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THE SINGLE-LEAF FROBENIUS THEOREM WITH APPLICATIONS

PAOLO PICCIONE AND DANIEL V. TAUSK

Dedicated to Prof. Serge Lang

ABSTRACT. Using the notion of Levi form of a smooth distribution, we dis-cuss the local and the global problem of existence ofonehorizontal section of asmooth vector bundle endowed with a horizontal distribution. The analysis willlead to the formulation of a “one-leaf” analogue of the classical Frobenius inte-grability theorem in elementary differential geometry. Several applications of theresult will be discussed. First, we will give a characterization of symmetric con-nections arising as Levi–Civita connections of semi-Riemannian metric tensors.Second, we will prove a general version of the classical Cartan–Ambrose–HicksTheorem giving conditions on the existence of an affine map with prescribeddifferential at one point between manifolds endowed with connections.

CONTENTS

1. Introduction 22. The Levi form and the “single leaf Frobenius Theorem” 52.1. The Levi form of a smooth distribution 52.2. Horizontal distributions and horizontal liftings 72.3. The single leaf Frobenius theorem 82.4. The higher order single leaf Frobenius theorem 93. The global “single leaf Frobenius Theorem” 113.1. Sprays on manifolds 113.2. The global single leaf Frobenius theorem 134. Levi–Civita connections 154.1. Levi form of the horizontal distribution of a connection 154.2. Connections arising from metric tensors 185. Affine maps 205.1. The Cartan–Ambrose–Hicks Theorem 205.2. Higher order Cartan–Ambrose–Hicks theorem 26Appendix A. A globalization principle 29Appendix B. A crash course on calculus with connections 32References 39

Date: October 25th, 2005.2000Mathematics Subject Classification.53B05, 53C05, 53C42, 55R25.Key words and phrases.Smooth distributions, Levi form, Frobenius theorem, affineconnections,

Levi–Civita connections, Cartan–Ambrose–Hicks theorem.1

http://arxiv.org/abs/math/0510555v1

THE SINGLE-LEAF FROBENIUS THEOREM 2

1. INTRODUCTION

The central theme of the paper is the study of conditions for the existence ofone integral leaf of (non integrable) smooth distributions satisfying a given initialcondition. The integrability condition given by Frobeniustheorem, a very classicalresult in elementary Differential Geometry, guarantees the existence of integralleaves withany initial condition. If on one hand such condition is very strong, onthe other hand the involutivity assumption in Frobenius theorem is very restrictive.For instance, the integrability of the horizontal distribution of a connection in avector bundle is equivalent to the flatness of the connection.

A measure of non integrability for a smooth distributionD on a manifoldE isprovided by the so-calledLevi formLD of D; this is a skew-symmetric bilineartensor defined on the distribution, taking values in the quotientTE/D. Forx ∈ Eandv,w ∈ Dx, the valueLD

x (v,w) is given by the projection onTxE/Dx of the Liebracket[X,Y ]x, whereX andY are arbitrary extensions ofv andw respectivelyto D-horizontal vector fields. IfΣ ⊂ E is an integral submanifold ofD, thenthe Levi form ofD vanishes on the points ofΣ. The first central observation thatis made in this paper is that, conversely, given an immersed submanifoldΣ of Ewith Tx0Σ = Dx0 for somex0 ∈ E, if Σ is ruled (in an appropriate sense) bycurves tangent toD, and ifLD vanishes alongΣ, thenΣ is an integral submanifoldof D. In particular, assume thatD ⊂ TE is a horizontal distribution of a vectorbundleπ : E → M over a manifoldM , and thatΣ is a local section ofπ whichis obtained by parallel lifting of a family of curves onM issuing from some fixedpoint x0. If the Levi form ofD vanishes alongΣ, thenΣ is a parallel section ofπ (Theorem 2.5); we call this result the(local) single leaf Frobenius theorem. Inthe real analytic case, a higher order version of this resultis given in Theorem 2.7;roughly speaking, the higher order derivatives of the Levi form LD are obtainedfrom iterated Lie brackets ofD-horizontal vector fields. The higher order single-leaf Frobenius theorem states that, in the real-analytic case, if at some pointx0 ofthe manifoldE all the iterated brackets of vector fields inD belong toDx0 , thenthere exists an integral submanifold ofD throughx0.

A global version of the single-leaf Frobenius theorem is discussed in Theo-rem 3.11; here, the base manifoldM has to be assumed simply-connected. As-sume that a spray is given onM , for instance, the geodesic spray of some Rie-mannian metric. The existence of a global parallel section of π through a pointe0 with π(e0) = x0 ∈ M is guaranteed by the following condition: every piece-wise solutionγ : [a, b] → M of S with γ(a) = x0 should admit a parallel liftingγ : [a, b] → E such thatγ(a) = e0 and such that the Levi form ofD vanishes atthe pointγ(b).

We also observe (Proposition 3.12) that in the real analyticcase, every localparallel section defined on a non empty open subset of a simplyconnected manifoldM extends to a global parallel section.


A huge number of problems in Analysis and in Geometry can be cast into thelanguage of distributions and integral submanifolds. As anapplication of the the-ory developed in this paper, we will consider two problems. First, we will char-acterize those symmetric connections that are Levi–Civitaconnections of somesemi-Riemannian metric (alternatively, this problem can be studied using holo-nomy theory, see [1]). Second, we will prove a very general version of anotherclassical result in Differential Geometry, which is the Cartan–Ambrose–Hicks the-orem (see [4]). We will prove a necessary and sufficient condition for the existenceof an affine map between manifolds endowed with arbitrary connections.

Let us describe briefly these two results.Consider the case of a distribution given by the horizontal space of a connec-

tion ∇ of a vector bundleπ : E → M . For ξ ∈ E, setm = π(ξ) ∈ M andEm = π−1(m); one can identifyDξ with TmM , and the quotientTξE/Dξ with thevertical subspaceTξ(Em) ∼= Em. Then, the Levi formLD

ξ : TmM ×TmM → Emis given by the curvature tensor of∇, up to a sign (Lemma 4.1). In this case,the single leaf Frobenius theorem tells us that a local parallel section ofπ throughsome pointe0 ∈ E exists provided that along each parallel lifting of a familyofcurves issuing fromπ(e0) ∈ M the curvature tensor vanishes (Corollary 4.2). Inthe real analytic case, the existence of a local parallel section through a pointξ ∈ Eis equivalent to the vanishing of all the covariant derivatives∇kR, k ≥ 0, of thecurvature tensorR at the pointm = π(ξ) (Proposition 4.5).

Assume that the vector bundleπ : E → M is endowed with a connection∇,and denote by∇bil the induced connection onE∗ ⊗ E∗. If g is a (local) sectionof E∗ ⊗ E∗, then vanishing of the curvature tensorRbil of ∇bil means that thebilinear mapg

(R(v,w)·, ·) is anti-symmetric for allv,w (formula (13)). From

this observation, we get the following result on the existence of parallel metrictensor relatively to a given connection∇ on a manifoldM : given a nondegenerate(symmetric) bilinear formg0 onTm0

M , assume that the tensorg obtained fromg0by ∇-parallel transport along a family of curves issuing fromm0 is such thatRis g-anti-symmetric. Then,g is ∇-parallel (Proposition 4.6). Similarly, in the realanalytic case, if∇kR atm0 is g0-anti-symmetric for allk ≥ 0, theng0 extends to asemi-Riemannian metric tensor whose Levi–Civita connection is∇. These resultshave been used in [3] to obtain characterizations of left-invariant semi-RiemannianLevi Civita connections in Lie groups.

As another application of our theory, in Section 5 we will study the problemof existence of an affine (i.e., connection preserving) mapf between two affinemanifolds(M,∇M ) and(N,∇N ), whose valuey0 ∈ N at some pointx0 ∈ M isgiven and whose differentialdf(x0) : Tx0M → Ty0N is prescribed. We prove ageneral version of the classical Cartan–Ambrose–Hicks theorem (Theorem 5.1 forthe local result, Theorem 5.3 for the global version), giving a necessary and suffi-cient condition for the existence of such a map; here, the connections∇M and∇N

are not assumed to be symmetric, and no assumption is made on the dimension ofthe manifoldsM andN , as well as on the linear mapdf(x0). The key observa-tion here (Lemma 5.6) is that, considering the vector bundleE = Lin(TM,TN)


over the productM × N , endowed with a natural connection induced by∇M

and∇N (see formula (15)), then a smooth mapf : U ⊂ M → N is an affinemap if and only if the differentialdf is a local parallel section ofE along themapU ∋ x 7→

(x, f(x)

)∈ M × N . WhenM andN are endowed with semi-

Riemannian metrics and∇M and∇N are the respective Levi–Civita connections,then our result gives a necessary and sufficient condition for the existence of atotally geodesic immersion ofM in N .

The proof of the Cartan–Ambrose–Hicks theorem is obtained as an applicationof the single-leaf Frobenius theorem, once the Levi form of the horizontal distribu-tion of the induced connection onE is computed (Lemma 5.14). The higher orderversion of this result (Theorem 5.16) is particularly interesting: in the real analyticcase, a (local) affine mapf : U ⊂ M → N with f(x0) = y0 anddf(x0) = σexists if and only ifσ relates covariant derivatives of all order of curvature andtorsion of∇M and∇N at the pointsx0 andy0 respectively.

As a nice corollary of the higher order Cartan–Ambrose–Hicks theorem, weget the following curious result (Corollary 5.19): ifM is a real-analytic manifoldendowed with a real-analytic connection∇, and letx0 ∈ M be fixed; there existsan affine symmetry aroundx0 if and only if ∇(2r)Tx0 = 0 and∇(2r+1)Rx0 = 0for all r ≥ 0.

A certain effort has been made in order to make the presentation of the materialself-contained. For this reason, we have found convenient to discuss, together withthe original material, some auxiliary topics needed for a more complete presenta-tion of our results. For instance, in Subsection 3.1, we discuss and give the basicproperties of the exponential map of a spray (this is needed in our statement of theglobal one-leaf Frobenius theorem). Similarly, in Appendix B we develop the ba-sic theory needed for making computations with covariant derivatives, curvaturesand torsions of connections on vector bundles obtained by functorial constructions;this kind of computations is heavily used throughout the paper. Finally, in Appen-dix A we discuss a globalization principle in a very general setting of pre-sheafson topological spaces. Such principle is used in the proof ofthe global versions ofthe single-leaf Frobenius theorem (see for instance the proofs of Theorem 3.11 andProposition 3.12). Typically, the globalization principle is employed in the follow-ing manner: given a vector bundleπ : E →M , a pre-sheafP is defined onM bydefining, for all open subsetU ⊂M , P(U) to be the set of all sectionss : U → Eof π satisfying some property (for instance, parallel sections). For V ⊂ U , ands ∈ P(U), the mapPU,V : P(U) → P(V ) is given by settingPU,V (s) = s|V .In this context, the existence of a global section ofπ with the required property isequivalent to the fact that the setP(M) should be non empty. The central resultof Appendix A (Proposition A.8) gives a sufficient conditionfor this, in terms ofthree properties of pre-sheaves, calledlocalization, uniquenessandextension.

Dedicatory. The proof of the single-leaf Frobenius theorem discussed here hastaken inspiration from the proof of the classical Frobeniustheorem presented inSerge Lang’s world famous book [2]. Since the very beginningof their mathe-matical careers, both authors have benefited very much from this and from other


beautiful books published by Prof. Lang. We want to thank himby dedicating thispaper to his memory.

2. THE LEVI FORM AND THE “ SINGLE LEAF FROBENIUS THEOREM”

Recall that asmooth distributionD on a smooth manifoldE is a smooth vectorsubbundle of the tangent bundleTE. Forx ∈ E we setDx = TxE ∩D, i.e.,Dx isthe fiber of the vector bundleD overx. A vector fieldX onE is calledhorizontalwith respect to a distributionD (or simplyD-horizontal) if X takes values inD,i.e., if X(x) ∈ Dx for all x ∈ E. An immersed submanifoldS of E is called anintegral submanifoldfor D if TxS = Dx, for all x ∈ S. The distributionD iscalledintegrableif through every point ofE passes an integral submanifold forD.

2.1. The Levi form of a smooth distribution.

Definition 2.1. LetE be a smooth manifold and letD be a distribution onE. TheLevi formof D at a pointx ∈ E is the bilinear map:

LDx : Dx ×Dx −→ TxE/Dx

defined byLDx (v,w) = [X,Y ](x) + Dx ∈ TxE/Dx, whereX andY areD-

horizontal smooth vector fields defined in an open neighborhood of x in E withX(x) = v andY (x) = w. By [X,Y ] we denote the Lie bracket of the vector fieldsX andY .

Below we show that the Levi form is well-defined, i.e.,[X,Y ](x)+Dx does notdepend on the choice of theD-horizontal vector fieldsX andY with X(x) = v,Y (x) = w. Let θ be a smoothRk-valued1-form on an open neighborhoodU ofx such thatKer(θx) = Dx for all x ∈ U . If X andY are vector fields on an openneighborhood ofx then Cartan’s formula for exterior differentiation gives:

dθ(X,Y ) = X(θ(Y )

)− Y

(θ(X)

)− θ

([X,Y ]

).

If X andY areD-horizontal then the equality above reduces to:

dθ(X,Y ) = −θ([X,Y ]

).

The formula above implies that ifX ′, Y ′ areD-horizontal vector fields such thatX ′(x) = X(x) and Y ′(x) = Y (x) then θ

([X,Y ] − [X ′, Y ′]

)(x) = 0, i.e.,

[X,Y ](x) − [X ′, Y ′](x) ∈ Dx. Hence the Levi form is well-defined. SettingX(x) = v andY (x) = w we obtain the following formula:

(1) θx(LDx (v,w)

)= −dθ(v,w), v, w ∈ Dx,

whereθx : TxE/Dx → R

k denotes the linear map induced byθx in the quotientspace.

Remark2.2. Clearly, by the classical Frobenius Theorem,D is integrable if andonly if its Levi form is identically zero. Moreover, the Leviform of D vanishesalong any integral submanifold ofD.


2.1. Example. LetU be an open subset ofRn = Rk ×Rn−k and let

F : U ∋ (x, y) 7−→ F(x,y) ∈ Lin(Rk,Rn−k)

be a smooth map. We consider the distributionD = Gr(F ) onU , i.e.,D(x,y) =

Gr(F(x,y)

), for all (x, y) ∈ U . GivenX ∈ Rk, we define aD-horizontal vector

field X on U by settingX(x,y) =(X,F(x,y)(X)

), for all (x, y) ∈ U . Given

X,Y ∈ Rk then:

[X, Y ] =(0, ∂xF (X,Y ) + ∂yF (F (X), Y )− ∂xF (Y,X) + ∂yF (F (Y ),X)

).

If we identifyD(x,y) withRk by the isomorphism(X,F (X)

)7→ X andRn/D(x,y)

withRn−k ∼= {0}k×Rn−k by the isomorphism(v,w)+D(x,y) 7→ w−F (v) thenthe Levi formLD : Rk ×Rk → R

n−k is given by:

(2) LD(X,Y ) = [X, Y ].

Lemma 2.3. LetE be a smooth manifold,D be a smooth distribution onE and let

U ∋ (t, s) 7−→ H(t, s) ∈ E

be a smooth map defined on an open subsetU ⊂ R

2. Let I ⊂ R be an intervaland lets0 ∈ R be such thatI × {s0} ⊂ U andLD

H(t,s0)= 0 for all t ∈ I. Assume

that ∂H∂t

(t, s) ∈ D for all (t, s) ∈ U . If ∂H∂s

(t0, s0) ∈ D for somet0 ∈ I then∂H∂s

(t, s0) ∈ D for all t ∈ I.

Proof. The set:

I ′ ={t ∈ I : ∂H

∂s(t, s0) ∈ D

}

is obviously closed inI because the mapI ∋ t 7→ ∂H∂s

(t, s0) ∈ TE is continuousandD is a closed subset ofTE. SinceI is connected andt0 ∈ I ′, the proof willbe complete once we show thatI ′ is open inI. Let t1 ∈ I ′ be fixed. Letθ be anR

k-valued smooth1-form defined in an open neighborhoodV of H(t1, s0) in Esuch that the linear mapθx : TxE → R

k is surjective andKer(θx) = Dx for allx ∈ V . Choose a distributionD′ on V such thatTxE = Dx ⊕ D′

x for all x ∈ V .Then, for eachx ∈ V , θx restricts to an isomorphism fromD′

x ontoRk. Let J bea connected neighborhood oft1 in I such thatH(t, s0) ∈ V for all t ∈ J . We willshow below that the map:

(3) J ∋ t 7−→ θH(t,s0)

(∂H∂s

(t, s0))∈ Rk

is a solution of a homogeneous linear ODE; sinceθH(t1,s0)

(∂H∂s

(t1, s0))= 0, it

will follow that θH(t,s0)

(∂H∂s

(t, s0))= 0 for all t ∈ J , i.e.,J ⊂ I ′.

We denote by∂∂t

and ∂∂s

the canonical basis ofR2 and we apply Cartan’s formulafor exterior differentiation to the1-formH∗θ obtaining:

d(H∗θ)(∂∂t, ∂∂s

)= ∂

∂t

((H∗θ)

(∂∂s

))− ∂

∂s

((H∗θ)

(∂∂t

))− (H∗θ)

([∂∂t, ∂∂s

]).


Sinced(H∗θ) = H∗(dθ) and[∂∂t, ∂∂s

]= 0 we get:

dθH(t,s0)

(∂H∂t

(t, s0),∂H∂s

(t, s0))= ∂

∂t

(θH(t,s0)

(∂H∂s

(t, s0)))

(4)

− ∂∂s

∣∣s=s0

(θH(t,s)

(∂H∂t

(t, s))), t ∈ J.

Observe that, since∂H∂t

(t, s) is in D, the last term on the righthand side of (4)vanishes. We can write∂H

∂s(t, s0) = u1(t)+u2(t) with u1(t) ∈ D andu2(t) ∈ D′.

Since the Levi form ofD vanishes at points of the formH(t, s0), equation (1)implies thatdθH(t,s0)(v,w) = 0 for all v,w ∈ DH(t,s0). We may thus replace∂H∂s

(t, s0) by u2(t) in the lefthand side of (4). Fort ∈ J we consider the linearmapL(t) : Rk → R

k defined by:

L(t) · z = dθH(t,s0)

(∂H∂t

(t, s0), σH(t,s0)(z)), z ∈ Rk,

where, forx ∈ V , σx : Rk → D′x denotes the inverse of the isomorphism

θx|D′x: D′

x −→ R

k.

Observe that:

dθH(t,s0)

(∂H∂t

(t, s0),∂H∂s

(t, s0))= dθH(t,s0)

(∂H∂t

(t, s0), u2(t))

= L(t) · θH(t,s0)

(u2(t)

)

= L(t) · θH(t,s0)

(∂H∂s

(t, s0)).

Equation (4) can now be rewritten as:

∂∂t

(θH(t,s0)

(∂H∂s

(t, s0)))

= L(t) · θH(t,s0)

(∂H∂s

(t, s0)), t ∈ J.

Hence the map (3) is a solution of a homogeneous linear ODE andwe are done. �

2.2. Horizontal distributions and horizontal liftings. If E,M are smooth man-ifolds andπ : E →M is a smooth submersion then a smooth distributionD onEis calledhorizontalwith respect toπ if

TxE = Ker(dπx)⊕Dx,

for all x ∈ E. Given a smooth horizontal distributionD on E then a piecewisesmooth curveγ : I → E is calledhorizontal if γ′(t) ∈ D for all t for which γ′(t)exists. Given a piecewise smooth curveγ : I →M then ahorizontal liftingof γ isa horizontal piecewise smooth curveγ : I → E such thatπ ◦ γ = γ.

By standard results of existence and uniqueness of solutions of ODE’s it fol-lows that givent0 ∈ I andx0 ∈ π−1

(γ(t0)

)then there exists a unique maximal

horizontal lifting γ of γ with γ(t0) = x0 defined in a subinterval ofI aroundt0.Let Λ be a smooth manifold. By aΛ-parametric family of curvesψ onM we

mean a smooth mapψ : Z ⊂ R× Λ →M defined on an open subsetZ of R× Λsuch that the set:

Iλ ={t ∈ R : (t, λ) ∈ Z

}⊂ R


is an interval containing the origin, for allλ ∈ Λ. By a local right inverseof ψ wemean a locally defined smooth mapα : V ⊂ M → Z such thatψ

(α(m)

)= m,

for all m ∈ V .

2.2. Example. LetM be a smooth manifold endowed with a connection∇. Givena pointx0 ∈ M we setΛ = Tx0M and we define aΛ-parametric family of curvesψ onM by settingψ(t, λ) = expx0(tλ); the domainZ ⊂ R × Λ of ψ is the setof pairs(t, λ) such thattλ is in the domain ofexpx0. A local right inverse ofψis defined as follows: letV0 be an open neighborhood of the origin inTx0M thatis mapped diffeomorphically byexpx0 onto an open neighborhoodV of x0 in M .We set:

α(m) =(1, (expx0 |V0)

−1(m)),

for all m ∈ V . We remark that the same construction holds if one replaces thegeodesic spray of a connection with an arbitrary spray (see Section 3).

A local sectionof a smooth submersionπ : E →M is a locally defined smoothmaps : U ⊂M → E such thatπ ◦ s = IdU . A local sections is calledhorizontalif the range ofds(m) is Ds(m), for allm ∈ U .

Lemma 2.4. Let s1 : U → E, s2 : U → E be local smooth horizontal sections ofE defined in an open connected subsetU ofM . If s1(x) = s2(x) for somex ∈ Uthens1 = s2.

Proof. Giveny ∈ U , there exists a piecewise smooth curveγ : [a, b] → U withγ(a) = x andγ(b) = y. Thens1 ◦ γ ands2 ◦ γ are both horizontal liftings ofγstarting at the same point ofE; hences1 ◦ γ = s2 ◦ γ ands1(y) = s2(y). �

2.3. Example. Consider the distributionD = Gr(F ) onU ⊂ R

n defined in Ex-ample 2.1. Then the first projectionπ1 : U → R

k is a submersion andD ishorizontal with respect toπ1. A horizontal sections : Rk ⊃ V → R

n of π1 is amaps(x) =

(x, f(x)

)wheref : V → R

n−k is a solution of thetotal differentialequation:

(5) df(x) = F(x, f(x)

), x ∈ V.

2.3. The single leaf Frobenius theorem.

Theorem 2.5(local single leaf Frobenius). Let E, M be smooth manifolds,π :E →M be a smooth submersion,D be a smooth horizontal distribution onE andψ : Z ⊂ R×Λ →M be aΛ-parametric family of curves onM with a local rightinverseα : V ⊂ M → Z. Let ψ : Z → E be aΛ-parametric family of curvesonE such thatt 7→ ψ(t, λ) is a horizontal lifting oft 7→ ψ(t, λ), for all λ ∈ Λ.Assume that:

(a) the Levi form ofD vanishes on the range ofψ;(b) ∂λψ(0, λ) : TλΛ → T

ψ(0,λ)E takes values inD for all λ ∈ Λ.

Thens = ψ ◦ α : V → E is a local horizontal section ofπ.


Proof. If ]−ε, ε[ ∋ s 7→ λ(s) is an arbitrary smooth curve onΛ then the map

H(t, s) = ψ(t, λ(s)

)

satisfies the hypotheses of Lemma 2.3 witht0 = 0 ands0 = 0. Thus:

∂H

∂s(t, 0) =

∂ψ

∂λ

(t, λ(0)

)λ′(0)

is inD for all t ∈ Iλ(0). It follows thatdψ(t,λ) takes values inD, for all (t, λ) ∈ Z.

Henceds(m) = dψ(s(m)

)◦ dα(m) also takes values inD, for allm ∈ V . �

Remark2.6. We observe that if the mapλ 7→ ψ(0, λ) is constant then hypothesis(b) of Theorem 2.5 is automatically satisfied. Theorem 2.5 istypically used as fol-lows: one considers theΛ-parametric family of curvesψ explained in Example 2.2,a fixed pointe0 ∈ π−1(x0) ⊂ E and for eachλ ∈ Λ one definest 7→ ψ(t, λ) to bethe horizontal lifting oft 7→ ψ(t, λ) with ψ(0, λ) = e0.

2.4. Example. The single leaf Frobenius theorem can be used to prove the exis-tence of solutions of the total differential equation (5) satisfying a initial conditionf(x0) = y0 as follows. LetV be a star-shaped open neighborhood ofx0 in Rk.SetΛ = R

k; we define aΛ-parametric family of curvesψ : Z ⊂ R × Λ → MonM = R

k by settingψ(t, λ) = x0 + tλ, whereZ is the set of pairs(t, λ) withx0 + tλ ∈ V . A horizontal lifting t 7→ ψ(t, λ) =

(ψ(t, λ),Ψ(t, λ)

)of the curve

t 7→ ψ(t, λ) is a solution of the ODE:

(6)d

dtΨ(t, λ) = F

ψ(t,λ)(λ).

We choose the solutiont 7→ ψ(t, λ) of the ODE (6) with initial conditionΨ(0, λ) =

y0. We can assume thatV is small enough so thatψ is well-defined onZ. Hy-pothesis (b) of Theorem 2.5 is then automatically satisfied and hypothesis (a) isequivalent to the condition that (2) vanishes on the points of the form ψ(t, λ),(t, λ) ∈ Z. Under these circumstances, the thesis of Theorem 2.5 guarantees thatf : V ∋ x 7→ Ψ(1, x − x0) ∈ R

n−k is a solution of the total differential equation(5) with f(x0) = y0.

2.4. The higher order single leaf Frobenius theorem.LetD be a smooth distri-bution on a smooth manifoldE. We denote byΓ(TE) the set of all smooth vectorfields onE, by Γ(D) the subspace ofΓ(TE) consisting ofD-horizontal vectorfields and byΓ∞(D) the Lie subalgebra ofΓ(TE) spanned byΓ(D). The LiealgebraΓ∞(D) can be alternatively described as follows; we define recursively asequence

Γ0(D) ⊂ Γ1(D) ⊂ Γ2(D) ⊂ · · ·

of subspaces ofΓ(TE) by settingΓ0(D) = Γ(D) andΓr+1(D) to be the subspaceof Γ(TE) spanned byΓr(D) and by the brackets[X,Y ], with X ∈ Γr(D) andY ∈ Γ(D). Then:

Γ∞(D) =

∞⋃

r=0

Γr(D).


GivenX ∈ Γ(TE) we denote byadX : Γ(TE) → Γ(TE) the operatoradX(Y ) =[X,Y ].

Theorem 2.7. Let E be a real-analytic manifold endowed with a real-analyticdistribution D. Given e0 ∈ E then there exists an integral submanifold ofDpassing throughe0 if and only ifX(e0) ∈ De0 , for all X ∈ Γ∞(D).

Proof. If there exists an integral submanifoldS of D passing throughe0 then itfollows immediately by induction onr thatX(S) ⊂ D, for all X ∈ Γr(D) andall r ≥ 0. Thus,X(e0) ∈ De0 , for all X ∈ Γ∞(D). Conversely, assume thatX(e0) ∈ De0, for allX ∈ Γ∞(D). By considering a convenient real-analytic localchart arounde0 we may assume without loss of generality thatE = U is an opensubset ofRn = R

k × Rn−k and thatD is of the formGr(F ) (see Example 2.1).Write e0 = (x0, y0); we will use the ideas explained in Example 2.4 to find asolutionf of the total differential equation (5) withf(x0) = y0. ThenGr(f) is therequired integral submanifold ofD passing throughe0. Observe that givenλ ∈ Rk

thent 7→ ψ(t, λ) is an integral curve of the constant vector fieldλ in Rk and thusthe horizontal liftt 7→ ψ(t, λ) is an integral curve of the vector fieldλ =

(λ, F (λ)

)

onRn passing throughe0 at t = 0. We now letλ ∈ Rk,X,Y ∈ Rk, be fixed andwe define a mapt 7→ φ(t) ∈ Rn−k by setting:

φ(t) = LD

ψ(t,λ)(X,Y ) = [X, Y ]ψ(t,λ).

The proof will be completed once we show thatφ is identically zero; sinceφ isreal-analytic, it suffices to proof that all derivatives ofφ at t = 0 vanish. Let usshow by induction onr that for allr ≥ 0 ther-th derivative ofφ is given by:

(7) φ(r)(t) = (adλ)r[X, Y ] + L(r)

((ad

λ)i[X, Y ]; i = 0, 1, . . . , r − 1

),

where the righthand side is computed at the pointψ(t, λ) andL(r) is a smooth mapthat associates to each(x, y) ∈ U ⊂ Rn a linear map:

L(r)(x,y) :

⊕

r

R

n−k −→ R

n−k.

From equality (7) the conclusion will follow; namely, for all i, (adλ)i[X, Y ] is in

{0}k × Rn−k and since((ad

λ)i[X, Y ]

)e0

∈ De0 , we get((ad

λ)i[X, Y ]

)e0

= 0.

Henceφ(r)(0) = 0, for all r ≥ 0. To prove (7) simply differentiate both sides withrespect tot, observing that:

d

dt(ad

λ)i[X, Y ] = d

((ad

λ)i[X, Y ]

)· λ = (ad

λ)i+1[X, Y ] + dλ

((ad

λ)i[X, Y ]

).

�

Remark2.8. Clearly, the hypotheses of Theorem 2.7 are local, i.e., ifU is an openneighborhood ofe0 in E thenX(e0) ∈ De0 for all X ∈ Γ∞(D|U ) if and only ifX(e0) ∈ De0 for all X ∈ Γ∞(D). ReplacingE with an open neighborhood ofe0,we may assume thatD admits a global referentialX1, . . . ,Xk. It is easy to see that


Γr(D) is theC∞(E)-module spanned byX1, . . . ,Xk and by the iterated brackets:

(8) [Xi1 , [Xi2 , . . . , [Xis ,Xis+1] · · · ]], i1, . . . , is = 1, . . . , k, s = 1, . . . , r.

Thus, in order to check the hypotheses of Theorem 2.7, it suffices to verify if thebrackets in (8) evaluated ate0 are inDe0 , for all s ≥ 1.

3. THE GLOBAL “ SINGLE LEAF FROBENIUS THEOREM”

3.1. Sprays on manifolds. Let M be a smooth manifold and letπ : TM → Mthe canonical projection of its tangent bundle. Denote bydπ : TTM → TMthe differential ofπ; we denote byπ : TTM → TM the natural projection ofTTM = T (TM). For eacha ∈ R we denote byma : TM → TM the operatorof multiplication bya.

Definition 3.1. A sprayonM is a smooth vector fieldS : TM → TTM on themanifoldTM satisfying the following two conditions:

(i) dπ ◦ S = π ◦ S;(ii) for all a ∈ R, adma ◦ S = S ◦ ma, i.e.,adma(v)S(v) = S(av), for all

v ∈ TM .

Remark3.2. Notice that property (b) on Definition 3.1 implies that a spray vanisheson the zero section ofTM . In particular, the integral curves ofS passing throughthe zero section are constant.

Lemma 3.3. LetS : TM → TTM be a smooth vector field onTM . ThenS is aspray onM if and only if the following conditions are satisfied:

(a) for every integral curveλ : I → TM of S, we haveλ = γ′, whereγ = π ◦ λ;

(b) if λ = γ′ : I → TM is an integral curve ofS then

I ∋ t 7−→d

dtγ(at) ∈ TM

is an integral curve ofS, for all a ∈ R.

Definition 3.4. A curveγ : I →M is called a (maximal) solution ofS if γ′ : I →TM is a (maximal) integral curve of the vector fieldS.

Obviously for everyx ∈ M , v ∈ TxM there exists a unique maximal solutionγ of S with γ(0) = x andγ′(0) = v.

3.1. Example (geodesic spray). If ∇ is a connection onM then one can define asprayS onM by takingS(v) to be the unique horizontal vector onTvTM suchthatdπv

(S(v)

)= v, for all v ∈ TM . The integral curves ofS are the curvesγ′,

with γ : I →M a geodesic of∇.

3.2. Example(one-parameter subgroup spray). LetG be a Lie group and denote byg its Lie algebra. Using left (resp., right) translations, one can identify the tangentbundleTG with the productG× g, so that

T (TG) ∼= T (G× g) ∼= (TG)× (Tg) ∼= (G× g)× (g× g).


The vector field onTG given byS(g,X) = (g,X,X, 0), g ∈ G,X ∈ g, is a sprayinG, whose solutions are left (resp., right) translations of one-parameter subgroupsof G. The sprayS is the geodesic spray of the connection whose Christoffel sym-bols vanish on a left (resp., right) invariant frame.

Let S be a fixed spray onM and denote by

F : Dom(F ) ⊂ R× TM −→ TM

its maximal flow. Theexponential mapassociated toS is the map:

exp(v) = π(F (1, v)

)∈M,

defined on the set:

Dom(exp) ={v ∈ TM : (1, v) ∈ Dom(F )

}.

SinceDom(F ) is open inR × TM , Dom(exp) is open inTM ; moreover, byRemark 3.2 the zero section ofTM is contained inDom(exp). In particular, foreachx ∈M , the intersection ofDom(exp) with TxM is an open neighborhood ofthe origin.

Lemma 3.5. For all t, s ∈ R, v ∈ TM , (t, sv) ∈ R × TM is in Dom(F ) if andonly if (ts, v) ∈ R× TM is in Dom(F ); moreover,F (t, sv) = sF (ts, v).

Corollary 3.6. For all s ∈ R, v ∈ TM , (s, v) ∈ R × TM is in Dom(F ) if andonly if sv is in Dom(exp); moreover,π

(F (s, v)

)= exp(sv).

Corollary 3.7. Givenx ∈ M , v ∈ TxM then the set{s ∈ R : sv ∈ Dom(exp)

}

is an open interval containing the origin; the mapγ(s) = exp(sv) defined on suchopen interval is the maximal solution ofS with γ(0) = x, γ′(0) = v.

For eachx ∈ M let us denote byexpx the restriction ofexp to Dom(exp) ∩TxM . It follows from Corollary 3.7 that the domain ofexpx is a star-shaped openneighborhood of the origin inTxM ; moreover,d expx(0) is the identity map ofTxM .

Definition 3.8. A normal neighborhoodof a pointx ∈M is an open neighborhoodV ⊂M of x such that there exists a star-shaped open neighborhoodU of the originin TxM such thatexpx |U : U → V is a diffeomorphism. An open subsetV of Mis callednormal1 if everyx ∈M has a normal neighborhood containingV .

It follows from the inverse function theorem that every point of M has a normalneighborhood. Moreover, we have the following:

Proposition 3.9. Every point ofM is contained in some normal open subset ofM .

Proof. Consider the mapφ : Dom(exp) ⊂ TM → M × M given byφ(v) =(exp(v), π(v)

). Givenx ∈ M and denote by0x ∈ TM the origin ofTxM . We

identify T0xTM with TxM ⊕ TxM , where the first summand corresponds to thetangent space of the zero section ofTM and the second summand corresponds to

1Observe that, according to this definition, a normal open subset ofM containing a pointx ∈ M

is not necessarily a normal neighborhood ofx!


the tangent space to the fiber ofTM containing0x. The differential ofφ at 0x iseasily computed as:

dφ0x(v,w) = (v + w, v), v, w ∈ TxM.

It follows from the inverse function theorem thatφ carries an open neighborhoodU of 0x in TM diffeomorphically onto an open neighborhood of(x, x) inM ×M .We can chooseU such thatU ∩ TyM is a star-shaped open neighborhood of theorigin of TyM , for all y ∈ π(U). Let V be an open neighborhood ofx in M suchthatV × V ⊂ φ(U). We claim thatV is a normal open subset ofM . Let y ∈ V befixed. ClearlyV ⊂ π(U), so thatU ∩ TyM is a star-shaped open neighborhood ofthe origin ofTyM ; thusexp(U ∩ TyM) is a normal neighborhood ofy. Moreover,givenz ∈ V then(z, y) ∈ V × V , so that there existsv ∈ U with φ(v) = (z, y);thenv ∈ U ∩ TyM and hencez ∈ exp(U ∩ TyM). �

Definition 3.10. A piecewise solutionof a sprayS is a curveγ : [a, b] → M forwhich there exists a partitiona = t0 < t1 < · · · < tk = b of [a, b] such thatγ|[ti,ti+1] is a solution ofS for all i.

3.2. The global single leaf Frobenius theorem.

Theorem 3.11(global single leaf Frobenius). Let E, M be smooth manifolds,π : E → M be a smooth submersion andD be a smooth horizontal distributiononE. Letx0 ∈ M , e0 ∈ π−1(x0) ⊂ E be given and letS be a fixed spray onM .Assume that:

(a) every piecewise solutionγ : [a, b] → M of S with γ(a) = x0 admits ahorizontal lifting γ : [a, b] → E with γ(a) = e0;

(b) if γ : [a, b] → E is the horizontal lifting of a piecewise solutionγ : [a, b] →M of S with γ(a) = e0 then the Levi form ofD vanishes at the pointγ(b) ∈ E;

(c) M is (connected and) simply-connected.

Then there exists a unique global smooth horizontal sections ofE withs(x0) = e0.

Proof. Uniqueness follows directly from Lemma 2.4. For the existence, we use theglobalization theory explained in Appendix A.

Let E′ denote the subset ofE consisting of the points of the formγ(b), whereγ(a) = e0 andγ : [a, b] → E is the horizontal lifting of some piecewise solutionγ : [a, b] → M of S with γ(a) = x0. We define a pre-sheafP over M asfollows: for each open subsetU of M , P(U) is the set of all smooth horizontalsectionss : U → E with s(U) ⊂ E′. Given open subsetsU, V ⊂ M withV ⊂ U thenPU,V is given byPU,V (s) = s|V , for all s ∈ P(U). The existenceof a global smooth horizontal section ofE is equivalent toP(M) 6= ∅. We willuse Proposition A.8. Using Theorem 2.5 (recall Remark 2.6) we get a smoothhorizontal sections : U → E defined in an open neighborhoodU of x0; it is clearby the construction ofs thats(U) ⊂ E′. Thus the pre-sheafP is nontrivial. Thelocalization property (Definition A.4) forP is trivial and the uniqueness property(Definition A.6) forP follows directly from Lemma 2.4. To conclude the proof, weshow thatP has the extension property (Definition A.6). We shall prove that every


normal open subset ofM has the extension property forP (recall Proposition 3.9).LetU be an open normal subset ofM , V be a nonempty open connected subset ofU ands ∈ P(V ) be a smooth horizontal section ofE with s(V ) ⊂ E′. Letx1 ∈ Vbe fixed. Sinces(x1) ∈ E′, there exists a piecewise solutionγ : [a, b] → M of Swith γ(a) = x0 and a horizontal liftingγ : [a, b] → E of γ with γ(a) = e0 andγ(b) = s(x1). LetW be a normal neighborhood ofx1 containingU andW0 be astar-shaped open neighborhood of the origin inTx1M such thatexpx1 : W0 →Wis a diffeomorphism. For eachx ∈ W let v ∈ W0 be such thatexpx1(v) = x; weclaim thatµx : [0, 1] ∋ t 7→ expx1(tv) ∈ M has a horizontal liftingµ : [0, 1] →E starting ats(x1) and that the Levi form ofD vanishes along the image ofµ.Namely, the concatenationγ · µ of γ with µ is a piecewise solution ofS starting atx0; by hypothesis (a),γ · µ has a horizontal lifting starting ate0. Such horizontallifting is of the form γ · µ, whereµ is a horizontal lifting ofµ starting ats(x1);moreover, hypothesis (b) implies that the Levi form ofD vanishes alongγ · µ.Observe that the image ofµ is contained inE′. We can now apply Theorem 2.5to obtain a smooth horizontal sections : W → E with s(x1) = s(x1). Thus, byLemma 2.4 and the connectedness ofV , s|V = s and hences|U ∈ P(U) is anextension ofs toU . �

Proposition 3.12. LetE, M be real-analytic manifolds,π : E → M be a real-analytic submersion andD be a real-analytic horizontal distribution onE. Assumethat:

(a) M is (connected and) simply-connected;(b) given a real analytic curveγ : I → M , t0 ∈ I ande0 ∈ π−1

(γ(t0)

)then

there exists a horizontal liftingγ : I → E of γ with γ(t0) = e0.

Then any local horizontal sections : U → E ofπ defined on a nonempty connectedopen subsetU ofM extends to a global horizontal section ofπ. In particular, ifDsatisfies the hypothesis of Theorem 2.7 at some pointe0 ofE, assumptions (a) and(b) imply thatπ admits a global horizontal section.

Proof. We use again the globalization theory explained in AppendixA. We define apre-sheafP overM as follows: for each open subsetU ofM , P(U) is the set of allsmooth horizontal sectionss : U → E; given open subsetsU, V ⊂M with V ⊂ UthenPU,V is given byPU,V (s) = s|V , for all s ∈ P(U). By Proposition A.8it suffices to show thatP has the localization property, the uniqueness propertyand the extension property. The localization property is trivial and the uniquenessproperty follows from Lemma 2.4. As to the extension property, it can be provedas follows. Letx0 ∈ M be fixed and letϕ : U → B0(r) be a real-analyticchart defined on an open neighborhoodU of x0, taking values in the open ballB0(r) ⊂ R

n of radiusr centered at the origin andϕ(x0) = 0. We will show thatV = ϕ−1

(B0(r/3)

)is an open neighborhood ofx0 having the extension property

for P. To this aim, letW be a nonempty connected open subset ofV and lets ∈ P(W ) be a local horizontal section defined onW . Choosex1 ∈ W . SetΛ = B0

(23r)

and letψ : Z ⊂ R× Λ →M be the one-parameter family of curvesdefined byψ(t, λ) = ϕ−1

(ϕ(x1)+ tλ

), whereZ is the set of pairs(t, λ) ∈ R×Λ


with ϕ(x1) + tλ ∈ B0(r). We define a local right inverse

α : ϕ−1(Bϕ(x1)(

23r)

)−→ Z ⊂ R× Λ

of ψ by settingα(x) =(1, ϕ(x) − ϕ(x1)

). By assumption (b), for eachλ ∈ Λ,

the curvet 7→ ψ(t, λ) has a horizontal liftingt 7→ ψ(t, λ) ∈ E with ψ(0, λ) =s(x1). Notice that, by the uniqueness of the horizontal lifting ofa curve, we haveψ(t, λ) = s

(ψ(t, λ)

)for smallt. Sinces is a horizontal section ofπ, its image is an

integral submanifold ofD and thus the Levi formLD vanishes along the image ofs.ThusLD vanishes at the pointψ(t, λ) for smallt; hence, sincet 7→ LD

(ψ(t, λ)

)is

real-analytic,LD must vanish along the entire curvet 7→ ψ(t, λ). By Theorem 2.5,ψ ◦ α is a horizontal section ofE with (ψ ◦ α)(x1) = s(x1); since the domainof α clearly containsV , Lemma 2.4 implies thatψ ◦ α extendss to (an openset containing)V . This proves the extension property ofP and concludes theproof. �

4. LEVI–CIVITA CONNECTIONS

4.1. Levi form of the horizontal distribution of a connection. Let π : E → Mbe a smooth vector bundle over a smooth manifoldM and let∇ be a connectiononE; for m ∈ M we denote byEm = π−1(m) the fiber ofE overm. We denotebyRm : TmM × TmM ×Em → Em thecurvature tensorof ∇ defined by:

R(X,Y )ξ = ∇X∇Y ξ −∇Y∇Xξ −∇[X,Y ]ξ,

for all smooth vector fieldsX, Y in M and every smooth sectionξ of E.Recall that there exists a unique distributionD on the manifoldE that is hori-

zontal with respect toπ and has the following property: ifγ : I →M is a smoothcurve onM then a curveγ : I → E is a horizontal lifting ofγ with respect toD if and only if γ is a∇-parallel section ofE alongγ. We callD thehorizontaldistribution of ∇. Givenm ∈ M andξ ∈ Em then the quotientTξE/Dξ can beidentified withTξ(Em) = Ker(dπξ); moreover, sinceEm is a vector space, weidentify Tξ(Em) with Em. We also identifyDξ with TmM usingdπξ. The Leviform of D at a pointξ ∈ E can thus be seen as a bilinear map:

LDξ : TmM × TmM −→ Em.

Lemma 4.1. The Levi form of the horizontal distributionD of a connection∇ isgiven by:

LDξ (v,w) = −Rm(v,w)ξ,

for all m ∈M , ξ ∈ Em.

Proof. Given a smooth vector fieldX onM we denote byXhor thehorizontal liftof X which is the unique horizontal vector field onE such thatdπξ(Xhor(ξ)

)=

X(π(ξ)

), for all ξ ∈ E. Given smooth vector fieldsX, Y onM , we have to show

that vertical component of[Xhor, Y hor] at a pointξ ∈ E is equal to−R(X,Y )ξ.Note that the horizontal component of[Xhor, Y hor] is [X,Y ]hor, sinceXhor and


Y hor areπ-related respectively withX andY . Thus, the proof will be concludedonce we show that:

α([Xhor, Y hor]− [X,Y ]hor

)= −α

(R(X,Y )ξ

),

for every smooth sectionα of the dual bundleE∗. Given one such sectionα, wedenote byfα : E → R the smooth map defined by:

fα(ξ) = α(ξ).

We claim that:Xhor(fα) = f

∇∗

Xα,

where∇∗ denotes the connection ofE∗. Namely, letγ : ]−ε, ε[ → M be anintegral curve ofX and lett 7→ ξ(t) be a parallel section ofE alongγ, so thatξ isan integral curve ofXhor; then:

Xhor(fα) =ddt

∣∣t=0

fα(ξ(t)

)= d

dt

∣∣t=0

αγ(t)(ξ(t)

)

=(∇∗

γ′(t)α)ξ(t)

∣∣t=0

= (∇∗Xα)ξ,

which proves the claim. Observe also that ifv ∈ TE is a vertical vector thenv(fα) = α(v); therefore:

(9) α([Xhor, Y hor]−[X,Y ]hor

)=

([Xhor, Y hor]−[X,Y ]hor

)(fα) = fR∗(X,Y )α,

whereR∗ denotes the curvature tensor of∇∗. A simple computation shows that:

R∗(X,Y )α = −α ◦R(X,Y ).

The conclusion follows from (9) by evaluating both sides at the pointξ. �

Corollary 4.2. Let π : E → M be a smooth vector bundle endowed with a con-nection∇, let ψ : Z ⊂ R × Λ → M be aΛ-parametric family of curves onMwith a local right inverseα : V ⊂ M → Z and let ψ : Z → E be a smoothsection ofE alongψ such thatt 7→ ψ(t, λ) is parallel for all λ ∈ Λ and such thatλ 7→ ψ(0, λ) is also parallel. If

Rψ(t,λ)(v,w)ψ(t, λ) = 0,

for all v,w ∈ Tψ(t,λ)M and all(t, λ) ∈ Z thens = ψ◦α is a parallel local sectionofE.

Proof. Follows readily from Theorem 2.5 and Lemma 4.1. �

Corollary 4.3. Let π : E → M be a smooth vector bundle endowed with a con-nection∇. Letx0 ∈M , e0 ∈ π−1(x0) ⊂ E be given and letS be a fixed spray onM . Assume that:

(a) if γ : [a, b] → M is a piecewise solution ofS with γ(a) = x0 andγ : [a, b] → E is a parallel section ofE along γ with γ(a) = e0 thenRγ(b)(v,w)γ(b) = 0, for all v,w ∈ Tγ(b)M ;

(b) M is (connected and) simply-connected.

Then there exists a unique global smooth parallel sections ofE with s(x0) = e0.

Proof. Follows directly from Lemma 4.1 and Theorem 3.11. �


Corollary 4.4. Letπ : E → M be a real-analytic vector bundle endowed with areal-analytic connection∇. Assume thatM is (connected and) simply-connected.Then any local parallel sections : U → E ofE defined on a nonempty connectedopen subsetU ofM extends to a global parallel section ofE.

Proof. It follows from Lemma 4.1 and Proposition 3.12. �

Proposition 4.5. Letπ : E → M be a real-analytic vector bundle endowed witha real-analytic connection∇. Givenx ∈M , e ∈ π−1(x), assume that:

(10) (∇kR)(v1, v2, . . . , vk+2)e = 0,

for all v1, . . . , vk+2 ∈ TxM and all k ≥ 0. Then there exists a parallel sectionsof E defined in an open neighborhood ofx in M with s(x) = e; in particular, byCorollary 4.4, ifM is (connected and) simply-connected then there exists a globalparallel sections ofE with s(x) = e.

Proof. Given a smooth vector fieldX onM , we denote byX the unique horizontalvector field onE that isπ-related withX. We show that condition (10) is equiva-lent to the condition that all iterated brackets of vector fieldsX are horizontal at thepoint e. The conclusion will then follow from Theorem 2.7. First, let us computethe bracket[X, Y ]. SinceX and Y areπ-related respectively withX andY , itfollows that the horizontal component of[X, Y ] is [X,Y ]; its vertical componentis computed in Lemma 4.1. Thus:

(11) [X, Y ]e =([X,Y ]x,−R(X,Y )e

),

where we write tangent vectors toE as pairs consisting of a horizontal componentand a vertical component. Given a smooth sectionL of the vector bundleLin(E),we denote byL the vertical vector field onE defined byL(e) =

(0, L(e)

). Given

a smooth vector fieldZ on M , let us compute the bracket[Z, L]. SinceZ isπ-related withZ and L is π-related with zero, it follows that[Z, L] is vertical.Given a smooth sectionα of E∗, we consider the mapfα : E → R defined byfα(e) = α(e) and we compute as follows:

L(fα)(e) = α(L(e)

)= fα◦L(e),

Z(fα)(e) =d

dtfα

(e(t)

)=

d

dtα(e(t)

)= (∇Zα)(e) = f∇Zα(e),

wheret 7→ e(t) is an integral curve ofZ, i.e., a parallel section ofE along anintegral curve ofZ. Then:

[Z, L](fα) = Z(L(fα)

)− L

(Z(fα)

)= f∇Z(α◦L) − f(∇Zα)◦L = fα◦∇ZL

= ∇ZL(fα),

so that:

(12) [Z, L] = ∇ZL.


Notice that (11) says that[X, Y ] is given by:

[X, Y ] = [X,Y ]− L,

whereL(e) = R(X,Y )e. Using the equality above and (12) it can be easily provedby induction that:

[Z1, [Z2, . . . [Zk, [X, Y ]] · · · ]] = [Z1, [Z2, . . . [Zk, [X,Y ]] · · · ]]− Lk,

where:Lk(e) = (∇Z1

(∇Z2(· · · ∇Zk

(R(X,Y )) · · · )))e.

The conclusion follows by observing thatLk(e) can be written in the form:

Lk(e) = (∇kR)(Z1, . . . , Zk,X, Y )e+

k−1∑

i=0

Lki,

whereLki is a term linear in(∇iR)(· · ·)e. �

4.2. Connections arising from metric tensors. Let π : E → M be a vectorbundle and letE∗ ⊗E∗ denote the vector bundle overM whose fiber atm ∈M isthe space of bilinear forms onEm. If ∇ is a connection onE then we can define ainduced connection∇bil onE∗ ⊗ E∗ by setting:

(∇bilX g)(ξ, η) = X

(g(ξ, η)

)− g(∇Xξ, η) − g(ξ,∇Xη),

whereX is a smooth vector field onM and ξ, η are smooth sections ofE. Astraightforward computation shows that the curvature tensor Rbil of ∇bil is givenby:

(13)(Rbil(X,Y )g

)(ξ, η) = −g

(R(X,Y )ξ, η

)− g

(ξ,R(X,Y )η

),

for any smooth vector fieldsX, Y onM , any smooth sectionsξ, η of E and anysmooth sectiong of E∗ ⊗ E∗. If γ : I → M is a smooth curve defined on anintervalI around0 and if g0 is a bilinear form onEγ(0) then the parallel transportI ∋ t 7→ gt of g0 alongγ relatively to the connection∇bil is given by:

gt(ξ, η) = g0(P−1t ξ, P−1

t η), ξ, η ∈ Eγ(t),

wherePt : Eγ(0) → Eγ(t) denotes the parallel transport alongγ.Given a smooth manifoldM then asemi-Riemannian metriconM is a smooth

sectiong of the vector bundleTM∗ ⊗ TM∗ such thatgm : TmM × TmM → R

is symmetric and nondegenerate; ifgm is positive definite for allm ∈ M , we callg a Riemannian metric. TheLevi-Civita connectionof g is the unique symmetricconnection∇ onTM such that∇bilg = 0.

We consider the following problem:given a symmetric connection∇ on asmooth manifoldM , when does there exist a semi-Riemannian metricg on Msuch that∇ is the Levi-Civita connection ofg?

Note that if∇ is the Levi-Civita connection of a semi-Riemannian metricg thenfor anym ∈ M and anyv,w ∈ TmM , the linear operatorRm(v,w) : TmM →TmM corresponding to the curvature tensor of∇ is anti-symmetric with respectto gm; moreover, given a smooth curveγ : [a, b] → M with γ(a) = m0 and


γ(b) = m then, denoting byP : Tm0M → TmM the parallel transport alongγ,

the linear operator:

P−1[Rm(v,w)

]P : Tm0

M −→ Tm0M

is anti-symmetric with respect togm0, for all v,w ∈ TmM . We will show below

that this anti-symmetry characterizes the connections arising from semi-Riemann-ian metrics.

Proposition 4.6. LetM be a smooth manifold,∇ be a symmetric connection onTM ,m0 ∈M andg0 be a nondegenerate symmetric bilinear form onTm0

M . Letψ : Z ⊂ R×Λ →M be aΛ-parametric family of curves onM with a local rightinverseα : V ⊂ M → Z; assume thatψ(0, λ) = m0, for all λ ∈ M . For each(t, λ) ∈ Z, we denote byP(t,λ) : Tm0

M → Tψ(t,λ)M the parallel transport alongt 7→ ψ(t, λ). Assume that for all(t, λ) ∈ Z the linear operator:

(14) P−1(t,λ)

[Rψ(t,λ)(v,w)

]P(t,λ) : Tm0

M −→ Tm0M

is anti-symmetric with respect tog0, for all v,w ∈ Tψ(t,λ)M , where

Rψ(t,λ)(v,w) : Tψ(t,λ)M −→ Tψ(t,λ)M

denotes the linear operator corresponding to the curvaturetensor of∇. Then∇ isthe Levi-Civita connection of the semi-Riemannian metricg onV ⊂M defined bysetting:

gm(·, ·) = g0(P−1α(m)·, P

−1α(m)·),

for all m ∈ V .

Proof. For each(t, λ) ∈ Z, let ψ(t, λ) ∈ TM∗ ⊗ TM∗ be the bilinear form onTψ(t,λ)M defined by:

ψ(t, λ)(·, ·) = g0(P−1(t,λ)·, P

−1(t,λ)·).

Thenψ satisfies the hypotheses of Corollary 4.2 withE = TM∗ ⊗ TM∗; namely,ψ(0, λ) = g0, for all λ ∈ Λ and by (13) and the anti-symmetry of (14), we haveRbilψ(t,λ)(v,w) = 0, for all v,w ∈ Tφ(t,λ)M . Henceg = ψ ◦α : V → TM∗⊗TM∗

is a parallel section ofTM∗⊗TM∗ and∇ is the Levi-Civita connection ofg. �

Theorem 4.7.LetM be a smooth manifold,∇ be a symmetric connection onTM ,m0 ∈ M andg0 be a nondegenerate symmetric bilinear form onTm0

M . LetS bea fixed spray onM . Assume that:

• for every piecewise solutionγ : [a, b] → M of S with γ(a) = m0 thelinear operatorP−1

γ Rγ(b)Pγ on Tm0M is g0-anti-symmetric, wherePγ :

Tm0M → Tγ(b)M denotes parallel transport alongγ;

• M is (connected and) simply-connected.

Theng0 extends to a Riemannian metric onM for which ∇ is the Levi-Civitaconnection.

Proof. It follows from (13) and Corollary 4.3. �


Proposition 4.8. LetM be a (connected and) simply-connected real-analytic man-ifold and let∇ be a real-analytic symmetric connection onTM . If there exists asemi-Riemannian metricg on a nonempty open connected subset ofM having∇as its Levi-Civita connection theng extends to a globally defined semi-Riemannianmetric onM having∇ as its Levi-Civita connection.

Proof. It follows from Corollary 4.4. �

Proposition 4.9. LetM be a real-analytic manifold and let∇ be a real-analyticsymmetric connection onTM . Given a pointx0 ∈ M and a nondegenerate sym-metric bilinear formg0 onTx0M , if:

(∇kR)(v1, . . . , vk+2) : TxM → TxM

is g0-anti-symmetric for allv1, . . . , vk+2 ∈ Tx0M and all k ≥ 0 theng0 extendsto a semi-Riemannian metric on an open neighborhood ofx0 whose Levi-Civitaconnection is∇. Moreover, ifM is (connected and) simply-connected theng0extends to a global semi-Riemannian metric onM having∇ as its Levi-Civitaconnection.

Proof. Follows easily from Proposition 4.5 and from formula (13). �

The above characterizations of Levi–Civita connections have been used in [3],where the authors study left-invariant (symmetric) connections in Lie groups.

5. AFFINE MAPS

Let us now discuss as an application of the “single leaf Frobenius Theorem” aclassical result in differential geometry.

5.1. The Cartan–Ambrose–Hicks Theorem. Consider the following setup. LetM ,N be smooth manifolds endowed respectively with connections∇M and∇N .We denote byTM , TN (resp.,RM , RN ) respectively the torsion tensors (resp.,curvature tensors) of∇M and∇N . A smooth mapf : M → N is calledaffine iffor everyx ∈M , v ∈ TxM and every smooth vector fieldX onM we have:

dfx(∇Mv X) = ∇N

v (df ◦X);

in the formula abovedf ◦X : M → TN is regarded as a vector field alongf onN , so that it makes sense to compute its covariant derivative∇N alongv ∈ TM .

Let x0 ∈ M , y0 ∈ N be given and letσ0 : Tx0M → Ty0N be a linear map.Given a geodesicγ : [a, b] →M with γ(a) = x0 then the geodesicµ : [a, b] → Nwith µ(a) = y0 andµ′(a) = σ

(γ′(a)

)is called inducedon N by the geodesic

γ and byσ0. We observe that the geodesicµ : [a, b] → N is well-defined onlyif (b − a)σ

(γ′(a)

)is in the domain of the exponential map ofN at the point

y0. Let σ : Tγ(b)M → Tµ(b)N be the linear map given by the composition ofparallel transport alongγ, σ0 and parallel transport alongµ; we callσ the linearmapinducedby γ andσ0.


Theorem 5.1. Let x0 ∈ M , y0 ∈ N be given and letσ0 : Tx0M → Ty0Nbe a linear map. LetU be an open subset ofTx0M which is star-shaped at theorigin and which is carried diffeomorphically onto an open subsetV ofM by theexponential map ofM at x0. Assume thatσ(U) is contained in the domain ofthe exponential map ofN at y0. For eachx ∈ V , let γx : [0, 1] → M be theunique geodesic such thatγ′x(0) ∈ U and γx(1) = x; let µx : [0, 1] → N andσx : TxM → Tµx(1)N be respectively the geodesic and the linear map induced byγx andσ0. Assume that for allx ∈ V the linear mapσx relatesTM with TN andRM withRN , i.e.:

σx(TM (·, ·)

)= TN

(σx(·), σx(·)

), σx

(RM (·, ·) ·

)= RN

(σx(·), σx(·)

)σx(·).

Then the smooth mapf : V → N defined byf(x) = µx(1) is affine anddf(x) =σx for all x ∈ V ; in particular, f(x0) = y0 anddf(x0) = σ0.

Remark5.2. In the statement of Theorem 5.1, if one assumes thatσ0 is an isomor-phism (resp., injective) then it follows thatf is a local diffeomorphism (resp., thatf is an immersion). Moreover, if∇M and∇N are the Levi-Civita connections ofRiemannian metrics onM andN respectively then, if one assumes thatσ0 is anisometry, it follows thatf is a local isometry.

In what follows we assume that∇N is geodesically complete, i.e., for ally ∈ Nthe exponential map ofN at y is defined on the whole tangent spaceTyN .

Let x0 ∈M , y0 ∈ N be given and letσ0 : Tx0M → Ty0N be a linear map. Letγ : [a, b] →M be apiecewise geodesicwith γ(a) = x0, i.e., there exists a partitiona = t0 < t1 < · · · < tk = b of [a, b] such thatγ|[ti,ti+1] is a geodesic for alli.Using the linear mapσ0 it is possible to define a piecewise geodesicµ : [a, b] → Nand a linear mapσ : Tγ(b)M → Tµ(b)N inducedby γ in the following way: wefirst define inductively a sequence of geodesicsµi : [ti, ti+1] → N and of linearmapsσi : Tγ(ti)M → Tµi(ti)N . Let µ0 andσ1 be respectively the geodesic andthe linear map induced by the geodesicγ|[t0,t1] and byσ0. Assuming thatµi andσi+1 are defined we letµi+1 andσi+2 be respectively the geodesic and the linearmap induced by the geodesicγ|[ti,ti+1] and byσi+1. Finally, we letµ : [a, b] → Nbe the piecewise geodesic such thatµ|[ti,ti+1] = µi for all i and we letσ = σk.

Theorem 5.3(Cartan–Ambrose–Hicks). LetM ,N be smooth manifolds endowedrespectively with connections∇M and∇N ; assume that∇N is geodesically com-plete and thatM is connected and simply-connected. Letx0 ∈ M , y0 ∈ N begiven and letσ0 : Tx0M → Ty0N be a linear map. For each piecewise ge-odesicγ : [a, b] → M with γ(a) = x0 denote byµγ : [a, b] → N and byσγ : Tγ(b)M → Tµγ(b)N respectively the piecewise geodesic and the linear mapinduced by the piecewise geodesicγ and byσ0. Assume that for every piecewisegeodesicγ the linear mapσγ relatesTM with TN andRM with RN . Then thereexists a smooth affine mapf : M → N such that for every piecewise geodesicγ : [a, b] →M we havef ◦γ = µγ anddf

(γ(b)

)= σγ ; in particular, f(x0) = y0

anddf(x0) = σ0.


Remark5.4. In the statement of the Cartan–Ambrose–Hicks Theorem, if one as-sumes in addition thatσ0 is an isomorphism, and that∇M is geodesically completethen it follows that the affine mapf :M → N is a covering map.

Corollary 5.5. Let (M,gM ), (N, gN ) be Riemannian manifolds with(N, gN )complete andM connected and simply-connected. Letx0 ∈ M , y0 ∈ N be givenand letσ0 : Tx0M → Ty0N be a linear isometry onto a subspace ofTy0N . Foreach piecewise geodesicγ : [a, b] →M with γ(a) = x0 denote byµγ : [a, b] → Nand byσγ : Tγ(b)M → Tµγ(b)N respectively the piecewise geodesic and the linearmap induced by the piecewise geodesicγ and byσ0. Assume that for every piece-wise geodesicγ the linear mapσγ relatesRM withRN . Then there exists a totallygeodesic isometric immersionf :M → N with f(x0) = y0 anddf(x0) = σ0.

Proof. It follows immediately from Theorem 5.3; observe that the condition thatfis totally geodesic follows from the fact thatf is affine. �

We now show how the proof of Theorems 5.1 and 5.3 can be obtained as an ap-plication of the local and the global version of the “single leaf Frobenius Theorem”(Theorems 2.5 and 3.11).

Consider the vector bundleE = Lin(TM,TN) overM × N whose fiber at apoint (x, y) ∈M ×N is the space of linear mapsLin(TxM,TyN). Notice thatEcoincides with the tensor bundleπ∗1(TM

∗)⊗π∗2(TN), whereπ1 andπ2 denote theprojections of the productM ×N . The connections∇M and∇N naturally inducea connection∇ onE given by:

(15) (∇(v,w)σ)(X) = ∇N(v,w)

(σ(X)

)− σ(∇M

v X),

wherev ∈ TM , w ∈ TN ,X is a smooth vector field onM andσ :M ×N → Eis a smooth section ofE. In the formula above,σ(X) :M ×N → TN is regardedas vector field along the projectionπ2 :M ×N → N onN .

Given a smooth mapf : U → N defined on an open subsetU of M thenthe differentialdf : U → E can be regarded as section ofE along the mapU ∋ x 7→

(x, f(x)

)∈ M × N , so that it makes sense to consider the covariant

derivative ofdf with respect to the connection∇.

Lemma 5.6. A smooth mapf : U → N defined on an open subset ofM is affineif and only ifdf is parallel with respect to∇.

Proof. Givenv ∈ TM and a smooth vector fieldX onU we compute:(∇v(df)

)(X) = ∇N

v

(df(X)

)− df(∇M

v X).

The conclusion follows. �

Lemma 5.7. Let λ : t 7→(γ(t), µ(t), σ(t)

)be a smooth curve onE, i.e.,γ is a

curve onM , µ is a curve onN andσ(t) is a linear map fromTγ(t)M to Tµ(t)Nfor all t. Thenλ is parallel with respect to∇ (or, equivalently,λ is tangent to thehorizontal distribution corresponding to∇) if and only if the following conditionholds: for every∇M -parallel vector fieldt 7→ v(t) ∈ TM alongγ, the vector fieldt 7→ σ(t)v(t) ∈ TN alongµ is ∇N -parallel.


Proof. Let t 7→ v(t) be a vector field alongγ. Let us denote byDdt ,DM

dt and DN

dtrespectively the covariant derivatives with respect to theparametert correspond-ing to the connections∇, ∇M and∇N . The conclusion follows easily from thefollowing formula:

DN

dt [σ(t)v(t)] =(

Ddt σ(t)

)v(t) + σ(t)D

M

dt v(t),

observing thatλ is∇-parallel if and only ifDdtσ(t) = 0. �

The geometric interpretation of Lemma 5.7 is given by the following:

Corollary 5.8. Letλ be as in the statement of Lemma 5.7 and lett0 in the domainof λ be fixed. Thenλ is parallel with respect to∇ if and only if the followingcondition holds: for allt, the linear mapσ(t) : Tγ(t)M → Tµ(t)N is given by thecomposition of∇M -parallel transport alongγ, σ(t0) and∇N -parallel transportalongµ. �

We now explain in which form the “single leaf Frobenius Theorem” (Theo-rem 2.5) is going to be applied. We consider the smooth submersionπ : E → Mgiven by the composition of the canonical projectionE → M × N with the firstprojectionπ1 : M ×N → M . Givenx ∈ M , y ∈ N , σ ∈ Lin(TxM,TyN) thenthe tangent spaceTσE is identified with the direct sum ofTxM ⊕ TyN (the hori-zontal space corresponding to the connection∇) andLin(TxM,TyN) (the tangentspace to the fiber). We will now define a distributionD on the manifoldE that ishorizontal with respect to the submersionπ : E →M . We set:

(16) Dσ = Gr(σ)⊕ {0} ⊂ TxM ⊕ TyN ⊕ Lin(TxM,TyN) ∼= TσE,

whereGr(σ) ⊂ TxM ⊕ TyN denotes the graph of the linear mapσ.

Lemma 5.9. Let s : U → E be a smooth section ofE defined on an open subsetU ofM ; we writes(x) =

(f(x), σ(x)

), wheref : U → N is a smooth map and

σ(x) ∈ Lin(TxM,Tf(x)N), for all x ∈ U . Thens is D-horizontal if and only ifσ(x) = df(x) for all x ∈ U andf is affine.

Proof. Givenx ∈ U , v ∈ TxM then the component ofdsx(v) in TxM ⊕ Tf(x)N

is equal to(v,dfx(v)

). Thus,s is D-horizontal if and only ifσ is ∇-parallel and

σ(x) = df(x), for all x ∈ U . The conclusion follows from Lemma 5.6. �

Lemma 5.10. Letλ be as in the statement of Lemma 5.7 and lett0 in the domainof λ be fixed. Assume thatγ is a geodesic onM . Thenλ is D-horizontal if andonly if the following conditions hold:

• µ is a geodesic onN ;• µ′(t0) = σ(t0)γ

′(t0);• for all t, the linear mapσ(t) : Tγ(t)M → Tµ(t)N is given by the compo-

sition of∇M -parallel transport alongγ, σ(t0) and∇N -parallel transportalongµ.


Proof. Clearlyλ is D-horizontal if and only ifλ is parallel with respect to∇ andµ′(t) = σ(t)γ′(t), for all t. The conclusion follows from Lemma 5.7 and Corol-lary 5.8. �

Corollary 5.11. Let x0 ∈ M , y0 ∈ N be fixed and letσ0 : Tx0M → Ty0Nbe a linear map. Letγ : [a, b] → M be a piecewise geodesic withγ(a) = x0.Thenλ : [a, b] ∋ t 7→

(γ(t), µ(t), σ(t)

)∈ E is the horizontal lift ofγ with

λ(a) = (x0, y0, σ0) if and only ifµ : [a, b] → N is the piecewise geodesic inducedbyγ andσ0 andσ(t) is the linear map induced byγ|[a,t] andσ0, for all t. �

Lemma 5.12. The curvature tensorRE of the connection∇ ofE is given by:

RE(x,y)((v1, w1), (v2, w2)

)σ = RNy (w1, w2) ◦ σ − σ ◦RMx (v1, v2),

for all (x, y) ∈M ×N , v1, v2 ∈ TxM ,w1, w2 ∈ TyN , σ ∈ Lin(TxM,TyN). �

Lemma 5.13.LetP ,Q be smooth manifolds,∇ a connection onQ andh : P → Qbe a smooth map. Given smooth vector fieldsX, Y in P then:

∇X

(dh(Y )

)−∇Y

(dh(X)

)− dh

([X,Y ]

)= T

(dh(X),dh(Y )

),

whereT denotes the torsion of∇.

Proof. It is a standard computation in calculus with connections (see Proposi-tion B.8). �

We will now compute the Levi form of the distributionD. Given x ∈ M ,y ∈ N , σ ∈ Lin(TxM,TyN), the Levi form ofD at the pointσ ∈ E is a bilinearmapLD

σ : Dσ × Dσ → TσE/Dσ . We identify the spaceDσ with TxM by theisomorphism:

TxM ∋ v 7−→(v, σ(v), 0

)∈ Dσ ⊂ TxM ⊕ TyN ⊕ Lin(TxM,TyN) ∼= TσE.

Moreover, the surjective linear map:

(17) TσE ∋ (v,w, τ) 7−→ (w − σ(v), τ) ∈ TyN ⊕ Lin(TxM,TyN)

has kernelDσ and thus induces an isomorphism from the spaceTσE/Dσ ontoTyN ⊕ Lin(TxM,TyN). Hence, the Levi form ofD atσ will be identified with abilinear map:

LDσ : TxM × TxM −→ TyN ⊕ Lin(TxM,TyN).

We now computeLDσ .

Lemma 5.14. Givenx ∈ M , y ∈ N , σ ∈ Lin(TxM,TyN), the Levi form ofD atthe pointσ ∈ E is given by:

LDσ (v1, v2) =

(σ(TM (v1, v2)

)− TN

(σ(v1), σ(v2)

),

σ ◦RMx (v1, v2)−RNy(σ(v1), σ(v2)

)◦ σ

),

for all v1, v2 ∈ TxM .


Proof. Given a smooth vector fieldX onM , we define a smooth vector fieldX onE by setting:(18)X(x, y, σ) =

(X(x), σ(X(x)), 0

)∈ TxM ⊕ TyN ⊕ Lin(TxM,TyN) ∼= TσE,

for all x ∈M , y ∈ N , σ ∈ Lin(TxM,TyN). Observe thatX isD-horizontal.Let x ∈ M , y ∈ N , σ ∈ Lin(TxM,TyN), v1, v2 ∈ TxM be fixed. Choose

smooth vector fieldsX1, X2 onM with X1(x) = v1, X2(x) = v2. In order tocompute the Levi form ofD at the pointσ it suffices to compute the Lie bracket[X1, X2] at the pointσ. The vector[X1, X2]σ is identified with an element ofTxM ⊕ TyN ⊕ Lin(TxM,TyN). The component inLin(TxM,TyN) of suchvector can be computed using Lemma 4.1, sinceX1 and X2 are both horizon-tal with respect to the connection∇ of E; thus, the component of[X1, X2]σ inLin(TxM,TyN) is equal to−RE

((v1, σ(v1)), (v2, σ(v2))

)σ. Let us now compute

the component of[X1, X2]σ in TxM⊕TyN ; this is justdπσ([X1, X2]σ

). Consider

the connection∇M×N onM ×N induced from∇M and∇N ; its torsionTM×N

is given by:

TM×N((v1, w1), (v2, w2)

)=

(TM (v1, v2), T

N (w1, w2)).

We now computedπσ([X1, X2]σ

)using Lemma 5.13 withP = E, Q = M ×N

andh = π. We get:

(19) ∇M×N

X1

(dπ(X2)

)−∇M×N

X2

(dπ(X1)

)− dπ

([X1, X2]

)

=(TM (X1,X2), T

N (σ(X1), σ(X2))).

We compute∇M×N

X1

(dπ(X2)

)as follows:

∇M×N

X1

(dπ(X2)

)= DM×N

dt dπ(X2(λ(t))

),

whereλ : ]−ε, ε[ → E is an integral curve ofX1 with λ(0) = σ. Thusλ(t) =(x(t), y(t), σ(t)

), wheret 7→ x(t) ∈ M is an integral curve ofX1, y′(t) =

σ(t)x′(t) andt 7→ σ(t) is∇-parallel. Hence:

DM×N

dt dπ(X2(λ(t))

)= DM×N

dt

(X2(x(t)), σ(t)X2(x(t))

)

=(DM

dt X2(x(t)),DN

dt [σ(t)X2(x(t))])

σ parallel=

(DM

dt X2(x(t)), σ(t)DM

dt X2(x(t))).

Evaluating att = 0 we obtain:

(20) ∇M×N

X1

(dπ(X2)

)= DM×N

dt dπ(X2(λ(t))

)∣∣∣t=0

=(∇MX1X2, σ(∇

MX1X2)

),

where the righthand side of (20) is evaluated at the pointx. Similarly:

(21) ∇M×N

X2

(dπ(X1)

)=

(∇MX2X1, σ(∇

MX2X1)

).


Using (19), (20) and (21) we get:

dπσ([X1, X2]σ

)

=([X1,X2]x, σ([X1,X2]x) + σ

(TM(X1,X2)

)− TN

(σ(X1), σ(X2)

)).

Hence, recalling Lemma 5.12:

(22)

[X1, X2]σ =([X1,X2]x, σ([X1,X2]x)+σ

(TM (X1,X2)

)−TN

(σ(X1), σ(X2)

),

σ ◦RMx (X1,X2)−RNy (σ(X1), σ(X2)) ◦ σ).

The conclusion follows recalling formula (17) that gives the identification betweenTσE/Dσ andTyN ⊕ Lin(TxM,TyN). �

Corollary 5.15. Givenx ∈ M , y ∈ N , σ ∈ Lin(TxM,TyN), then the Levi formof D at the pointσ ∈ E vanishes if and only if the linear mapσ : TxM → TyNrelatesTM with TN andRM withRN .

Proof. It follows from Lemma 5.14. �

Proof of Theorems 5.1 and 5.3.It follows from Theorems 2.5 and 3.11, keeping inmind Examples 2.2 and 3.1, Lemmas 5.9 and 5.10, and Corollary5.15. �

5.2. Higher order Cartan–Ambrose–Hicks theorem. Given a tensor fieldτ ona manifold endowed with a connection∇, we denote by∇(r)τ its r-th covariantderivative, forr ≥ 1; we set∇(0)τ = τ .

Theorem 5.16. LetM , N be real-analytic manifolds endowed with real-analyticconnections∇M and∇N , respectively. Letx0 ∈ M , y0 ∈ N be given and letσ0 : Tx0M → Ty0N be a linear map. If for allr ≥ 0 the linear mapσ0 relates∇(r)TMx0 with∇(r)TNy0 and∇(r)RMx0 with∇(r)RNy0 then there exists a real-analyticaffine mapf : U → N defined on an open neighborhoodU of x0 in M satisfyingf(x0) = y0 anddf(x0) = σ0.

Proof. We will apply Theorem 2.7 to the distributionD onE defined in (16). Asbefore, forx ∈M , y ∈ N , σ ∈ Lin(TxM,TyN), we use the identification:

TσE ∼= TxM ⊕ TyN ⊕ Lin(TxM,TyN).

Given a smooth vector fieldX onM , we define aD-horizontal vector fieldX onEas in (18). Recall that forX1,X2 ∈ Γ(TM), the bracket[X1, X2] was computedin the proof of Lemma 5.14 (see (22)). By Remark 2.8, the thesis will follow oncewe show that the iterated brackets:

(23) [Xr+1, . . . , X1]def= [Xr+1, [Xr, . . . , [X2, X1] · · · ]],


evaluated atσ0 ∈ E are inDσ0 , for all X1, . . . ,Xr ∈ Γ(TM) and allr ≥ 1. Forr ≥ 0, x ∈M , y ∈ N , σ ∈ Lin(TxM,TyN) we set:

T(r)σ (X1, . . . ,Xr+2) = σ

(∇(r)TM (X1, . . . ,Xr+2)

)

−∇(r)TN(σ(X1), . . . , σ(Xr+2)

)∈ TyN,

R(r)σ (X1, . . . ,Xr+2) = σ ◦ ∇(r)RM(X1, . . . ,Xr+2)

−∇(r)RN(σ(X1), . . . , σ(Xr+2)

)◦ σ ∈ Lin(TxM,TyN),

for all X1, . . . ,Xr+2 ∈ TxM . The hypotheses of the theorem say thatT(r)σ0 and

R(r)σ0 vanish for allr ≥ 0. Observe thatT(r) andR(r) are, respectively, sections

along the mapπ : E →M ×N of the vector bundles overM ×N given by:(⊗

r+2

(π∗1TM)∗)⊗ π∗2TN and

(⊗

r+2

(π∗1TM)∗)⊗ (π∗1TM)∗ ⊗ π∗2TN,

whereπ1 andπ2 denote the projections of the productM ×N .Our plan is to show that the iterated bracket (23) can be written in the form:

(24) [Xr+1, . . . , X1] =(0,T(r−1)(Xr+1, . . . ,X1),R

(r−1)(Xr+1, . . . ,X1))

+(0,L(r)(T(0),R(0), . . . ,T(r−2),R(r−2))

)+ terms inΓ(r−1)(D),

for all r ≥ 1, whereL(r) is a section along the mapπ : E →M ×N of the vectorbundle overM ×N given by:

Lin

( r−2⊕

i=0

[(⊗

i+2

(π∗1TM)∗)⊗π∗2TN⊕

(⊗

i+2

(π∗1TM)∗)⊗(π∗1TM)∗⊗π∗2TN

],

π∗2TN ⊕ (π∗1TM)∗ ⊗ π∗2TN

).

Once formula (24) is proven, the conclusion follows easily by induction onr. Wewill now conclude the proof by showing formula (24) by induction onr. Forr = 1,we have (recall (22)):

[X2, X1] =(0, σ

(TM (X2,X1)

)− TN

(σ(X2), σ(X1)

),

σ ◦RM (X2,X1)−RN (σ(X2), σ(X1)) ◦ σ)+ terms inΓ0(D)

=(0,T(0)(X2,X1),R

(0)(X2,X1))+ terms inΓ0(D),

proving the base of the induction. The induction step can be proven by applyingad

Xr+2to both sides of (24), keeping in mind Lemma 5.17 below and thefollowing

formulas:

(∇horT(i))σ

(Z, σ(Z)

)= T(i+1)(Z, · · · ),

(∇horR(i))σ

(Z, σ(Z)

)= R(i+1)(Z, · · · ),

whereZ is a vector field onM . This concludes the proof. �


Lemma 5.17. LetA be a section along the mapπ2 ◦ π : E → N of the tangentbundle ofN and letB be a section along the mapπ : E → M ×N of the vectorbundleE, so that(0, A,B) is a vector field on the manifoldE. LetZ be a vectorfield onM . Then:

[Z, (0, A,B)]σ =(0,∇N

horA(Z, σ(Z)

)−B(Z)− TN

(σ(Z), A

),

∇horB(Z, σ(Z)

)−RN

(σ(Z), A

)◦ σ

), σ ∈ E,

where∇NhorA (resp.,∇horB) denotes the restriction of∇NA (resp., of∇B) to the

horizontal subbundle ofTE determined by the connection ofE.

Proof. We compute the horizontal component of[Z, (0, A,B)] using Lemma 5.13with P = E,Q =M ×N andh = π. We have:

dπσ[(Z, σ(Z), 0), (0, A,B)] = ∇M×N(Z,σ(Z),0)(0, A)−∇M×N

(0,A,B)

(Z, σ(Z)

)

− TM×N((Z, σ(Z)), (0, A)

).

Clearly:

TM×N((Z, σ(Z)), (0, A)

)=

(TM (Z, 0), TN (σ(Z), A)

)=

(0, TN (σ(Z), A)

)

and:

∇M×N(Z,σ(Z),0)(0, A) =

(0,∇N

horA(Z, σ(Z))).

Let t 7→(x(t), y(t), σ(t)

)be an integral curve of(0, A,B), i.e., t 7→ x(t) is

constant,y′ = A and Ddtσ = B. We compute:

∇M×N(0,A,B)

(Z, σ(Z)

)= DM×N

dt

(Zx(t), σ(t)Zx(t)

)=

(0, B(Z)

).

Let us now compute the vertical component of[Z, (0, A, 0)]. Since bothZ and(0, A, 0) are in the horizontal subbundle ofTE determined by the connection ofE, the vertical component of[Z, (0, A, 0)] can be directly computed using Lem-mas 4.1 and 5.12, as follows:

vertical component of[Z, (0, A, 0)]σ = −RE((Z, σ(Z)), (0, A)

)σ

= σ ◦RM (Z, 0) −RN(σ(Z), A

)◦ σ = −RN

(σ(Z), A

)◦ σ.

Finally, we compute the vertical component of[Z, (0, 0, B)]. LetW be a vectorfield onM andα be a1-form onN ; we define a mapfW,α : E → R by setting:

fW,α(σ) = α(σ(W )

).

Let x ∈ M , y ∈ N , σ ∈ Lin(TxM,TyN) be fixed and assume that∇MW (x) =

0, ∇Nα(y) = 0, so thatdfW,α(σ) annihilates the horizontal subspace ofTσE


determined by the connection ofE. We compute:

(0, 0, B)(fW,α) = α(B(W )

),

Z(fW,α) = (∇σ(Z)α)(σ(W )

)+ α

(σ(∇ZW )

),

Z((0, 0, B)(fW,α)

)σ= (∇σ(Z)α)x

(B(W )

)+ α

(∇horBσ(Z, σ(Z))(W )

)

+ α(B(∇ZxW )

)= α


),

(0, 0, B)(Z(fW,α)

)σ= (∇B(Z)α)x

(σ(W )

)+ (∇σ(Z)α)x

(B(W )

)

+ α(B(∇ZxW )

)= 0,

so that:[Z, (0, 0, B)](fW,α) = α


).

Hence, the vertical component of[Z, (0, 0, B)]σ is equal to∇horBσ(Z, σ(Z)

).

This concludes the proof. �

Proposition 5.18.LetM ,N be real-analytic manifolds endowed with real-analyticconnections∇M and ∇N , respectively. Assume that∇N is geodesically com-plete and thatM is (connected and) simply-connected. Then every affine mapf : U → N defined on a nonempty connected open subsetU of M extends toan affine map fromM to N . In particular, if in addition x0 ∈ M , y0 ∈ N ,σ0 ∈ Lin(Tx0M,Ty0N) satisfy the hypotheses of Theorem 5.16 then there existsan affine mapf :M → N with f(x0) = y0 anddf(x0) = σ0.

Proof. If D is the distribution onE defined in (16) then, by Lemma 5.9,s(x) =(f(x),df(x)

)is aD-horizontal section ofπ : E →M defined inU . The geodesi-

cal completeness of∇N guarantees that hypothesis (b) of Proposition 3.12 is sat-isfied; hence, such proposition gives a global horizontal section ofπ. �

An affine symmetryaround a pointx0 ∈M is an affine mapf : U →M definedin an open neighborhoodU of x0 with f(x0) = x0 anddf(x0) = −Id.

Corollary 5.19. LetM be a real-analytic manifold endowed with a real-analyticconnection∇. Letx0 ∈ M be fixed. Then there exists an affine symmetry aroundx0 if and only if:

(25) ∇(2r)Tx0 = 0, and ∇(2r+1)Rx0 = 0, for all r ≥ 0.

Moreover, ifM is (connected and) simply-connected and complete then condition(25) is equivalent to the existence of a globally defined affine symmetryf : M →M aroundx0.

Proof. Apply Theorem 5.16 withM = N , y0 = x0 andσ0 = −Id. For the globalresult apply Proposition 5.18. �

APPENDIX A. A GLOBALIZATION PRINCIPLE

Definition A.1. Let X, X be topological spaces andπ : X → X be a map. Anopen subsetU ⊂ X is called afundamental open subsetof X if π−1(U) equalsa disjoint union

⋃i∈I Ui of open subsetsUi of X such thatπ|Ui

: Ui → U is


a homeomorphism for alli ∈ I. We say thatπ is a covering mapif X can becovered by fundamental open subsets.

Obviously every covering map is a local homeomorphism.Given a local homeomorphismπ : X → X then by alocal sectionof π we mean

a continuous maps : U → X defined on an open subset ofX with π ◦ s = IdU .

Lemma A.2. LetX, X be topological spaces andπ : X → X be a local home-omorphism. Assume thatX is Hausdorff. LetU be a connected open subset ofXsatisfying the following property:

(∗) for everyx ∈ U and everyx ∈ X with π(x) = x there exists a localsections : U → X of π with s(x) = x.

ThenU is a fundamental open subset ofX.

Proof. Let S be the set of all local sections ofπ defined inU . We claim that:

π−1(U) =⋃

s∈S

s(U).

Indeed, ifs ∈ S then obviouslys(U) ⊂ π−1(U); moreover, givenx ∈ π−1(U)thenx = π(x) ∈ U and by property (∗) there existss ∈ S with s(x) = x. Thusx ∈ s(U). This proves the claim. Now observe thats(U) is open inX for alls ∈ S; moreover,π|s(U) : s(U) → U is a homeomorphism, being the inverseof s : U → s(U). To complete the proof, we show that the union

⋃s∈S s(U) is

disjoint. Picks, s′ ∈ S with s(U) ∩ s′(U) 6= ∅. Then there existsx, y ∈ U withs(x) = s′(y). Observe that:

x = π(s(x)

)= π

(s′(y)

)= y,

and thuss(x) = s′(x). SinceU is connected andX is Hausdorff it follows thats = s′. �

Corollary A.3. Let X, X be topological spaces andπ : X → X be a localhomeomorphism. Assume thatX is Hausdorff and thatX is locally connected.If every point ofX has an open neighborhood satisfying property (∗) thenπ is acovering map. �

LetX be a topological space. Apre-sheafonX is a mapP that assigns to eachopen subsetU ⊂ X a setP(U) and to each pair of open subsetsU, V ⊂ X withV ⊂ U a mapPU,V : P(U) → P(V ) such that the following properties hold:

• for every open subsetU ⊂ X the mapPU,U is the identity map of the setP(U);

• given open sets,U, V,W ⊂ X with W ⊂ V ⊂ U then:

PV,W ◦PU,V = PU,W .

We say that the pre-sheafP is nontrivial if there exists a nonempty open subsetUof X with P(U) 6= ∅.


A.1. Example. For each open subsetU of X letP(U) be the set of all continuousmapsf : U → R. Given open subsetsU , V of X with V ⊂ U we setPU,V (f) =f |V , for all f ∈ P(U). ThenP is a pre-sheaf overX.

A sheaf over a topological spaceX is a pair(S, π), whereS is a topologicalspace andπ : S → X is a local homeomorphism.

LetP be a pre-sheaf over a topological spaceX. Given a pointx ∈ X, considerthe disjoint union of all setsP(U), whereU is an open neighborhood ofx inX. We define an equivalence relation∼ on such disjoint union as follows; givenf1 ∈ P(U1), f2 ∈ P(U2), whereU1, U2 are open neighborhoods ofx in X thenf1 ∼ f2 if and only if there exists an open neighborhoodV of x contained inU1 ∩ U2 such thatPU1,V (f1) = PU2,V (f2). If U is an open neighborhood ofx inX andf ∈ P(U) then the equivalence class off corresponding to the equivalencerelation∼ will be denote by[f ]x and will be called thegermof f at the pointx.We set:

Sx ={[f ]x : f ∈ P(U), for some open neighborhoodU of x in X

}.

Let S denote the disjoint union of allSx, with x ∈ X. Let π : S → X denotethe map that carriesSx to the pointx. Our goal now is to define a topology onS.Given an open subsetU ⊂ X and an elementf ∈ P(U) we set:

V(f) ={[f ]x : x ∈ U

}⊂ S.

The set: {V(f) : f ∈ P(U), U an open subset ofX

}

is a basis for a topology onS; moreover, ifS is endowed with such topology, themapπ : S → X is a local homeomorphism, so that(S, π) is a sheaf overX. Wecall (S, π) thesheaf of germscorresponding to the pre-sheafP. Observe that ifUis an open subset ofX andf ∈ P(U) then the mapf : U → S defined by

f(x) = [f ]x, x ∈ U,

is a local section of the sheaf of germs defined inU .

Definition A.4. We say that the pre-sheafP has thelocalization propertyif, givena family (Ui)i∈I of open subsets ofX and settingU =

⋃i∈I Ui then the map:

(26) P(U) ∋ f 7−→(PU,Ui

(f))i∈I

∈∏

i∈I

P(Ui)

is injective and its image consists of all the families(fi)i∈I in∏i∈I P(Ui) such

thatPUi,Ui∩Uj(fi) = PUj ,Ui∩Uj

(fj), for all i, j ∈ I.

RemarkA.5. If the pre-sheafP has the localization property then for every localsections : U → S of its sheaf of germsS there exists a uniquef ∈ P(U) suchthat f = s.

Definition A.6. We say that the pre-sheafP has theuniqueness propertyif forevery connected open subsetU ⊂ X and every nonempty open subsetV ⊂ Uthe mapPU,V is injective. We say that an open subsetU ⊂ X has theextension


property with respect to the pre-sheafP if for every connected nonempty opensubsetV of U the mapPU,V is surjective. We say that the pre-sheafP has theextension propertyif X can be covered by open sets having the extension propertywith respect toP.

RemarkA.7. If the pre-sheafP has the uniqueness property and ifX is locallyconnected and Hausdorff then the spaceS is Hausdorff. IfX is locally connectedand ifU is an open subset ofX having the extension property with respect to thepre-sheafP thenU has the property (∗) with respect to the local homeomorphismπ : S → X. It follows from Lemma A.2 that ifX is Hausdorff and locallyconnected and if the pre-sheafP has the uniqueness property and the extensionproperty then the mapπ : S → X is a covering map.

Proposition A.8. Assume thatX is Hausdorff, locally arc-connected, connected,and simply-connected. IfP is a pre-sheaf overX satisfying the localization prop-erty, the uniqueness property and the extension property then the open setX hasthe extension property forP, i.e., for every nonempty open connected subsetV ofX the mapPX,V : P(X) → P(V ) is surjective. In particular, ifP is nontrivialthen the setP(X) is nonempty.

Proof. Let V be a nonempty open connected subset ofX and letf ∈ P(V ) befixed. We will show thatf is in the image ofPX,V . Let π : S → X denote thesheaf of germs ofP. By Remark A.7π is a covering map. Choose an arbitrarypointx0 ∈ V and letS0 be the arc-connected component of[f ]x0 in S. SinceXis locally arc-connected, the restriction ofπ toS0 is again a covering map. By theconnectedness and simply-connectedness ofX, π|S0

: S0 → X is a homeomor-phism. The inverse ofπ|S0

is therefore a global sections : X → S and, by Re-mark A.5, there existsg ∈ P(X) with g = s. Now [g]x0 = g(x0) = s(x0) = [f ]x0and hence, by the uniqueness property,PX,V (g) = f . �

APPENDIX B. A CRASH COURSE ON CALCULUS WITH CONNECTIONS

Given a smooth vector bundleπ : E →M over a smooth manifoldM , we willdenote byΓ(E) the space of all smooth sectionss : M → E of E. Observe thatΓ(E) is a real vector space and it is a module over the commutative ringC∞(M)of all smooth mapsf :M → R. Given an open subsetU of M , we denote byE|Uthe restriction of the vector bundleE toU , i.e.,E|U = π−1(U).

Definition B.1. A connectionon a vector bundleπ : E →M is aR-bilinear map:

∇ : Γ(TM)× Γ(E) ∋ (X, s) 7−→ ∇Xs ∈ Γ(E)

that isC∞(M)-linear in the variableX and satisfies the Leibnitz derivative rule:

∇X(fs) = X(f)s+ f∇Xs,

for all X ∈ Γ(TM), s ∈ Γ(E), f ∈ C∞(M).

B.1. Example. If E0 is a fixed real finite-dimensional vector space andE =M ×E0 is a trivial vector bundle overM then a sections of E can be identified with a


maps :M → E0 and a connection onE can be defined by:

(27) ∇Xs = ds(X),

for all X ∈ Γ(TM). We call (27) thestandard connectionof the trivial bundleE.

It follows from theC∞(TM)-linearity of ∇ in the variableX that ∇Xs(x)depends only of the value ofX at the pointx ∈ M , i.e., if X(x) = X ′(x) then∇Xs(x) = ∇X′s(x). Givens ∈ Γ(E), x ∈M andv ∈ TxM , we set:

∇vs = ∇Xs(x),

whereX ∈ Γ(TM) is an arbitrary vector field withX(x) = v. For allx ∈ M wedenote by∇s(x) : TxM → Ex the linear map given byv 7→ ∇vs. Thus, givens ∈ Γ(E), we obtain a smooth section∇s of TM∗ ⊗ E.

It follows from the Leibnitz rule that ifU ⊂ M is an open subset then therestriction of∇Xs to U depends only of the restriction ofs to U . Thus, given anopen subsetU of M , a connection∇ on E induces a unique connection∇U onE|U such that:

(28) ∇Uv (s|U ) = ∇vs,

for all s ∈ Γ(E), v ∈ TU .

RemarkB.2. Given connections∇ and∇′ on a vector bundleπ : E → M thentheir difference is a tensor; more explicitly:

t(X, s) = ∇Xs−∇′Xs ∈ Γ(E), X ∈ Γ(TM), s ∈ Γ(E),

is C∞(M)-bilinear and hence defines a smooth sectiont of the vector bundleTM∗ ⊗ E∗ ⊗ E. Moreover, if∇ is a connection onE and t is a smooth sec-tion of TM∗ ⊗ E∗ ⊗ E then∇ + t is also a connection onE. If t is a sectionof TM∗ ⊗ E∗ ⊗ E then, givenx ∈ M , v ∈ TxM , we identify t(v) with a linearoperator on the fiberEx.

Given vector bundlesπ : E →M , π : E →M over the same base manifoldMthen avector bundle morphismis a smooth mapL : E → E such thatπ ◦ L = π

and such thatL|Ex : Ex → Ex is a linear map, for allx ∈ M . We will denote therestriction ofL toEx byLx. If L : E → E is a vector bundle morphism such thatLx is an isomorphism for allx ∈ M then we callL a vector bundle isomorphism.If A is an open subset ofE andL : A→ E is a smooth map such thatπ ◦L = π|Athen we callL a fiber bundle morphism. Givenx ∈ M , we writeAx = A ∩ ExandLx = L|Ax : Ax → Ex.

Definition B.3. Given vector bundlesπ : E → M , π : E → M over the samebase manifoldM , a vector bundle morphismL : E → E and connections∇, ∇onE andE respectively then we say that∇ and∇ areL-related if:

∇X

(L(s)

)= L(∇Xs),

for all X ∈ Γ(TM), s ∈ Γ(E).


In what follows, we will deal with several constructions involving connectionson different vector bundles. In order to avoid heavy notations, we will usuallydenote all these connections by the symbol∇; it should be clear from the con-text which connection the symbol∇ refers to. For instance, formula (28) will berewritten in the following simpler form:

∇v(s|U ) = ∇vs.

Definition B.4. Given a connection∇ on a vector bundleπ : E → M , then thecurvature tensorof ∇ is defined by:

(29) R(X,Y )s = ∇X∇Y s−∇Y∇Xs−∇[X,Y ]s ∈ Γ(E),

for all X,Y ∈ Γ(TM), s ∈ Γ(E).

Since the righthand side of (29) isC∞(M)-linear in the variablesX, Y ands, it follows thatR can be identified with a smooth section of the vector bundleTM∗ ⊗ TM∗ ⊗E∗ ⊗ E. Clearly,R(X,Y )s is anti-symmetric in the variablesXandY .

The notion of torsion is usually defined only for connection on tangent bundles.We will present a slight generalization of this notion.

Definition B.5. Let π : E → M be a smooth vector bundle and letι : TM → Ebe a vector bundle morphism. Given a connection∇ onE then theι-torsionof ∇is defined by:

(30) T ι(X,Y ) = ∇X

(ι(Y )

)−∇Y

(ι(X)

)− ι

([X,Y ]

)∈ Γ(E),

for all X,Y ∈ Γ(TM). If E = TM and ι is the identity map ofTM , we willwrite simplyT and call it thetorsionof ∇.

Again, the righthand side of (30) isC∞(M)-linear on the variablesX andY ,so thatT ι can be identified with a smooth section of the vector bundleTM∗ ⊗TM∗ ⊗ E. Clearly,T ι(X,Y ) is anti-symmetric inX andY .

In what follows we will study some natural constructions with vector bundlesendowed with connections and we will present some formulas for the computationof torsions and curvatures. We will consider constructionsthat act on the basis ofthe vector bundles and constructions that act on their fibers.

Given smooth manifolds,M , N , a smooth vector bundleπ : E → M overMand a smooth mapf : N → M then we denote byf∗E the pull-back ofE by fwhich is a vector bundle overN whose fiber at a pointx ∈ N is equal toEf(x).Observe that there is a natural identification of smooth sections of the bundlef∗Ewith smoothsections ofE along f , i.e., smooth mapss : N → E such thatπ ◦ s = f . Notice that every smooth sections : M → E of E gives rise to asmooth section ofE alongf given bys ◦ f : N → E; we may thus identifys ◦ fwith a section off∗E.

Proposition B.6. Given smooth manifoldsM , N , a smooth vector bundleE overM endowed with a connection∇ and a smooth mapf : N →M then there existsa unique connectionf∗∇ on the pull-back bundlef∗E such that:

(31) (f∗∇)v(s ◦ f) = ∇df(v)s,


for all s ∈ Γ(E) and allv ∈ TN .

The next result follows easily from Proposition B.6.

Proposition B.7. Let P , N , M be smooth manifolds,E be a vector bundle overM endowed with a connection∇ andg : P → N , f : N → M be smooth maps.Then:

(32) (f ◦ g)∗∇ = f∗(g∗∇);

moreover, ifi : U →M denotes the inclusion map of an open subsetU ofM theni∗E can be naturally identified with the bundleE|U and i∗∇ coincides with theinduced connection∇U .

Identity (32) can be interpreted as achain ruleas follows; given a sections :N → E of E alongf andv ∈ TP then:

((f ◦ g)∗∇

)v(s ◦ g)

by (32)=

(g∗(f∗∇)

)v(s ◦ g)

by (31)= (f∗∇)dg(v)s.

We have the following natural formula to compute the curvature and the torsionof a pull-back connection.

Proposition B.8. Given smooth manifoldsM , N , a smooth vector bundleE overM endowed with a connection∇ and a smooth mapf : N → M then the curva-ture tensor off∗∇ is given by:

Rf∗∇x (v,w)e = R∇

f(x)

(df(x)v,df(x)w

)e,

for all x ∈ N , v,w ∈ TxN , e ∈ (f∗E)x = Ef(x). Moreover, given a smoothvector bundle morphismι : TM → E, thenι ◦ df : TN → E is identified with avector bundle morphismι : TN → f∗E and the the following formula holds:

(33) T ιx(v,w) = T ιf(x)(df(x)v,df(x)w

),

for all x ∈ N , v,w ∈ TxN .

Observe that ifE = TM andι is the identity ofTM then formula (31) meansthat:

(f∗∇)X(df(Y )

)− (f∗∇)Y

(df(X)

)− df

([X,Y ]

)= T

(df(X),df(Y )

),

for all X,Y ∈ Γ(TN).Now we consider constructions acting on the fibers of the vector bundles. To

this aim, we need some categorical language. Given an integer n ≥ 1, we denotebyVecn the category whose objects aren-tuples(Vi)ni=1 of real finite-dimensionalvector spaces and whose morphisms from(Vi)

ni=1 to (Wi)

ni=1 aren-tuples(Ti)ni=1

of vector space isomorphismsTi : Vi → Wi. We setVec1 = Vec. A functorF : Vecn → Vec is calledsmoothif for any object(Vi)ni=1 of Vecn the map:

(34) F : GL(V1)× · · · ×GL(Vn) −→ GL(F(V1, . . . , Vn)

)

is smooth. Observe that (34) is a Lie group homomorphism; itsdifferential at theidentity is a Lie algebra homomorphism that will be denoted by:

F : gl(V1)× · · · × gl(Vn) −→ gl(F(V1, . . . , Vn)

).


Given vector bundlesE1, . . . ,En over a smooth manifoldM we obtain naturally anew vector bundleF(E1, . . . , En) overM whose fiber at a pointx ∈M is equal toF(E1

x, . . . , Enx ). Given a smooth manifoldN and a smooth mapf : N → M , we

may identify vector bundlesf∗(F(E1, . . . , En)

)andF(f∗E1, . . . , f∗En). Given

vector bundle isomorphismsLi : Ei → Ei, i = 1, . . . , n, then we obtain a vectorbundle isomorphismL = F(T 1, . . . , T n) from F(E1, . . . , En) to F(E1, . . . , En)by setting:

Lx = F(L1x, . . . , L

nx),

for all x ∈M .We have the following functorial construction for connections.

Proposition B.9. Given an integern ≥ 1 and a smooth functorF : Vecn → Vec

then there exists a unique rule that associates to each smooth manifoldM , eachn-tuple of vector bundles(E1, . . . , En) overM and eachn-tuple of connections(∇1, . . . ,∇n) on(E1, . . . , En) respectively, a connection∇ = F(∇1, . . . ,∇n) onF(E1, . . . , En) satisfying the following properties:

(a) (naturality with pull-backs) given smooth manifoldsN , M and a smoothmapf : N →M thenf∗

(F(∇1, . . . ,∇n)

)= F(f∗∇1, . . . , f∗∇n);

(b) (naturality with morphisms) given vector bundle isomorphismsLi : Ei →Ei, i = 1, . . . , n, if ∇i is a connection onEi which isLi-related with aconnection∇i on Ei thenF(∇1, . . . ,∇n) is F(L1, . . . , Ln)-related withF(∇1, . . . , ∇n);

(c) given connections∇i and∇i onEi with∇i−∇i = ti, i = 1, . . . , n, then:

F(∇1, . . . ,∇n)Xs− F(∇1, . . . , ∇n)Xs = F(t1(X), . . . , tn(X)

)s,

for all s ∈ Γ(F(E1, . . . , En)

);

(d) (trivial bundle property) If ∇i is the standard connection of the trivialbundleM × Ei0 thenF(∇1, . . . ,∇n) is the standard connection of thetrivial bundleM × F(E1

0 , . . . , En0 ).

Let F = (F1, . . . ,Fm) be anm-tuple of functorsFi : Vecn → Vec and letG : Vecm → Vec be a functor; we denote byG ◦ F : Vecn → Vec the smoothfunctor defined2 by:

(G ◦ F)(V1, . . . , Vn) = G(F1(V1, . . . , Vn), . . . F

m(V1, . . . , Vn)),

for all objectsV1, . . . ,Vn of Vec.

Proposition B.10. Let F = (F1, . . . ,Fm) be anm-tuple of smooth functorsFi :Vecn → Vec and letG : Vecm → Vec be a smooth functor. Given vector bundlesE1, . . . ,En over a smooth manifoldM endowed respectively with connections∇1,. . . ,∇n then:

(G ◦ F)(∇1, . . . ,∇n) = G(F1(∇1, . . . ,∇n), . . . ,Fm(∇1, . . . ,∇n)

).

2We will usually only describe functors on objects; the action of the functor on morphisms shouldbe clear.


Moreover, ifI : Vec → Vec denotes the identity functor ofVec then, given aconnection∇ on a vector bundleE, we have:

I(∇) = ∇.

Proposition B.11. Given a smooth functorF : Vecn → Vec and smooth vectorbundlesπi : Ei → M endowed with connections∇i, i = 1, . . . , n, then thecurvature tensor of the connectionF(∇1, . . . ,∇n) is given by:

Rx(v,w) = F(R1x(v,w), . . . , R

nx(v,w)

),

for all x ∈ M , v,w ∈ TxM , whereRi denotes the curvature tensor of∇i, i =1, . . . , n.

Definition B.12. Given a positive integern and smooth functorsF : Vecn → Vec

andF′ : Vecn → Vec then asmooth natural transformationρ from F to F′ is arule that associates to each object(Vi)

ni=1 of Vecn an open subsetA(V1,...,Vn) of

F(V1, . . . , Vn) and a smooth map:

ρV1,...,Vn : AV1,...,Vn −→ F′(V1, . . . , Vn)

such that, given objects(Vi)ni=1, (Wi)ni=1 of Vecn and a morphism(Ti)ni=1 from

(Vi)ni=1 to (Wi)

ni=1 thenF(T1, . . . , Tn) carriesAV1,...,Vn toAW1,...,Wn and the dia-

gram:

AV1,...,VnρV1,...,Vn

//

F(T1,...,Tn)

��

F′(V1, . . . , Vn)

F′(T1,...,Tn)��

AW1,...,Wn ρW1,...,Wn

// F′(W1, . . . ,Wn)

commutes.

Given a smooth natural transformationρ from F to F′ and given vector bundlesπi : Ei → M , i = 1, . . . , n, we obtain a fiber bundle morphismρE1,...,En : A →F′(E1, . . . , En) defined on an open subsetA of F(E1, . . . , En) by setting:

Ax = AE1x,...,E

nx, (ρE1,...,En)x = ρE1

x,...,Enx,

for all x ∈M .

Proposition B.13. Given a positive integern, smooth functorsF : Vecn → Vec,F′ : Vecn → Vec, a smooth natural transformationρ fromF to F′, vector bundlesπi : Ei →M endowed with connections∇i, i = 1, . . . , n, then:

F′(∇1, . . . ,∇n)v(ρE1,...,En ◦ s) = dρE1x,...,E

nx

(s(x)

)(F(∇1, . . . ,∇n)vs

),

for all x ∈M , v ∈ TxM and every smooth sections of F(E1, . . . , En) with rangecontained in the domain ofρE1,...,En .

B.2. Example. Let n be a positive integer and consider the smooth functorS :Vecn → Vec defined by:

S(V1, . . . , Vn) = V1 ⊕ · · · ⊕ Vn.


Given vector bundlesE1, . . . ,En over a smooth manifoldM thenS(E1, . . . , En)is the Whitney sum ofE1, . . . ,En. Let ∇i be a connection onEi, i = 1, . . . , n.For eachi = 1, . . . , n, consider the smooth functorPi : Vecn → Vec defined by:

Pi(V1, . . . , Vn) = Vi.

We have a smooth natural transformationρi from S toPi given by:

ρi : V1 ⊕ · · · ⊕ Vn ∋ (v1, . . . , vn) 7−→ vi ∈ Vi.

Set∇ = S(∇1, . . . ,∇n). Proposition B.13 implies that:

∇v(s1, . . . , sn) = (∇1vs1, . . . ,∇

nvsn),

for all s1 ∈ Γ(E1), . . . ,sn ∈ Γ(En), v ∈ TM .

B.3. Example. Consider the smooth functorsF : Vec2 → Vec andG : Vec2 →Vec defined as follows; letV1, V2, W1, W2 be objects ofVec and letT1 : V1 →W1, T2 : V2 → W2 be isomorphisms. We set:

F(V1, V2) = Lin(V1, V2), F(T1, T2)L = T2 ◦ L ◦ T−11 ,

G(V1, V2) = Lin(V ∗2 , V

∗1 ) G(T1, T2)R = (T ∗

1 )−1 ◦R∗ ◦ T ∗

2 ,

for L ∈ Lin(V1, V2), R ∈ Lin(V ∗2 , V

∗1 ). We have a natural transformationρ from

F toG defined by:

ρ : Lin(V1, V2) ∋ t 7−→ t∗ ∈ Lin(V ∗2 , V

∗1 ).

Let E1, E2 be vector bundles over a smooth manifoldM endowed with connec-tions∇1 and∇2 respectively. We denote by∇ both the connectionsF(∇1,∇2)andG(∇1,∇2) on the bundlesLin(E1, E2) andLin((E2)∗, (E1)∗) respectively.Proposition B.13 tells us that, given a smooth sectionL of Lin(E1, E2) then:

∇vL∗ = (∇vL)

∗,

for all v ∈ TM .

B.4. Example. Consider the smooth functorF : Vec → Vec defined as follows;let V ,W be objects ofVec and letT : V →W be an isomorphism. We set:

F(V ) = Bilin(V, V ;V )⊕ V ⊕ V,

F(T )(B, v1, v2) =(T ◦B(T−1·, T−1·), T (v1), T (v2)

),

for every bilinear mapB : V × V → V and allv1, v2 ∈ V . We have a smoothnatural transformationρ from F to the identity functorI of Vec defined by:

ρ : Bilin(V, V ;V )⊕ V ⊕ V ∋ (B, v1, v2) 7−→ B(v1, v2) ∈ V.

Let E be a vector bundle over a smooth manifoldM endowed with a connection∇. We will also denote by∇ the connectionF(∇) on Bilin(E,E;E). Proposi-tion B.13 tells us that, given a smooth sectionB of Bilin(E,E;E) and smoothsectionss1, s2 of E then:

∇v

(B(s1, s2)

)= (∇vB)(s1, s2) +B(∇vs1, s2) +B(s1,∇vs2),

for all v ∈ TM .


B.5. Example. Consider the smooth functorF : Vec → Vec defined as follows;let V ,W be objects ofVec and letT : V →W be an isomorphism. We set:

F(V ) = Lin(V ),

F(T )L = T ◦ L ◦ T−1,

for all L ∈ Lin(V ). We have a smooth natural transformationρ from F to itselfdefined by:

ρ : Lin(V ) ⊃ GL(V ) ∋ L 7−→ L−1 ∈ Lin(V ).

Let E be a vector bundle over a smooth manifoldM endowed with a connection∇. We will also denote by∇ the connectionF(∇) onLin(E). LetL be a smoothsection ofLin(E) such thatLx is an isomorphism ofEx, for all x ∈ M . Proposi-tion B.13 tells us that:

∇v(L−1) = −L−1(∇vL)L

−1,

for all v ∈ TM .

REFERENCES

[1] S. Kobayashi and K. Nomizu,Foundations of differential geometry. Vol I, Interscience Publish-ers, New York-London, 1963.

[2] S. Lang,Fundamentals of differential geometry, Graduate Texts in Mathematics. 191. NewYork, NY: Springer, 2001.

[3] P. Piccione, D. V. Tausk,Connections compatible with tensors. A characterization of left-invariant Levi–Civita connections in Lie groups, arXiv math.DG/0509656.

[4] J. A. Wolf, Spaces of constant curvature. New York, Mc-Graw-Hill, 1967.

DEPARTAMENTO DEMATEMATICA ,UNIVERSIDADE DE SAO PAULO , BRAZIL

E-mail address: [email protected], [email protected]

URL: http://www.ime.usp.br/˜piccione,http://www.ime.usp.br/˜tausk

arXiv:math/0510555v1 [math.DG] 26 Oct 2005 · 2019. 3. 26. · grability theorem in elementary...

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