) surface, M, with a Riemannian metric. Pick any chart, (U,ϕ), and choose an orthonormal coframe of one-forms, (θ1,θ2), such that Vol = θ θ on U.Then,wehave 1 ∧ 2 dθ = a θ θ 1 1 1 ∧ 2 dθ = a θ θ 2 2 1 ∧ 2 362 CHAPTER 11. CONNECTIONS AND CURVATURE IN VECTOR BUNDLES

for some functions, a1,a2,andwelet

ω12 = a1θ1 + a2θ2.

Clearly,

0 ω θ 0 a θ + a θ θ dθ 12 1 = 1 1 2 2 1 = 1 ω12 0 θ (a1θ1 + a2θ2)0 θ dθ − 2 − 2 2 which shows that 0 ω12 ω = ω∗ = ω 0 − 12 corresponds to the Levi-Civita connection on M.LetΩ=dω ω ω,weseethat − ∧ 0 dω Ω= 12 . dω 0 − 12 As M is oriented and as M has a metric, the transition functions are in SO(2). We easily check that cos t sin t 0 dω cos t sin t 0 dω 12 = 12 , sin t cos t dω 0 sin t −cos t dω 0 − − 12 − 12 which shows that Ωis a global two-form called the Gauss-Bonnet 2-form of M.Then,there is a function, κ,theGaussian curvature of M such that

dω = κVol, 12 − where Vol is the oriented volume form on M.TheGauss-Bonnet Theorem for orientable surfaces asserts that

dω12 =2πχ(M), M where χ(M)istheEuler characteristic of M.

Remark: The Levi-Civita connection induced by a Riemannian metric, g,canalsobede- ﬁned in terms of the Lie derivative of the metric, g.ThisistheapproachfollowedinPetersen [121] (Chapter 2). If θX is the one-form given by

θX = iX g, that is, (iX g)(Y )=g(X, Y )forallX, Y X(M)andifLX g is the Lie derivative of the symmetric (0, 2) tensor, g,deﬁnedsothat∈

(L g)(Y,Z)=X(g(Y,Z)) g(L Y,Z) g(Y,L Z) X − X − X 11.4. CONNECTIONS COMPATIBLE WITH A METRIC 363

(see Proposition 8.18), then, it is proved in Petersen [121] (Chapter 2, Theorem 1) that the Levi-Civita connection is defined implicitly by the formula 2g( Y,Z)=(L g)(X, Z)+(dθ )(X, Z). ∇X Y Y We conclude this section with various useful facts about torsion-free or metric connections. First, there is a nice characterization for the Levi-Civita connection induced by a Riemannian manifold over a submanifold. The proof of the next proposition is left as an exercise. Proposition 11.22 Let M be any Riemannian manifold and let N be any submanifold of M equipped with the induced metric. If M and N are the Levi-Civita connections on M and N, respectively, induced by the metric∇ on M,∇ then for any two vector fields, X and Y in X(M) with X(p),Y(p) T N, for all p N, we have ∈ p ∈ N M Y =( Y ), ∇X ∇X M M where ( Y )(p) is the orthogonal projection of Y (p) onto T N, for every p N. ∇X ∇X p ∈ In particular, if γ is a curve on a surface, M R3,thenavectorfield,X(t), along γ is ⊆ parallel iff X(t)isnormaltothetangentplane,Tγ(t)M. For any manifold, M,andanyconnection, ,onM,if is torsion-free, then the Lie derivative of any (p, 0)-tensor can be expressed in∇ terms of ∇(see Proposition 8.18). ∇ q Proposition 11.23 For every (0,q)-tensor, S Γ(M,(T ∗M)⊗ ), we have ∈ q (L S)(X ,...,X )=X[S(X ,...,X )] + S(X ,..., X ,...,X ), X 1 q 1 q 1 ∇X i q i=1 for all X ,...,X ,X X(M). 1 q ∈ Proposition 11.23 is proved in Gallot, Hullin and Lafontaine [60] (Chapter 2, Proposition 2.61). Using Proposition 8.13 it is also possible to give a formula for dω(X0 ...,Xk)interms of the ,whereω is any k-form, namely ∇Xi k i dω(X0 ...,Xk)= ( 1) X ω(X1,...,Xi 1,X0,Xi+1,...,Xk). − ∇ i − i=0 Again, the above formula in proved in Gallot, Hullin and Lafontaine [60] (Chapter 2, Propo- sition 2.61). If is a metric connection, then we can say more about the parallel transport along a curve.∇ Recall from Section 11.3, Definition 11.4, that a vector field, X,alongacurve,γ,is parallel iff DX =0. dt The following proposition will be needed: 364 CHAPTER 11. CONNECTIONS AND CURVATURE IN VECTOR BUNDLES

Proposition 11.24 Given any Riemannian manifold, M, and any metric connection, , on M, for every curve, γ :[a, b] M, on M, if X and Y are two vector fields along γ, then∇ → d DX DY X(t),Y(t) = ,Y + X, . dt dt dt Proof . (After John Milnor.) Using Proposition 11.13, we can pick some parallel vector fields, Z1,...,Zn,alongγ,suchthatZ1(a),...,Zn(a)formanorthogonalframe.Then,asinthe proof of Proposition 11.12, in any chart, (U,ϕ), the vector fields X and Y along the portion of γ in U can be expressed as n ∂ n ∂ X = X (t) ,Y= Y (t) , i ∂x i ∂x i=1 i i=1 i and n dui ∂ γ(t0)= , dt ∂xi i=1 γ(t0) with ui = pri ϕ γ.LetX and Y be two parallel vector fields along γ. As the vector fields, ∂ , can be extended◦ ◦ over the whole space, M,as is a metric connection and as X and Y ∂xi ∇ are parallel along γ,weget d( X,Y )(γ)=γ[ X,Y ]= X,Y + X, Y =0. ∇γ ∇γ So, X,Y is constant along the portion of γ in U.Butthen, X,Y is constant along γ. Applying this to the Zi(t), we see that Z1(t),...,Zn(t)isanorthogonalframe,forevery t [a, b]. Then, we can write ∈

X = xiZi,Y= yjZj, i=1 j=1 where xi(t)andyi(t)aresmoothreal-valuedfunctions.Itfollowsthat n X(t),Y(t) = x y i i i=1 and that DX dx DZ dx DY dy DZ dy = i Z + x i = i Z , = i Z + y i = i Z . dt dt i i dt dt i dt dt i i dt dt i Therefore, DX DY n dx dy d ,Y + X, = i y + x i = X(t),Y(t) , dt dt dt i i dt dt i=1 as claimed. Using Proposition 11.24 we get 11.5. DUALITY BETWEEN VECTOR FIELDS AND DIFFERENTIAL FORMS 365

Proposition 11.25 Given any Riemannian manifold, M, and any metric connection, , on M, for every curve, γ :[a, b] M, on M, if X and Y are two vector ﬁelds along γ that∇ are parallel, then → X, Y = C, for some constant, C. In particular, X(t) is constant. Furthermore, the linear isomorphism, P : T T , is an isometry. γ γ(a) → γ(b) Proof .FromProposition11.24,wehave

d DX DY X(t),Y(t) = ,Y + X, . dt dt dt As X and Y are parallel along γ, we have DX/dt =0andDY/dt =0,so

d X(t),Y(t) =0, dt

which shows that X(t),Y(t) is constant. Therefore, for all v, w Tγ(a),ifX and Y are the unique vector ﬁelds parallel along γ such that X(a)=v and Y (a)=∈ w given by Proposition 11.13, we have

P (v),P (w) = X(b),Y(b) = X(a),Y(a) = u, v , γ γ

which proves that Pγ is an isometry.

In particular, Proposition 11.25 shows that the holonomy group, Holp( ), based at p,is asubgroupofO(n). ∇

11.5 Duality between Vector Fields and Diﬀerential Forms and their Covariant Derivatives

If (M, , )isaRiemannianmanifold,thentheinnerproduct, , ,onT M,estab- − − − −p p lishes a canonical duality between TpM and Tp∗M, as explained in Section 22.1. Namely, we have the isomorphism, : TpM Tp∗M,deﬁnedsuchthatforeveryu TpM,thelinear → ∈ form, u T ∗M,isgivenby ∈ p u(v)= u, v v T M. p ∈ p

The inverse isomorphism, : Tp∗M TpM,isdeﬁnedsuchthatforeveryω Tp∗M,the → ∈ vector, ω ,istheuniquevectorinTpM so that

ω,v = ω(v),vT M. p ∈ p 366 CHAPTER 11. CONNECTIONS AND CURVATURE IN VECTOR BUNDLES

The isomorphisms and induce isomorphisms between vector fields, X X(M), and one- ∈ forms, ω 1(M): A vector field, X X(M), yields the one-form, X 1(M), given by ∈A ∈ ∈A (X )p =(Xp) and a one-form, ω 1(M), yields the vector field, ω X(M), given by ∈A ∈ (ω )p =(ωp) ,

so that ω (v)= (ω ),v ,vT M, p M. p p p ∈ p ∈ In particular, for every smooth function, f C∞(M), the vector ﬁeld corresponding to the one-form, df ,isthegradient,gradf,off.Thegradientof∈ f is uniquely determined by the condition (grad f) ,v = df (v),vT M, p M. p p p ∈ p ∈

Recall from Proposition 11.5 that the covariant derivative, X ω,ofanyone-form, ω 1(M), is the one-form given by ∇ ∈A ( ω)(Y )=X(ω(Y )) ω( Y ). ∇X − ∇X If is a metric connection, then the vector ﬁeld, ( X ω) ,correspondingto X ω is nicely expressed∇ in terms of ω:Indeed,wehave ∇ ∇

( ω) = ω. ∇X ∇X The proof goes as follows:

( ω)(Y )=X(ω(Y )) ω( Y ) ∇X − ∇X = X( ω,Y ) ω, Y − ∇X = ω,Y + ω, Y ω, Y ∇X ∇X − ∇X = ω,Y , ∇X where we used the fact that the connection is compatible with the metric in the third line and so, ( ω) = ω, ∇X ∇X as claimed.

11.6 Pontrjagin Classes and Chern Classes, a Glimpse

This section can be omitted at first reading. Its purpose is to introduce the reader to Pon- trjagin Classes and Chern Classes which are fundamental invariants of real (resp. complex) 11.6. PONTRJAGIN CLASSES AND CHERN CLASSES, A GLIMPSE 367 vector bundles. We focus on motivations and intuitions and omit most proofs but we give precise pointers to the literature for proofs. Given a real (resp. complex) rank n vector bundle, ξ =(E,π,B,V ), we know that locally, ξ “looks like” a trivial bundle, U V ,forsomeopensubset,U,ofthebasespace,B,but globally, ξ can be very twisted and× one of the main issues is to understand and quantify “how twisted” ξ really is. Now, we know that every vector bundle admit a connection, say ,and ∇ the curvature, R∇, of this connection is some measure of the twisting of ξ. However, R∇ depends on ,socurvatureisnotintrinsictoξ,whichisunsatisfactoryasweseekinvariants that depend∇ only on ξ. Pontrjagin, Stiefel and Chern (starting from the late 1930’s) discovered that invariants with “good” properties could be defined if we took these invariants to belong to various cohomology groups associated with B.Suchinvariantsareusuallycalledcharacteristic classes. Roughly, there are two main methods for defining characteristic classes, one using topology and the other, due to Chern and Weil, using differential forms. A masterly exposition of these methods is given in the classic book by Milnor and Stasheff[110]. Amazingly, the method of Chern and Weil using differential forms is quite accessible for someone who has reasonably good knowledge of differential forms and de Rham cohomology as long as one is willing to gloss over various technical details. As we said earlier, one of the problems with curvature is that is depends on a connection. The way to circumvent this difficuty rests on the simple, yet subtle observation that locally, given any two overlapping local trivializations (Uα,ϕα)and(Uβ,ϕβ), the transformation rule for the curvature matrices Ωα and Ωβ is

1 Ωβ = gαβΩαgαβ− , where gαβ : Uα Uβ GL(V )isthetransitionfunction.Thematricesoftwo-forms,Ωα,are local, but the stroke∩ → of genius is to glue them together to form a global form using invariant polynomials.

Indeed, the Ωα are n n matrices so, consider the algebra of polynomials, R[X1,...,Xn2 ] × 2 (or C[X1,...,Xn2 ]inthecomplexcase)inn variables, considered as the entries of an n n × matrix. It is more convenient to use the set of variables Xij 1 i, j n , and to let X be the n n matrix X =(X ). { | ≤ ≤ } × ij

Deﬁnition 11.7 Apolynomial,P R[ Xij 1 i, j n ](orP C[ Xij 1 i, j n ]) is invariant iﬀ ∈ { | ≤ ≤ } ∈ { | ≤ ≤ } 1 P (AXA− )=P (X), for all A GL(n, R)(resp.A GL(n, C)). The algebra of invariant polynomials over n n ∈ ∈ × matrices is denoted by In.

Examples of invariant polynomials are, the trace, tr(X), and the determinant, det(X), of the matrix X.WewillcharacterizeshortlythealgebraIn. 368 CHAPTER 11. CONNECTIONS AND CURVATURE IN VECTOR BUNDLES

Now comes the punch line: For any homogeneous invariant polynomial, P I ,ofdegree ∈ n k, we can substitute Ωα for X,thatis,substituteωij for Xij,andevaluateP (Ωα). This is because Ωis a matrix of two-forms and the wedge product is commutative for forms of even degree. Therefore, P (Ω ) 2k(U ). But, the formula for a change of trivialization yields α ∈A α 1 P (Ωα)=P (gαβΩαgαβ− )=P (Ωβ),

so the forms P (Ωα)andP (Ωβ)agreeonoverlapsandthus,theydefineaglobal form denoted 2k P (R∇) (B). ∈A 2k Now, we know how to obtain global 2k-forms, P (R∇) (B), but they still seem to depend on the connection and how do they define a cohomology∈A class? Both problems are settled thanks to the following Theorems: Theorem 11.26 For every real rank n vector bundle, ξ, for every connection, , on ξ, for ∇ 2k every invariant homogeneous polynomial, P , of degree k, the 2k-form, P (R∇) (B), is 2k ∈A closed. If ξ is a complex vector bundle, then the 2k-form, P (R∇) (B; C), is closed. ∈A 2k Theorem 11.26 implies that the 2k-form, P (R∇) (B), defines a cohomology class, 2k ∈A [P (R∇)] H (B). We will come back to the proof of Theorem 11.26 later. ∈ DR Theorem 11.27 For every real (resp. complex) rank n vector bundle, for every invariant 2k homogeneous polynomial, P , of degree k, the cohomology class, [P (R∇)] HDR(B) (resp. 2k ∈ [P (R∇)] HDR(B; C)) is independent of the choice of the connection . ∈ ∇

The cohomology class, [P (R∇)], which does not depend on is denoted P (ξ)andis called the characteristic class of ξ corresponding to P . ∇ The proof of Theorem 11.27 involves a kind of homotopy argument, see Madsen and Tornehave [100] (Lemma 18.2), Morita [114] (Proposition 5.28) or see Milnor and Stasheff [110] (Appendix C). The upshot is that Theorems 11.26 and 11.27 give us a method for producing invariants of a vector bundle that somehow reflect how curved (or twisted) the bundle is. However, it appears that we need to consider infinitely many invariants. Fortunately, we can do better because the algebra, In,ofinvariantpolynomialsisfinitelygeneratedandinfact,hasvery nice sets of generators. For this, we recall the elementary symmetric functions in n variables.

Given n variables, λ1,...,λn,wecanwrite n (1 + tλ )=1+σ t + σ t2 + + σ tn, i 1 2 ··· n i=1 where the σi are symmetric, homogeneous polynomials of degree i in λ1,...,λn called elementary symmetric functions in n variables. For example, n σ = λ ,σ= λ λ ,σ= λ λ . 1 i 1 i j n 1 ··· n i=1 1 i

To be more precise, we write σi(λ1,...,λn)insteadofσi. Given any n n matrix, X =(X ), we deﬁne σ (X) by the formula × ij i det(I + tX)=1+σ (X)t + σ (X)t2 + + σ (X)tn. 1 2 ··· n We claim that

σi(X)=σi(λ1,...,λn),

where λ1,...,λn are the eigenvalues of X.Indeed,λ1,...,λn are the roots the the polynomial det(λI X)=0,andas − n n n i n i det(λI X)= (λ λ )=λ + ( 1) σ (λ ,...,λ )λ − , − − i − i 1 n i=1 i=1 n 1 1 by factoring λ and replacing λ− by λ− ,weget − n 1 1 n det(I +( λ− )X)=1+ σ (λ ,...,λ )( λ− ) , − i 1 n − i=1 which proves our claim. Observe that

σ1(X)=tr(X),σn(X)=det(X).

Also, σk(X)=σk(X), since det(I + tX)=det((I + tX))=det(I + tX). It is not very diﬃcult to prove the following theorem:

2 Theorem 11.28 The algebra, In, of invariant polynomials in n variables is generated by σ1(X),...,σn(X), that is

In ∼= R[σ1(X),...,σn(X)] (resp. In ∼= C[σ1(X),...,σn(X)]).

For a proof of Theorem 11.28, see Milnor and Stasheﬀ[110] (Appendix C, Lemma 6), Madsen and Tornehave [100] (Appendix B) or Morita [114] (Theorem 5.26). The proof uses the fact that for every matrix, X,thereisanupper-triangularmatrix,T ,andaninvertible matrix, B,sothat 1 X = BTB− . Then, we can replace B by the matrix diag(,2,...,n)B,where is very small, and make the oﬀdiagonal entries arbitrarily small. By continuity, it follows that P (X)dependsonly 1 on the diagonal entries of BTB− ,thatis,ontheeigenvaluesofX.So,P (X)mustbe a symmetric function of these eigenvalues and the classical theory of symmetric functions completes the proof. 370 CHAPTER 11. CONNECTIONS AND CURVATURE IN VECTOR BUNDLES

It turns out that there are situations where it is more convenient to use another set of generators instead of σ1,...,σn.Deﬁnesi(X)by i si(X)=tr(X ). Of course, s (X)=λi + + λi , i 1 ··· n where λ1,...,λn are the eigenvalues of X. Now, the σi(X)andsi(X)arerelatedtoeach other by Newton’s formula,namely: i 1 i si(X) σ1(X)si 1(X)+σ2(X)si 2(X)+ +( 1) − σi 1(X)s1(X)+( 1) iσi(X)=0 − − − ··· − − − with 1 i n. A “cute” proof of the Newton formulae is obtained by computing the derivative≤ of≤ log(h(t)), where n h(t)= (1 + tλ )=1+σ t + σ t2 + + σ tn, i 1 2 ··· n i=1 see Madsen and Tornehave [100] (Appendix B) or Morita [114] (Exercise 5.7).

Consequently, we can inductively compute si in terms of σ1,...,σi and conversely, σi in terms of s1,...,si.Forexample, s = σ ,s= σ2 2σ ,s= σ3 3σ σ +3σ . 1 1 2 1 − 2 3 1 − 1 2 3 It follows that

In ∼= R[s1(X),...,sn(X)] (resp. In ∼= C[s1(X),...,sn(X)]). Using the above, we can give a simple proof of Theorem 11.26, using Theorem 11.28.

Proof of Theorem 11.26.Sinces1,...,sn generate In,itisenoughtoprovethatsi(R∇)is closed. We need to prove that dsi(R∇)=0andforthis,itisenoughtoproveitinevery i local trivialization, (Uα,ϕα). To simplify notation, we write ΩforΩ α. Now, si(Ω) = tr(Ω ), so i i dsi(Ω) = dtr(Ω )=tr(dΩ ), and we use Bianchi’s identity (Proposition 11.10), dΩ=ω Ω Ω ω. ∧ − ∧ We have i i 1 i 2 k i k 1 i 1 dΩ = dΩ Ω − +Ω dΩ Ω − + +Ω dΩ Ω − − + +Ω− dΩ ∧ ∧ ∧i 1 ··· ∧ ∧ i 2 ··· ∧ =(ω Ω Ω ω) Ω − +Ω (ω Ω Ω ω) Ω − ∧ − k ∧ ∧ ∧ ∧i k 1− ∧k+1 ∧ i k 2 + +Ω (ω Ω Ω ω) Ω − − +Ω (ω Ω Ω ω) Ω − − ··· i ∧1 ∧ − ∧ ∧ ∧ ∧ − ∧ ∧ + +Ω− (ω Ω Ω ω) ··· i ∧ ∧i 1− ∧ i 1 2 i 2 = ω Ω Ω ω Ω − +Ω ω Ω − Ω ω Ω − + + k∧ − i∧k ∧ k+1 ∧ i ∧k 1 −k+1 ∧ ∧ i k 1 ···k+2 i k 2 Ω ω Ω − Ω ω Ω − − +Ω ω Ω − − Ω ω Ω − − ∧ ∧ i 1 − ∧ i ∧ ∧ ∧ − ∧ ∧ + +Ω− ω Ω Ω ω ··· ∧ ∧ − ∧ = ω Ωi Ωi ω. ∧ − ∧ 11.6. PONTRJAGIN CLASSES AND CHERN CLASSES, A GLIMPSE 371

However, the entries in ω are one-forms, the entries in Ωare two-forms and since η θ = θ η ∧ ∧ for all η 1(B)andallθ 2(B)andtr(XY )=tr(YX)forallmatricesX and Y with commuting∈A entries, we get ∈A tr(dΩi)=tr(ω Ωi Ωi ω)=tr(ω Ωi) tr(Ωi ω)=0, ∧ − ∧ ∧ − ∧ as required. Amoreelegantproof(alsousingBianchi’sidentity)canbefoundinMilnorandStasheff [110] (Appendix C, page 296-298). For real vector bundles, only invariant polynomials of even degrees matter. Proposition 11.29 If ξ is a real vector bundle, then for every homogeneous invariant polynomial, P , of odd degree, k, we have P (ξ)=0 H2k (B). ∈ DR Proof . As In ∼= R[s1(X),...,sn(X)] and si(X)ishomogeneousofdegreei,itisenoughto prove Proposition 11.29 for si(X)withi odd. By Theorem 11.27, we may assume that we pick a metric connection on ξ,sothatΩα is skew-symmetric in every local trivialization. i Then, Ωα is also skew symmetric and i tr(Ωα)=0, since the diagonal entries of a real skew-symmetric matrix are all zero. It follows that i si(Ωα)=tr(Ωα)=0. Proposition 11.29 implies that for a real vector bundle, ξ,non-zerocharacteristicclasses 4k can only live in the cohomology groups HDR(B)ofdimension4k.Thispropertyisspecific to real vector bundles and generally fails for complex vector bundles. Before defining Pontrjagin and Chern classes, we state another important properties of the homology classes, P (ξ):

Proposition 11.30 If ξ =(E,π,B,V ) and ξ =(E,π,B,V) are real (resp. complex) vector bundles, for every bundle map

fE E E

π π B B, f for every homogeneous invariant polynomial, P , of degree k, we have

2k 2k P (ξ)=f ∗(P (ξ)) HDR(B)(resp. P (ξ)=f ∗(P (ξ)) HDR(B; C)). ∈ ∈ In particular, for every smooth map, f : N B, we have → 2k 2k P (f ∗ξ)=f ∗(P (ξ)) HDR(N)(resp. P (f ∗ξ)=f ∗(P (ξ)) HDR(N; C)). ∈ ∈ 372 CHAPTER 11. CONNECTIONS AND CURVATURE IN VECTOR BUNDLES

The above proposition implies that isomorphic vector bundles have identical characteristic classes. We ﬁnally deﬁne Pontrjagin classes and Chern classes.

Deﬁnition 11.8 If ξ be a real rank n vector bundle, then the kth Pontrjagin class of ξ, denoted p (ξ), where 1 2k n,isthecohomologyclass k ≤ ≤

1 4k p (ξ)= σ (R∇) H (B), k (2π)2k 2k ∈ DR for any connection, ,onξ. ∇ th If ξ be a complex rank n vector bundle, then the k Chern class of ξ,denotedck(ξ), where 1 k n,isthecohomologyclass ≤ ≤ k 1 2k c (ξ)= − σ (R∇) H (B), k 2πi k ∈ DR for any connection, ,onξ.Wealsosetp (ξ)=1andc (ξ) = 1 in the complex case. ∇ 0 0

The strange coefficient in pk(ξ)ispresentsothatourexpressionmatchesthetopological definition of Pontrjagin classes. The equally strange coefficient in ck(ξ)istheretoinsurethat 2k ck(ξ)actuallybelongstothereal cohomology group HDR(B), as stated (from the definition 2k we can only claim that ck(ξ) HDR(B; C)). This requires a proof which can be found in Morita [114] (Proposition 5.30)∈ or in Madsen and Tornehave [100] (Chapter 18). One can use the fact that every complex vector bundle admits a Hermitian connection. Locally, the curvature matrices are skew-Hermitian and this easily implies that the Chern classes are real since if Ωis skew-Hermitian, then iΩis Hermitian. (Actually, the topological version of 2k Chern classes shows that ck(ξ) H (B; Z).) ∈ If ξ is a real rank n vector bundle and n is odd, say n =2m +1,thenthe“top” Pontrjagin class, pm(ξ), corresponds to σ2m(R∇), which is not det(R∇). However, if n is even, say n =2m,thenthe“top”Pontrjaginclasspm(ξ)correspondstodet(R∇).

It is also useful to introduce the Pontrjagin polynomial, p(ξ)(t) H• (B)[t], given by ∈ DR

t 2 n p(ξ)(t)= det I + R∇ =1+p1(ξ)t + p2(ξ)t + + p n (ξ)t 2 2π ··· 2

and the Chern polynomial, c(ξ)(t) H• (B)[t], given by ∈ DR

t 2 n c(ξ)(t)= det I R∇ =1+c (ξ)t + c (ξ)t + + c (ξ)t . − 2πi 1 2 ··· n If a vector bundle is trivial, then all its Pontrjagin classes (or Chern classes) are zero for all k 1. If ξ is the real tangent bundle, ξ = TB,ofsomemanifoldofdimensionn, then n≥ the Pontrjagin classes of TB are denoted p1(B),...,p n (B). 4 4 11.6. PONTRJAGIN CLASSES AND CHERN CLASSES, A GLIMPSE 373

For complex vector bundles, the manifold, B,isoftentherealmanifoldcorresponding to a complex manifold. If B has complex dimension, n,thenB has real dimension 2n. In this case, the tangent bundle, TB,isarankn complex vector bundle over the real manifold of dimension, 2n,andthus,ithasn Chern classes, denoted c1(B),...,cn(B). The determination of the Pontrjagin classes (or Chern classes) of a manifold is an important step for the study of the geometric/topological structure of the manifold. For example, it n is possible to compute the Chern classes of complex projective space, CP (as a complex manifold). The Pontrjagin classes of a real vector bundle, ξ,arerelatedtotheChernclassesofits complexiﬁcation, ξ = ξ 1 (where 1 is the trivial complex line bundle B C). C ⊗ C C ×

1 Proposition 11.31 For every real rank n vector bundle, ξ =(E,π,B,V ), if ξC = ξ C is the complexiﬁcation of ξ, then ⊗

p (ξ)=( 1)kc (ξ ) H4k (B) k 0. k − 2k C ∈ DR ≥

Basically, the reason why Proposition 11.31 holds is that

1 1 2k =( 1)k − (2π)2k − 2πi

We conclude this section by stating a few more properties of Chern classes.

Proposition 11.32 For every complex rank n vector bundle, ξ, the following properties hold:

(1) If ξ has a Hermitian metric, then we have a canonical isomorphism, ξ∗ ∼= ξ.

(2) The Chern classes of ξ, ξ∗ and ξ satisfy:

k c (ξ∗)=c (ξ)=( 1) c (ξ). k k − k

(3) For any complex vector bundles, ξ and η,

ck(ξ η)= ci(ξ)ck i(η) ⊕ − i=0 or equivalently c(ξ η)(t)=c(ξ)(t)c(η)(t) ⊕ and similarly for Pontrjagin classes when ξ and η are real vector bundles. 374 CHAPTER 11. CONNECTIONS AND CURVATURE IN VECTOR BUNDLES

To prove (2) we can use the fact that ξ can be given a Hermitian metric. Then, we saw earlier that if ω is the connection matrix of ξ over U then ω = ω is the connection matrix k − k of ξ over U. However, it is clear that σ ( Ω)=( 1) σ (Ω )andso,c (ξ)=( 1) c (ξ). k − α − k α k − k

Remark: For a real vector bundle, ξ,itiseasytoseethat(ξC)∗ =(ξ∗)C,whichimpliesthat ck((ξC)∗)=ck(ξC)(asξ ∼= ξ∗)and(2)impliesthatck(ξC)=0fork odd. This proves again that the Pontrjagin classes exit only in dimension 4k.

Acomplexrankn vector bundle, ξ,canalsobeviewedasarank2n vector bundle, ξR and we have:

Proposition 11.33 For every rank n complex vector bundle, ξ, if pk = pk(ξR) and ck = ck(ξ), then we have

1 p + p + +( 1)np =(1+c + c + + c )(1 c + c + +( 1)nc ). − 1 2 ··· − n 1 2 ··· n − 1 2 ··· − n 11.7 Euler Classes and The Generalized Gauss-Bonnet Theorem

Let ξ =(E,π,B,V )bearealvectorbundleofrankn =2m and let be any metric connection on ξ.Then,ifξ is orientable (as defined in Section 7.4, see∇ Definition 7.12 2m and the paragraph following it), it is possible to define a global form, eu(R∇) (B), ∈2Am which turns out to be closed. Furthermore, the cohomology class, [eu(R∇)] HDR(B), is independent of the choice of .Thiscohomologyclass,denotede(ξ), is called∈ the Euler ∇ 2 class of ξ and has some very interesting properties. For example, pm(ξ)=e(ξ) . As is a metric connection, in a trivialization, (U ,ϕ ), the curvature matrix, Ω ,isa ∇ α α α skew symmetric 2m 2m matrix of 2-forms. Therefore, we can substitute the 2-forms in Ωα for the variables of the× Pfaffian of degree m (see Section 22.20) and we obtain the 2m-form, 2m Pf(Ωα) (B). Now, as ξ is orientable, the transition functions take values in SO(2m), so by Proposition∈A 11.9, since 1 Ωβ = gαβΩαgαβ− , we conclude from Proposition 22.38 (ii) that

Pf(Ωα)=Pf(Ωβ).

2m Therefore, the local 2m-forms, Pf(Ω ), patch and deﬁne a global form, Pf(R∇) (B). α ∈A The following propositions can be shown:

Proposition 11.34 For every real, orientable, rank 2m vector bundle, ξ, for every metric 2m connection, , on ξ the 2m-form, Pf(R∇) (B), is closed. ∇ ∈A 11.7. EULER CLASSES AND THE GENERALIZED GAUSS-BONNET THEOREM 375

Proposition 11.35 For every real, orientable, rank 2m vector bundle, ξ, the cohomology 2m class, [Pf(R∇)] H (B), is independent of the metric connection, , on ξ. ∈ DR ∇

Proofs of Propositions 11.34 and 11.35 can be found in Madsen and Tornehave [100] (Chapter 19) or Milnor and Stasheﬀ[110] (Appendix C) (also see Morita [114], Chapters 5 and 6).

Deﬁnition 11.9 Let ξ =(E,π,B,V )beanyreal,orientable,rank2m vector bundle. For any metric connection, ,onξ the Euler form associated with is the closed form ∇ ∇

1 2m eu(R∇)= Pf(R∇) (B) (2π)n ∈A

and the Euler class of ξ is the cohomology class,

2m e(ξ)= eu(R∇) H (B), ∈ DR which does not depend on . ∇ Some authors, including Madsen and Tornehave [100], have a negative sign in front of R∇ in their deﬁnition of the Euler form, that is, they deﬁne eu(R∇)by 1 eu(R∇)= Pf( R∇). (2π)n −

However these authors use a Pfaffian with the opposite sign convention from ours and this Pfaffian differs from ours by the factor ( 1)n (see the warning in Section 22.20). Madsen and Tornehave [100] seem to have overlooked− this point and with their definition of the Pfaffian (which is the one we have adopted) Proposition 11.37 is incorrect.

Here is the relationship between the Euler class, e(ξ), and the top Pontrjagin class, pm(ξ):

Proposition 11.36 For every real, orientable, rank 2m vector bundle, ξ =(E,π,B,V ), we have p (ξ)=e(ξ)2 H4m(B). m ∈ DR

Proof .ThetopPontrjaginclass,pm(ξ), is given by

1 p (ξ)= det(R∇) , m (2π)2m for any (metric) connection, and ∇

e(ξ)= eu(R∇) 376 CHAPTER 11. CONNECTIONS AND CURVATURE IN VECTOR BUNDLES

with 1 eu(R∇)= Pf(R∇). (2π)n From Proposition 22.38 (i), we have

2 det(R∇)=Pf(R∇) ,

which yields the desired result. A rank m complex vector bundle, ξ =(E,π,B,V ), can be viewed as a real rank 2m

vector bundle, ξR,byviewingV as a 2m dimensional real vector space. Then, it turns out that ξR is naturally orientable. Here is the reason.

For any basis, (e1,...,em), of V over C,observethat(e1, ie1,...,em, iem)isabasisofV m m m over R (since v = (λi + iµi)ei = λiei + µiiei). But, any m m invertible i=1 i=1 i=1 × matrix, A,overC becomes a real 2m 2m invertible matrix, AR,obtainedbyreplacingthe entry a + ib in Aby the real 2 2matrix× jk jk ×

ajk bjk b −a . jk jk Indeed, if v = m a e + m b ie ,theniv = m b e + m a ie and when we k j=1 jk j j=1 jk j k j=1 − jk j j=1 jk j express vk and ivk over the basis (e1, ie1,...,em, iem), we get a matrix AR consisting of 2 2 blocks as above. Clearly, the map r : A A is a continuous injective homomorphism from× → R GL(m, C)toGL(2m, R). Now, it is known GL(m, C)isconnected,thusIm(r)=r(GL(m, C)) is connected and as det(I2m)=1,weconcludethatallmatricesinIm(r)havepositive 1 determinant. Therefore, the transition functions of ξR which take values in Im(r)have positive determinant and ξR is orientable. We can give ξR an orientation by ﬁxing some basis of V over R.Then,wehavethefollowingrelationshipbetweene(ξR)andthetopChern class, cm(ξ):

Proposition 11.37 For every complex, rank m vector bundle, ξ =(E,π,B,V ), we have

c (ξ)=e(ξ) H2m(B). m ∈ DR Proof .Picksomemetricconnection, .Recallthat ∇ m m 1 m 1 c (ξ)= − det(R∇) = i det(R∇) . m 2πi 2π On the other hand, 1 e(ξ)= Pf(R∇) . (2π)m R 1One can also prove directly that every matrix in Im(r) has positive determinant by expressing r(A) as a product of simple matrices whose determinants are easily computed. 11.7. EULER CLASSES AND THE GENERALIZED GAUSS-BONNET THEOREM 377

Here, RR∇ denotes the global 2m-form wich, locally, is equal to ΩR,whereΩisthem m curvature matrix of ξ over some trivialization. By Proposition 22.39, ×

n Pf(ΩR)=i det(Ω),

so cm(ξ)=e(ξ), as claimed.

The Euler class enjoys many other nice properties. For example, if f : ξ1 ξ2 is an orientation preserving bundle map, then →

e(f ∗ξ2)=f ∗(e(ξ2)),

where f ∗ξ2 is given the orientation induced by ξ2. Also, the Euler class can be defined by topological means and it belongs to the integral cohomology group H2m(B; Z). Although this result lies beyond the scope of these notes we cannot resist stating one of the most important and most beautiful theorems of differential geometry usually called the Generalized Gauss-Bonnet Theorem or Gauss-Bonnet-Chern Theorem. For this, we need the notion of Euler characteristic. Since we haven’t discussed triangu- lations of manifolds, we will use a defintion in terms of cohomology. Although concise, this definition is hard to motivate and we appologize for this. Given a smooth n-dimensional manifold, M, we define its Euler characteristic, χ(M), as

n χ(M)= ( 1)i dim(Hi ). − DR i=0 i 2 The integers, bi =dim(HDR), are known as the Betti numbers of M.Forexample,χ(S )=2. It turns out that if M is an odd dimensional manifold, then χ(M)=0.Thisexplains partially why the Euler class is only deﬁned for even dimensional bundles. The Generalized Gauss-Bonnet Theorem (or Gauss-Bonnet-Chern Theorem) is a generalization of the Gauss-Bonnet Theorem for surfaces. In the general form stated below it was ﬁrst proved by Allendoerfer and Weil (1943), and Chern (1944).

Theorem 11.38 (Generalized Gauss-Bonnet Formula) Let M be an orientable, smooth, compact manifold of dimension 2m. For every metric connection, , on TM, (in particular, the Levi-Civita connection for a Riemannian manifold) we have ∇

eu(R∇)=χ(M). M A proof of Theorem 11.38 can be found in Madsen and Tornehave [100] (Chapter 21), but beware of some sign problems. The proof uses another famous theorem of diﬀerential topology, the Poincar´e-Hopf Theorem.AsketchoftheproofisalsogiveninMorita[114], Chapter 5. 378 CHAPTER 11. CONNECTIONS AND CURVATURE IN VECTOR BUNDLES

Theorem 11.38 is remarkable because it establishes a relationship between the geometry of the manifold (its curvature) and the topology of the manifold (the number of “holes”), somehow encoded in its Euler characteristic. Characteristic classes are a rich and important topic and we’ve only scratched the surface. We refer the reader to the texts mentioned earlier in this section as well as to Bott and Tu [19] for comprehensive expositions. Chapter 12

Geodesics on Riemannian Manifolds

12.1 Geodesics, Local Existence and Uniqueness

If (M,g)isaRiemannianmanifold,thentheconceptoflength makes sense for any piecewise smooth (in fact, C1)curveonM.Then,itpossibletodeﬁnethestructureofametricspaceon M,whered(p, q)isthegreatestlowerboundofthelengthofallcurvesjoiningp and q.Curves on M which locally yield the shortest distance between two points are of great interest. These curves called geodesics play an important role and the goal of this chapter is to study some of their properties. Since geodesics are a standard chapter of every diﬀerential geometry text, we will omit most proofs and instead give precise pointers to the literature. Among the many presentations of this subject, in our opinion, Milnor’s account [106] (Part II, Section 11) is still one of the best, certainly by its clarity and elegance. We acknowledge that our presentation was heavily inspired by this beautiful work. We also relied heavily on Gallot, Hulin and Lafontaine [60] (Chapter 2), Do Carmo [50], O’Neill [119], Kuhnel [91] and class notes by Pierre Pansu (see http://www.math.u-psud.fr/%7Epansu/web dea/resume dea 04.html in http://www.math.u-psud.fr˜pansu/). Another reference that is remarkable by its clarity and the completeness of its coverage is Postnikov [125].

Given any p M,foreveryv TpM,the(Riemannian) norm of v,denoted v ,is deﬁned by ∈ ∈ v = g (v, v). p The Riemannian inner product, gp(u, v), of two tangent vectors, u, v TpM,willalsobe denoted by u, v ,orsimply u, v .Recallthefollowingdeﬁnitionsregardingcurves:∈ p

Definition 12.1 Given any Riemannian manifold, M,asmooth parametric curve (for short, curve)onM is a map, γ : I M,whereI is some open interval of R.Foraclosedinterval, → [a, b] R,amapγ :[a, b] M is a smooth curve from p = γ(a) to q = γ(b)iffγ can be extended⊆ to a smooth curve→γ :(a , b + ) M,forsome>0. Given any two points, p, q M, a continuous map, γ :[a,− b] M,isa→ piecewise smooth curve from p to q iff ∈ → 379 380 CHAPTER 12. GEODESICS ON RIEMANNIAN MANIFOLDS

(1) There is a sequence a = t0

The set of all piecewise smooth curves from p to q is denoted Ω(M; p, q), or brieﬂy Ω(p, q) (or even Ω, when p and q are understood).

The set Ω(M; p, q)isanimportantobjectsometimescalledthepath space of M (from p to q). Unfortunately it is an inﬁnite-dimensional manifold, which makes it hard to investigate its properties.

Observe that at any junction point, γi 1(ti)=γi(ti), there may be a jump in the velocity − vector of γ.Weletγ((ti)+)=γi 1(ti)andγ((ti) )=γi(ti). − − Given any curve, γ Ω(M; p, q), the length, L(γ), of γ is defined by ∈ k 1 k 1 − ti+1 − ti+1 L(γ)= γ(t) dt = g(γ (t),γ(t)) dt. i=0 ti i=0 ti It is easy to see that L(γ) is unchanged by a monotone reparametrization (that is, a map h:[a, b] [c, d], whose derivative, h,hasaconstantsign). → Let us now assume that our Riemannian manifold, (M,g), is equipped with the Levi- D Civita connection and thus, for every curve, γ,onM,letdt be the associated covariant derivative along γ,alsodenoted ∇γ Definition 12.2 Let (M,g)beaRiemannianmanifold.Acurve,γ : I M,(whereI R → ⊆ is any interval) is a geodesic iff γ(t)isparallelalongγ,thatis,iff

Dγ = γ =0. dt ∇γ

If M was embedded in Rd,ageodesicwouldbeacurve,γ,suchthattheacceleration Dγ vector, γ = dt ,isnormaltoTγ(t)M.

By Proposition 11.25, γ(t) = g(γ(t),γ(t)) is constant, say γ(t) = c.Ifwedefine the arc-length function, s(t), relative to a,wherea is any chosen point inI,by t s(t)= g(γ(t),γ(t)) dt = c(t a),tI, a − ∈ we conclude that for a geodesic, γ(t), the parameter, t,isanaffinefunctionofthearc-length. When c =1,whichcanbeachievedbyanaffinereparametrization,wesaythatthegeodesic is normalized. 12.1. GEODESICS, LOCAL EXISTENCE AND UNIQUENESS 381

The geodesics in Rn are the straight lines parametrized by constant velocity. The geodesics of the 2-sphere are the great circles, parametrized by arc-length. The geodesics of the Poincar´ehalf-plane are the lines x = a and the half-circles centered on the x-axis. The geodesics of an ellipsoid are quite fascinating. They can be completely characterized and they are parametrized by elliptic functions (see Hilbert and Cohn-Vossen [75], Chapter 4, Section and Berger and Gostiaux [17], Section 10.4.9.5). If M is a submanifold of Rn, geodesics are curves whose acceleration vector, γ =(Dγ)/dt is normal to M (that is, for every p M, γ is normal to T M). ∈ p In a local chart, (U,ϕ), since a geodesic is characterized by the fact that its velocity vector ﬁeld, γ(t), along γ is parallel, by Proposition 11.13, it is the solution of the following system of second-order ODE’s in the unknowns, uk: d2u du du k + Γk i j =0,k=1,...,n, dt2 ij dt dt ij with u = pr ϕ γ (n =dim(M)). i i ◦ ◦ The standard existence and uniqueness results for ODE’s can be used to prove the following proposition (see O’Neill [119], Chapter 3): Proposition 12.1 Let (M,g) be a Riemannian manifold. For every point, p M, and every tangent vector, v T M, there is some interval, ( η,η), and a unique geodesic,∈ ∈ p − γ :( η,η) M, v − → satisfying the conditions

γv(0) = p, γv (0) = v.

The following proposition is used to prove that every geodesic is contained in a unique maximal geodesic (i.e,withlargestpossibledomain).Foraproof,seeO’Neill[119],Chapter 3orPetersen[121](Chapter5,Section2,Lemma7). Proposition 12.2 For any two geodesics, γ : I M and γ : I M, if γ (a)=γ (a) 1 1 → 2 2 → 1 2 and γ (a)=γ (a), for some a I I , then γ = γ on I I . 1 2 ∈ 1 ∩ 2 1 2 1 ∩ 2 Propositions 12.1 and 12.2 imply that for every p M and every v T M,thereisa ∈ ∈ p unique geodesic, denoted γv,suchthatγ(0) = p, γ(0) = v,andthedomainofγ is the largest possible, that is, cannot be extended. We call γv a maximal geodesic (with initial conditions γv(0) = p and γv (0) = v). Observe that the system of diﬀerential equations satisﬁed by geodesics has the following homogeneity property: If t γ(t)isasolutionoftheabovesystem,thenforeveryconstant, c,thecurvet γ(ct)isalsoasolutionofthesystem.Wecanusethisfacttogetherwith→ standard existence→ and uniqueness results for ODE’s to prove the proposition below. For proofs, see Milnor [106] (Part II, Section 10), or Gallot, Hulin and Lafontaine [60] (Chapter 2). 382 CHAPTER 12. GEODESICS ON RIEMANNIAN MANIFOLDS

Proposition 12.3 Let (M,g) be a Riemannian manifold. For every point, p M, there is 0 ∈ an open subset, U M, with p0 U, and some >0, so that: For every p U and every tangent vector, v ⊆T M, with v∈ <, there is a unique geodesic, ∈ ∈ p γ :( 2, 2) M, v − → satisfying the conditions

γv(0) = p, γv (0) = v.

If γv :( η,η) M is a geodesic with initial conditions γv(0) = p and γv (0) = v =0, for any constant,− →c =0,thecurve,t γ (ct), is a geodesic deﬁned on ( η/c, η/c)(or → v − (η/c, η/c)ifc<0) such that γ(0) = cv.Thus, − γ (ct)=γ (t),ct( η,η). v cv ∈ − This fact will be used in the next section.

Given any function, f C∞(M), for any p M and for any u T M,thevalueof ∈ ∈ ∈ p the Hessian, Hessp(f)(u, u), can be computed using geodesics. Indeed, for any geodesic, γ :[0,] M,suchthatγ(0) = p and γ(0) = u,wehave → Hess (u, u)=γ(γ(f)) ( γ)(f)=γ(γ(f)) p − ∇γ since γ =0becauseγ is a geodesic and ∇γ d d2 γ(γ(f)) = γ(df (γ)) = γ f(γ(t)) = f(γ(t)) , dt dt2 t=0 t=0 and thus, d2 Hess (u, u)= f(γ(t)) . p dt2 t=0 12.2 The Exponential Map

The idea behind the exponential map is to parametrize a Riemannian manifold, M,locally near any p M in terms of a map from the tangent space TpM to the manifold, this map being deﬁned∈ in terms of geodesics.

Deﬁnition 12.3 Let (M,g)beaRiemannianmanifold.Foreveryp M,let (p)(or simply, )betheopensubsetofT M given by ∈ D D p (p)= v T M γ (1) is deﬁned , D { ∈ p | v }

where γv is the unique maximal geodesic with initial conditions γv(0) = p and γv (0) = v. The exponential map is the map, exp : (p) M,givenby p D →

expp(v)=γv(1). 12.2. THE EXPONENTIAL MAP 383

It is easy to see that (p)isstar-shaped, which means that if w (p), then the line segment tw 0 t 1 Dis contained in (p). In view of the remark∈ madeD at the end of the previous{ section,| ≤ ≤ the} curve D

t exp (tv),tv(p) → p ∈D

is the geodesic, γv,throughp such that γv (0) = v.Suchgeodesicsarecalledradial geodesics. The point, expp(tv), is obtained by running along the geodesic, γv,anarclengthequalto t v ,startingfromp. In general, (p)isapropersubsetofTpM.Forexample,ifU is a bounded open subset n D n of R ,sincewecanidentifyTpU with R for all p U,then (p) U,forallp U. ∈ D ⊆ ∈ Definition 12.4 ARiemannianmanifold,(M,g), is geodesically complete iff (p)=T M, D p for all p M, that is, iffthe exponential, expp(v), is defined for all p M and for all v T M.∈ ∈ ∈ p

Equivalently, (M,g) is geodesically complete iﬀevery geodesic can be extended indeﬁ- nitely. Geodesically complete manifolds have nice properties, some of which will be investi- gated later. Observe that d(exp ) =id .Thisisbecause,foreveryv (p), the map t exp (tv) p 0 TpM ∈D → p is the geodesic, γv,and d d (γ (t)) = v = (exp (tv)) = d(exp ) (v). dt v |t=0 dt p |t=0 p 0

It follows from the inverse function theorem that expp is a diﬀeomorphism from some open ball in TpM centered at 0 to M.Thefollowingslightlystrongerpropositioncanbeshown (Milnor [106], Chapter 10, Lemma 10.3):

Proposition 12.4 Let (M,g) be a Riemannian manifold. For every point, p M, there is an open subset, W M, with p W and a number >0, so that ∈ ⊆ ∈

(1) Any two points q1,q2 of W are joined by a unique geodesic of length <.

(2) This geodesic depends smoothly upon q1 and q2, that is, if t expq1 (tv) is the geodesic joining q and q (0 t 1), then v T M depends smoothly→ on (q ,q ). 1 2 ≤ ≤ ∈ q1 1 2

(3) For every q W , the map expq is a diﬀeomorphism from the open ball, B(0,) TqM, to its image,∈U =exp(B(0,)) M, with W U and U open. ⊆ q q ⊆ ⊆ q q

For any q M,anopenneighborhoodofq of the form, U =exp(B(0,)), where exp ∈ q q q is a diﬀeomorphism from the open ball B(0,)ontoUq,iscalledanormal neighborhood. 384 CHAPTER 12. GEODESICS ON RIEMANNIAN MANIFOLDS

Deﬁnition 12.5 Let (M,g)beaRiemannianmanifold.Foreverypoint,p M,theinjectivity radius of M at p, denoted i(p), is the least upper bound of the numbers,∈ r>0, such that expp is a diﬀeomorphism on the open ball B(0,r) TpM.Theinjectivity radius, i(M), of M is the greatest lower bound of the numbers, i(p),⊆ where p M. ∈

1 For every p M,wegetachart,(U ,ϕ), where U =exp(B(0,i(p))) and ϕ =exp− , ∈ p p p called a normal chart.Ifwepickanyorthonormalbasis,(e1,...,en), of TpM,thenthexi’s, 1 with xi = pri exp− and pri the projection onto Rei,arecallednormal coordinates at p ◦ (here, n =dim(M)). These are deﬁned up to an isometry of TpM.Thefollowingproposition shows that Riemannian metrics do not admit any local invariants of order one. The proof is left as an exercise.

Proposition 12.5 Let (M,g) be a Riemannian manifold. For every point, p M, in normal coordinates at p, ∈ ∂ ∂ g , = δ and Γk (p)=0. ∂x ∂x ij ij i j p

For the next proposition, known as Gauss Lemma,weneedtodeﬁnepolar coordinates n 1 on T M.Ifn =dim(M), observe that the map, (0, ) S − T M 0 ,givenby p ∞ × −→ p −{ } n 1 (r, v) rv, r > 0,v S − → ∈ n 1 is a diﬀeomorphism, where S − is the sphere of radius r =1inTpM.Then,themap, n 1 f :(0,i(p)) S − U p ,givenby × → p −{ } n 1 (r, v) exp (rv), 0

Proposition 12.6 (Gauss Lemma) Let (M,g) be a Riemannian manifold. For every point, p M, the images, expp(S(0,r)), of the spheres, S(0,r) TpM, centered at 0 by the exponential∈ map, exp , are orthogonal to the radial geodesics,⊆r exp (rv), through p, for p → p all r

Sketch of proof . (After Milnor, see [106], Chapter II, Section 10.) Pick any curve, t v(t) n 1 → on the unit sphere, S − .WemustshowthatthecorrespondingcurveonM,

t exp (rv(t)), → p 12.2. THE EXPONENTIAL MAP 385

with r ﬁxed, is orthogonal to the radial geodesic, r exp (rv(t)), → p with t ﬁxed, 0 r

f(0,t)=expp(0) = p, hence ∂f/∂t(0,t)=0 and thus, ∂f ∂f , =0 ∂r ∂t for all r, t,whichconcludestheproofoftheﬁrststatement.Fortheproofofthesecond statement, see Pansu’s class notes, Chapter 3, Section 3.5. Consider any piecewise smooth curve ω :[a, b] U p . → p −{ } We can write each point ω(t)uniquelyas

ω(t)=expp(r(t)v(t)), with 0

Proposition 12.7 Let (M,g) be a Riemannian manifold. We have

b ω(t) dt r(b) r(a) , ≥| − | a where equality holds only if the function r is monotone and the function v is constant. Thus,

the shortest path joining two concentric spherical shells, expp(S(0,r1)) and expp(S(0,r2)), is a radial geodesic.

Proof . (After Milnor, see [106], Chapter II, Section 10.) Again, let f(r, t)=expp(rv(t)), so that ω(t)=f(r(t),t). Then, dω ∂f ∂f = r(t)+ . dt ∂r ∂t The proof of the previous proposition showed that the two vectors on the right-hand side are orthogonal and since ∂f/∂r =1,thisgives 2 2 dω 2 ∂f 2 = r(t) + r(t) dt | | ∂t ≥| | where equality holds only if ∂f/∂t =0;henceonlyif v(t)=0.Thus, b dω b dt r(t) dt r(b) r(a) dt ≥ | | ≥| − | a a where equality holds only if r(t)ismonotoneand v(t)isconstant. We now get the following important result from Proposition 12.6 and Proposition 12.7: Theorem 12.8 Let (M,g) be a Riemannian manifold. Let W and be as in Proposition 12.4 and let γ :[0, 1] M be the geodesic of length <joining two points q ,q of W . For → 1 2 any other piecewise smooth path, ω, joining q1 and q2, we have 1 1 γ(t) dt ω(t) dt ≤ 0 0 where equality can holds only if the images ω([0, 1]) and γ([0, 1]) coincide. Thus, γ is the shortest path from q1 to q2. Proof . (After Milnor, see [106], Chapter II, Section 10.) Consider any piecewise smooth path, ω,fromq1 = γ(0) to some point q =exp (rv) U , 2 q1 ∈ q1 where 0

Corollary 12.9 Let (M,g) be a Riemannian manifold. If ω :[0,b] M is any curve parametrized by arc-length and ω has length less than or equal to the→ length of any other curve from ω(0) to ω(b), then ω is a geodesic.

Proof . Consider any segment of ω lying within an open set, W ,asabove,andhavinglength <. By Theorem 12.8, this segment must be a geodesic. Hence, the entire curve is a geodesic.

Deﬁnition 12.6 Let (M,g) be a Riemannian manifold. A geodesic, γ :[a, b] M,is minimal iﬀits length is less than or equal to the length of any other piecewise smooth→ curve joining its endpoints.

Theorem 12.8 asserts that any sufficiently small segment of a geodesic is minimal. On the other hand, a long geodesic may not be minimal. For example, a great circle arc on the unit sphere is a geodesic. If such an arc has length greater than π,thenitisnotminimal. Minimal geodesics are generally not unique. For example, any two antipodal points on a sphere are joined by an infinite number of minimal geodesics. A broken geodesic is a piecewise smooth curve as in Definition 12.1, where each curve segment is a geodesic.

Proposition 12.10 A Riemannian manifold, (M,g), is connected iﬀany two points of M can be joined by a broken geodesic.

Proof . Assume M is connected, pick any p M,andletSp M be the set of all points that can be connected to p by a broken geodesic.∈ For any q M,chooseanormalneighborhood,⊆ ∈ U,ofq.Ifq Sp, then it is clear that U Sp.Ontheotherhand,ifq/Sp,then U M S .Therefore,∈ S = is open and closed,⊆ so S = M.Theconverseisobvious.∈ ⊆ − p p ∅ p In general, if M is connected, then it is not true that any two points are joined by a geodesic. However, this will be the case if M is geodesically complete, as we will see in the next section. Next, we will see that a Riemannian metric induces a distance on the manifold whose induced topology agrees with the original metric.

12.3 Complete Riemannian Manifolds, the Hopf-Rinow Theorem and the Cut Locus

Every connected Riemannian manifold, (M,g), is a metric space in a natural way. Fur- thermore, M is a complete metric space iﬀ M is geodesically complete. In this section, we explore brieﬂy some properties of complete Riemannian manifolds. 388 CHAPTER 12. GEODESICS ON RIEMANNIAN MANIFOLDS

Proposition 12.11 Let (M,g) be a connected Riemannian manifold. For any two points, p, q M, let d(p, q) be the greatest lower bound of the lengths of all piecewise smooth curves joining∈ p to q. Then, d is a metric on M and the topology of the metric space, (M,d), coincides with the original topology of M.

A proof of the above proposition can be found in Gallot, Hulin and Lafontaine [60] (Chapter 2, Proposition 2.91) or O’Neill [119] (Chapter 5, Proposition 18). The distance, d,isoftencalledtheRiemannian distance on M.Foranyp M and any >0, the metric ball of center p and radius is the subset, B (p) M,givenby∈ ⊆ B (p)= q M d(p, q) < . { ∈ | }

The next proposition follows easily from Proposition 12.4 (Milnor [106], Section 10, Corol- lary 10.8).

Proposition 12.12 Let (M,g) be a connected Riemannian manifold. For any compact subset, K M, there is a number δ>0 so that any two points, p, q K, with distance d(p, q) <⊆δare joined by a unique geodesic of length less than δ. Furthermore,∈ this geodesic is minimal and depends smoothly on its endpoints.

Recall from Definition 12.4 that (M,g) is geodesically complete iffthe exponential map, v expp(v), is defined for all p M and for all v TpM.Wenowprovethefollowing important→ theorem due to Hopf and∈ Rinow (1931): ∈

Theorem 12.13 (Hopf-Rinow) Let (M,g) be a connected Riemannian manifold. If there is a point, p M, such that expp is deﬁned on the entire tangent space, TpM, then any point, q M, can∈ be joined to p by a minimal geodesic. As a consequence, if M is geodesically complete,∈ then any two points of M can be joined by a minimal geodesic.

Proof .WefollowMilnor’sproofin[106],Chapter10,Theorem10.9.Pickanytwopoints, p, q M and let r = d(p, q). By Proposition 12.4, there is some open subset, W ,with p ∈W and some >0sothatanytwopointsofW are joined by a unique geodesic and the∈ exponential map is a diﬀeomorphism between the open ball, B(0,), and its image,

Up =expp(B(0,)). For δ<,letS =expp(S(0,δ)), where S(0,δ) is the sphere of radius δ. Since S U is compact, there is some point, ⊆ p p =exp(δv), with v =1, 0 p on S for which the distance to q is minimized. We will prove that

expp(rv)=q, 12.3. COMPLETE RIEMANNIAN MANIFOLDS, HOPF-RINOW, CUT LOCUS 389

which will imply that the geodesic, γ,givenbyγ(t)=expp(tv) is actually a minimal geodesic from p to q (with t [0,r]). Here, we use the fact that the exponential exp is deﬁned ∈ p everywhere on TpM. The proof amounts to showing that a point which moves along the geodesic γ must get closer and closer to q.Infact,foreacht [δ, r], we prove ∈ d(γ(t),q)=r t. ( ) − ∗t We get the proof by setting t = r. First, we prove ( ). Since every path from p to q must pass through S,bythechoiceof ∗δ p0,wehave r = d(p, q) = min d(p, s)+d(s, q) = δ + d(p0,q). s S ∈ { } Therefore, d(p ,q)=r δ and since p = γ(δ), this proves ( ). 0 − 0 ∗δ Deﬁne t [δ, r]by 0 ∈ t =sup t [δ, r] d(γ(t),q)=r t . 0 { ∈ | − }

As the set, t [δ, r] d(γ(t),q)=r t ,isclosed,itcontainsitsupperbound,t0,sothe equation ( {)∈ also holds.| We claim that− } if t

As we did with p,thereissomesmallδ > 0sothatifS =expγ(t0)(B(0,δ)), then there is some point, p0 ,onS with minimum distance from q and p0 is joined to γ(t0)byamimimal geodesic. We have

r t0 = d(γ(t0),q) = min d(γ(t0),s)+d(s, q) = δ + d(p0 ,q), s S − ∈ { } hence d(p ,q)=r t δ. ( ) 0 − 0 − † We claim that p0 = γ(t0 + δ). By the triangle inequality and using ( )(recallthatd(p, q)=r), we have †

d(p, p ) d(p, q) d(p ,q)=t + δ. 0 ≥ − 0 0

But, a path of length precisely t0 + δ from p to p0 is obtained by following γ from p to γ(t0), and then following a minimal geodesic from γ(t0)top0 .Sincethisbrokengeodesichas minimal length, by Corollary 12.9, it is a genuine (unbroken) geodesic, and so, it coincides with γ.Butthen,asp = γ(t + δ), equality ( ) becomes ( ), namely 0 0 † ∗t0+δ

d(γ(t + δ),q)=r (t + δ), 0 − 0 contradicting the maximality of t0.Therefore,wemusthavet0 = r and q =expp(rv), as desired. 390 CHAPTER 12. GEODESICS ON RIEMANNIAN MANIFOLDS

Remark: Theorem 12.13 is proved is every decent book on Riemannian geometry. Among those, we mention Gallot, Hulin and Lafontaine [60], Chapter 2, Theorem 2.103 and O’Neill [119], Chapter 5, Lemma 24. Theorem 12.13 implies the following result (often known as the Hopf-Rinow Theorem):

Theorem 12.14 Let (M,g) be a connected, Riemannian manifold. The following state- ments are equivalent: (1) The manifold (M,g) is geodesically complete, that is, for every p M, every geodesic ∈ through p can be extended to a geodesic defined on all of R. (2) For every point, p M, the map exp is defined on the entire tangent space, T M. ∈ p p (3) There is a point, p M, such that exp is defined on the entire tangent space, T M. ∈ p p (4) Any closed and bounded subset of the metric space, (M,d), is compact. (5) The metric space, (M,d), is complete (that is, every Cauchy sequence converges).

Proofs of Theorem 12.14 can be found in Gallot, Hulin and Lafontaine [60], Chapter 2, Corollary 2.105 and O’Neill [119], Chapter 5, Theorem 21. In view of Theorem 12.14, a connected Riemannian manifold, (M,g), is geodesically complete iﬀthe metric space, (M,d), is complete. We will refer simply to M as a complete Riemannian manifold (it is understood that M is connected). Also, by (4), every compact, Riemannian manifold is complete. If we remove any point, p,fromaRiemannianmanifold, M,thenM p is not complete since every geodesic that formerly went through p yields ageodesicthatcan’tbeextended.−{ } Assume (M,g)isacompleteRiemannianmanifold.Givenanypoint,p M,itis ∈ interesting to consider the subset, p TpM,consistingofallv TpM such that the geodesic U ⊆ ∈ t exp (tv) → p is a minimal geodesic up to t =1+,forsome>0. The subset p is open and star-shaped and it turns out that exp is a diﬀeomorphism from onto itsU image, exp ( ), in M. p Up p Up The left-over part, M expp( p)(ifnonempty),isactuallyequaltoexpp(∂ p)anditis an important subset of−M calledU the cut locus of p.ThefollowingpropositionisneededtoU establish properties of the cut locus:

Proposition 12.15 Let (M,g) be a complete Riemannian manifold. For any geodesic, γ :[0,a] M, from p = γ(0) to q = γ(a), the following properties hold: → (i) If there is no geodesic shorter than γ between p and q, then γ is minimal on [0,a]. (ii) If there is another geodesic of the same length as γ between p and q, then γ is no longer minimal on any larger interval, [0,a+ ]. 12.3. COMPLETE RIEMANNIAN MANIFOLDS, HOPF-RINOW, CUT LOCUS 391

(iii) If γ is minimal on any interval, I, then γ is also minimal on any subinterval of I.

Proof . Part (iii) is an immediate consequence of the triangle inequality. As M is complete, by the Hopf-Rinow Theorem, there is a minimal geodesic from p to q,soγ must be minimal too. This proves part (i). Part (ii) is proved in Gallot, Hulin and Lafontaine [60], Chapter 2, Corollary 2.111. Again, assume (M,g)isacompleteRiemannianmanifoldandletp M be any point. For every v T M,let ∈ ∈ p

Iv = s R the geodesic t expp(tv) is minimal on [0,s] . { ∈ ∪{∞} | → }

It is easy to see that Iv is a closed interval, so Iv =[0,ρ(v)] (with ρ(v)possiblyinﬁnite).It can be shown that if w = λv,thenρ(v)=λρ(w), so we can restrict our attention to unit n 1 n 1 vectors, v.Itcanalsobeshownthatthemap,ρ: S − R,iscontinuous,whereS − is → the unit sphere of center 0 in TpM,andthatρ(v)isboundedbelowbyastrictlypositive number.

Deﬁnition 12.7 Let (M,g)beacompleteRiemannianmanifoldandletp M be any point. Deﬁne by ∈ Up v = v T M ρ > v = v T M ρ(v) > 1 Up ∈ p v { ∈ p | } and the cut locus of p by n 1 Cut(p)=exp(∂ )= exp (ρ(v)v) v S − . p Up { p | ∈ } The set is open and star-shaped. The boundary, ∂ ,of in T M is sometimes Up Up Up p called the tangential cut locus of p and is denoted Cut(p).

Remark: The cut locus was ﬁrst introduced for convex surfaces by Poincar´e(1905) under the name ligne de partage. According to Do Carmo [50] (Chapter 13, Section 2), for Rie- mannian manifolds, the cut locus was introduced by J.H.C. Whitehead (1935). But it was Klingenberg (1959) who revived the interest in the cut locus and showed its usefuleness.

Proposition 12.16 Let (M,g) be a complete Riemannian manifold. For any point, p M, the sets exp ( ) and Cut(p) are disjoint and ∈ p Up M =exp( ) Cut(p). p Up ∪ Proof . From the Hopf-Rinow Theorem, for every q M,thereisaminimalgeodesic, t exp (vt)suchthatexp(v)=q. This shows that∈ρ(v) 1, so v and → p p ≥ ∈ Up M =exp( ) Cut(p). p Up ∪ 392 CHAPTER 12. GEODESICS ON RIEMANNIAN MANIFOLDS

It remains to show that this is a disjoint union. Assume q exp ( ) Cut(p). Since ∈ p Up ∩ q expp( p), there is a geodesic, γ,suchthatγ(0) = p, γ(a)=q and γ is minimal on [0,a∈ + ],U for some >0. On the other hand, as q Cut(p), there is some geodesic, γ, with γ(0) = p, γ(b)=q, γ minimal on [0,b], but γ not∈ minimal after b. As γ and γ are both minimal from p to q,theyhavethesamelengthfromp to q. But then, as γ and γ are distinct, by Proposition 12.15 (ii), the geodesic γ can’t be minimal after q,acontradiction. Observe that the injectivity radius, i(p), of M at p is equal to the distance from p to the cut locus of p: i(p)=d(p, Cut(p)) = inf d(p, q). q Cut(p) ∈ Consequently, the injectivity radius, i(M), of M is given by

i(M)= inf d(p, Cut(p)). p M ∈ If M is compact, it can be shown that i(M) > 0. It can also be shown using Jacobi fields that exp is a diffeomorphism from onto its image, exp ( ). Thus, exp ( )isdiffeomorphic p Up p Up p Up to an open ball in Rn (where n =dim(M)) and the cut locus is closed. Hence, the manifold, M,isobtainedbygluingtogetheranopenn-ball onto the cut locus of a point. In some sense the topology of M is “contained” in its cut locus. n 1 Given any sphere, S − ,thecutlocusofanypoint,p,isitsantipodalpoint, p . For more examples, consult Gallot, Hulin and Lafontaine [60] (Chapter 2, Section{ 2C7),− } Do Carmo [50] (Chapter 13, Section 2) or Berger [16] (Chapter 6). In general, the cut locus is very hard to compute. In fact, according to Berger [16], even for an ellipsoid, the determination of the cut locus of an arbitrary point is still a matter of conjecture!

12.4 The Calculus of Variations Applied to Geodesics; The First Variation Formula

Given a Riemannian manifold, (M,g), the path space, Ω(p, q), was introduced in Definition 12.1. It is an “infinite dimensional” manifold. By analogy with finite dimensional manifolds, we define a kind of tangent space to Ω(p, q)ata“point”ω.

Definition 12.8 For every “point”, ω Ω(p, q), we define the “tangent space”, TωΩ(p, q), of Ω(p, q)atω,tobethespaceofallpiecewisesmoothvectorfields,∈ W ,alongω,forwhich W (0) = W (1) = 0 (we may assume that our paths, ω,areparametrizedover[0, 1]).

Now, if F :Ω(p, q) R is a real-valued function on Ω(p, q), it is natural to ask what the induced “tangent map”,→ dFω : TωΩ(p, q) R, → 12.4. THE CALCULUS OF VARIATIONS APPLIED TO GEODESICS 393

should mean (here, we are identifying TF (ω)R with R). Observe that Ω(p, q)isnoteven atopologicalspacesotheanswerisfarfromobvious!Inthecasewheref : M R is a function on a manifold, there are various equivalent ways to define df ,oneofwhichinvolves→ curves. For every v TpM,ifα:( ,) M is a curve such that α(0) = p and α(0) = v, then we know that ∈ − → d(f(α(t))) df (v)= . p dt t=0 We may think of α as a small variation of p.Recallthat p is a critical point of f iff df p(v)=0, for all v T M. ∈ p Rather than attempting to define dFω (which requires some conditions on F ), we will mimic what we did with functions on manifolds and define what is a critical path of a function, F :Ω(p, q) R,usingthenotionofvariation. Now, geodesics from p to q are special paths in Ω(p,→ q)andtheyturnouttobethecriticalpathsoftheenergy function,

b b 2 E (ω)= ω(t) dt, a a where ω Ω(p, q), and 0 a0, such that − → (1) α(0) = ω (2) There is a subdivision, 0 = t0

Clearly, W TωΩ(p, q). In particular, W(0) = W (1) = 0. The vector ﬁeld, W ,isalsocalled the variation∈ vector ﬁeld associated with the variation α. 394 CHAPTER 12. GEODESICS ON RIEMANNIAN MANIFOLDS

Besides the curves in the variation, α(u)(withu ( ,)), for every t [0, 1], we have acurve,α :( ,) M,calledatransversal curve of∈ the− variation,deﬁnedby∈ t − → αt(u)=α(u)(t),

and W is equal to the velocity vector, α (0), at the point ω(t)=α (0). For sufficiently t t t small, the vector field, Wt,isaninfinitesimalmodelofthevariationα.

We can show that for any W TωΩ(p, q)thereisavariation,α:( ,) Ω(p, q), which satisfies the conditions ∈ − → dα α(0) = ω, (0) = W. du Here is a sketch of the proof: By the compactness of ω([0, 1]), it is possible to find a δ>0 so that exp is defined for all t [0, 1] and all v T M,with v <δ.Then,if ω(t) ∈ ∈ ω(t)

N =max Wt , t [0,1] ∈ δ for any such that 0 << N ,itcanbeshownthat

α(u)(t)=expω(t)(uWt)

works (for details, see Do Carmo [50], Chapter 9, Proposition 2.2). As we said earlier, given a function, F :Ω(p, q) R,wedonotattempttodeﬁnethe → diﬀerential, dFω,butinstead,thenotionofcriticalpath.

Definition 12.10 Given a function, F :Ω(p, q) R,wesaythatapath,ω Ω(p, q), is a critical path for F iff → ∈ dF (α(u)) =0, du u=0 dF (α(u)) for every variation, α,ofω (which implies that the derivative du is defined for every u=0 variation, α,ofω). dF (α(u)) For example, if F takes on its minimum on a path ω0 and if the derivatives du are all defined, then ω0 is a critical path of F . We will apply the above to two functions defined on Ω(p, q):

(1) The energy function (also called action integral):

b b 2 E (ω)= ω(t) dt. a a 1 (We write E = E0 .) 12.4. THE CALCULUS OF VARIATIONS APPLIED TO GEODESICS 395

(2) The arc-length function, b b L (ω)= ω(t) dt. a a

b b The quantities Ea(ω)andLa(ω)canbecomparedasfollows:ifweapplytheCauchy- Schwarz’s inequality,

b 2 b b f(t)g(t)dt f 2(t)dt g2(t)dt ≤ a a a with f(t) 1andg(t)= ω(t) ,weget ≡ (Lb (ω))2 (b a)Eb, a ≤ − a where equality holds iﬀ g is constant; that is, iﬀthe parameter t is proportional to arc-length. Now, suppose that there exists a minimal geodesic, γ,fromp to q.Then,

E(γ)=L(γ)2 L(ω)2 E(ω), ≤ ≤ where the equality L(γ)2 = L(ω)2 holds only if ω is also a minimal geodesic, possibly reparametrized. On the other hand, the equality L(ω)=E(ω)2 can hold only if the parameter is proportional to arc-length along ω.ThisprovesthatE(γ)

Proposition 12.17 Let (M,g) be a complete Riemannian manifold. For any two points, p, q M, if d(p, q)=δ, then the energy function, E :Ω(p, q) R, takes on its minimum, δ2, precisely∈ on the set of minimal geodesics from p to q. →

Next, we are going to show that the critical paths of the energy function are exactly the geodesics. For this, we need the ﬁrst variation formula. Let α:( ,) Ω(p, q)beavariationofω and let − → ∂α Wt = (0,t) ∂u be its associated variation vector ﬁeld. Furthermore, let dω V = = ω(t), t dt the velocity vector of ω and

∆tV = Vt+ Vt , − − the discontinuity in the velocity vector at t,whichisnonzeroonlyfort = ti,with0

Theorem 12.18 (First Variation Formula) For any path, ω Ω(p, q), we have ∈ 1 dE(α(u)) 1 D = W , ∆ V W , V dt, 2 du − t t − t dt t u=0 i 0 where α:( ,) Ω(p, q) is any variation of ω. − → Proof . (After Milnor, see [106], Chapter II, Section 12, Theorem 12.2.) By Proposition 11.24, we have ∂ ∂α ∂α D ∂α ∂α , =2 , . ∂u ∂t ∂t ∂u ∂t ∂t Therefore, dE(α(u)) d 1 ∂α ∂α 1 D ∂α ∂α = , dt =2 , dt. du du ∂t ∂t ∂u ∂t ∂t 0 0 Now, because we are using the Levi-Civita connection, which is torsion-free, it is not hard to prove that D ∂α D ∂α = , ∂t ∂u ∂u ∂t so dE(α(u)) 1 D ∂α ∂α =2 , dt. du ∂t ∂u ∂t 0 We can choose 0 = t0

t t=(ti) t i D ∂α ∂α ∂α ∂α − i ∂α D ∂α , dt = , , dt. ∂t ∂u ∂t ∂u ∂t − ∂u ∂t ∂t ti 1 t=(ti 1)+ ti 1 − − − ∂α Adding up these formulae for i =1,...k 1andusingthefactthat ∂u =0fort =0and t =1,weget −

k 1 1 dE(α(u)) − ∂α ∂α 1 ∂α D ∂α = , ∆ , dt. 2 du − ∂u ti ∂t − ∂u ∂t ∂t i=1 0 Setting u =0,weobtaintheformula

1 dE(α(u)) 1 D = W , ∆ V W , V dt, 2 du − t t − t dt t u=0 i 0 12.4. THE CALCULUS OF VARIATIONS APPLIED TO GEODESICS 397

as claimed. Intuitively, the ﬁrst term on the right-hand side shows that varying the path ω in the direction of decreasing “kink” tends to decrease E.Thesecondtermshowsthatvaryingthe D curve in the direction of its acceleration vector, dt ω(t), also tends to reduce E. Ageodesic,γ,(parametrizedover[0, 1]) is smooth on the entire interval [0, 1] and its D acceleration vector, dt γ(t), is identically zero along γ.Thisgivesushalfof

Theorem 12.19 Let (M,g) be a Riemanian manifold. For any two points, p, q M,a path, ω Ω(p, q) (parametrized over [0, 1]), is critical for the energy function, E, iﬀ∈ ω is a geodesic.∈

Proof .Fromtheﬁrstvariationformula,itisclearthatageodesicisacriticalpathofE. Conversely, assume ω is a critical path of E.Thereisavariation,α,ofω such that its associated variation vector ﬁeld is of the form D W (t)=f(t) ω(t), dt

with f(t) smooth and positive except that it vanishes at the ti’s. For this variation, we get

1 dE(α(u)) 1 D D = f(t) ω(t), γ(t) dt. 2 du − dt dt u=0 0 This expression is zero iﬀ D ω(t)=0 on[0, 1]. dt

Hence, the restriction of ω to each [ti,ti+1]isageodesic. It remains to prove that ω is smooth on the entire interval [0, 1]. For this, pick a variation α such that

W (ti)=∆ti V. Then, we have 1 dE(α(u)) k = ∆ V,∆ V . 2 du − ti ti u=0 i=1 If the above expression is zero, then ∆ V =0fori =1,...,k 1, which means that ω is C1 ti everywhere on [0, 1]. By the uniqueness theorem for ODE’s, ω−must be smooth everywhere on [0, 1], and thus, it is an unbroken geodesic.

Remark: If ω Ω(p, q)isparametrizedbyarc-length,itiseasytoprovethat ∈ dL(α(u)) 1 dE(α(u)) = . du 2 du u=0 u=0 398 CHAPTER 12. GEODESICS ON RIEMANNIAN MANIFOLDS

As a consequence, a path, ω Ω(p, q)iscriticalforthearc-lengthfunction,L,iﬀitcanbe reparametrized so that it is a∈ geodesic (see Gallot, Hulin and Lafontaine [60], Chapter 3, Theorem 3.31). In order to go deeper into the study of geodesics we need Jacobi ﬁelds and the “second variation formula”, both involving a curvature term. Therefore, we now proceed with a more thorough study of curvature on Riemannian manifolds.