Geometric characterization of linearizable second-order

differential equations

Eduardo Martınez† and Jose F. Carinena‡†Dpto. de Matematica Aplicada, Universidad de Zaragoza

E-50015 Zaragoza, Spain‡Departamento de Fısica Teorica, Universidad de Zaragoza

E-50009 Zaragoza, Spain

Abstract

Given an Ehresmann connection on the tangent bundle τ :TM → M wedefine a linear connection on the pull-back bundle τ∗(TM). With the aid of thistool, necessary and sufficient conditions for the existence of local coordinates inwhich a system of second-order differential equations is linear are derived.

1 Introduction

In previous papers [8, 9] we developed a theory of derivations of scalar and vector-valued forms along the tangent bundle projection τ :TM → M of a manifold M .This theory was used to geometrically characterize when a system of second-orderdifferential equations is separable, i. e. there exists a coordinate system on the basemanifold in which the system decouples as a sum of one-dimensional independentdifferential equations (see [10]). Two fundamental operators associated with everyvector field X along τ did appear in these papers. They were denoted DV

X andDHX and called the horizontal and vertical covariant derivative operators. Their

usefulness is partially based on the fact that they are derivations (of degree zero)and that the maps X 7→ DH

X and X 7→ DVX are C∞(TM)-linear, so that there is a

clear resemblance with a covariant derivative operator.In this paper we show that this resemblence is not a coincidence but a conse-

quence of the existence of a covariant derivative on the pull-back bundle by τ of thetangent bundle. With the aid of this tool, we will geometrically characterize thosesecond order differential equations which are linearizable under a point transforma-tion of coordinates.

In order to the paper be selfcontained, we give in Section 1 a summary of theresults of the theory of derivations which are relevant for subsequent sections. In Sec-tion 2 we will introduce the linear connection associated to a non-linear connection

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on the tangent bundle and we will show the relation between the curvature of thelinear connection and the tensors Rie and θ, that were previously defined in termsof the horizontal and vertical covariant derivatives. Some relations between Rie andθ which are just the expression of the Bianchi identities for the linear connection arealso given. Finally, in Section 3 we study the problem of the linearization of a secondorder differential equations. Our results will be related to previous ones [1, 13]

2 Preliminaries

Let π:E → M be a fibre bundle and φ:N → M a smooth map. A section of Ealong φ is a map σ:N → E such that π ◦ σ = φ. Equivalently, a section along φ canbe considered as a section σ of the pull-back bundle φ∗π:φ∗E → N (see e. g. [11]).The relation between σ and σ is σ = φ[π]◦σ, and we have the following commutativediagram

N M

φ∗E E

φ

φ[π]

φ∗π σ σ π

-

-

6

? ?�������>

If E is a vector bundle then the set of sections along φ is a C∞(N)-module. Themost interesting cases are E = TM , (T ∗M)∧p or any other tensor bundle, and thena section of E along φ is called a vector field along φ, a p-form along φ, or a tensorfield along φ (respectively). The set of vector fields along φ is denoted by X (φ), andthe set of p-forms along φ by

∧p(φ). In particular, we are interested in the case in

which φ is the projection τ of the tangent bundle τ :TM →M . The easiest exampleof tensor field along τ is the composition W ◦ τ for W a tensor field on the basemanifold M . We will then say that W is a basic tensor field and we will simplify thenotation writing just W instead of W ◦ τ . Another example is the canonical vectorfield along τ denoted by T and defined by the identity map on TM thought of as asection of TM along τ .

In two recent papers [8, 9] we have studied the algebra of derivations of scalarand vector valued forms along τ . A complete classification of the set of derivationsof forms along τ was given with the aid of an Ehresmann connection in the tangentbundle, i. e. a horizontal subbundle of T (TM) (see [11, 2, 3]). In presence of aconnection, any vector field Z ∈ X (TM) can be decomposed as a sum Z = XH

1 +XV2

for X1, X2 ∈ X (τ), where the indices H and V mean horizontal and vertical lift,respectively. In coordinates, if Z has the expression Z = Xi∂/∂xi+Y i∂/∂vi, thenX1

and X2 are X1 = Xi∂/∂xi and X2 = (Y i + ΓijXj)∂/∂xi. Here, the functions Γij are

the coefficients of the connection, defined by the horizontal liftHi = ∂/∂xi−Γji∂/∂vj

of the coordinate vector field ∂/∂xi. If W is a vertical vector field on TM , the vector

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fieldX along τ such thatW = XV will be denotedW ↓. Note that ifW is a horizontalvector field on TM then the vector field X along τ such that XH = W is just theprojection of W , i. e. X = Tτ ◦W .

With the above decomposition in mind, we define two fundamental derivationsDHX and DV

X for every X ∈ X (τ), called respectively the horizontal and verticalcovariant derivatives, by mean of the equation

[XH , Y V ] = {DHXY }V − {DV

YX}H .

They are called covariant because they are C∞(TM)-linear in the subscript argu-ment. We extend the action of these derivations to 1-forms along τ by imposing theduality rule

D〈X,α〉 = 〈DX,α〉+ 〈X,Dα〉,

and to general tensor fields along τ by imposing the Leibnitz rule for the tensorproduct. Derivations satisfying these properties are called self-dual. If X, Y arevector fields along τ with local expressions X = Xi∂/∂xi and Y = Y i∂/∂xi, thenthe coordinate expressions of DH

XY and DVXY are

DHXY = Xi(HiY

j + Y kΓjik)∂

∂xj

DVXY = Xi(ViY j)

∂

∂xj,

where Vi = ∂/∂vi and Γijk = Vk(Γij). It can be easily shown that a tensor field Walong τ is basic if and only if DVW = 0.

The commutator of two horizontal lifts and two vertical lifts is given by

[XV , Y V ] = [X,Y ]VV[XH , Y H ] = [X,Y ]HH + {R(X,Y )}V

where [X,Y ]V = DVXY − DV

YX, [X,Y ]H = DHXY − DH

YX − T (X,Y ) and where Rand T are, respectively, the curvature and the torsion of the non-linear connection.It is easy to see (e. g. in coordinates) that if X and Y are vector fields on the basemanifold M , then [X,Y ] = [X,Y ]H .

The commutators of the covariant derivatives define two important 2-covarianttensor fields along τ , θ and Rie, taking values in the set of (1,1) tensor fields, by

[DVX , D

VY ]Z = DV

[X,Y ]VZ

[DVX , D

HY ]Z = DH

DVXY−DV

DHY X

Z + θ(X,Y )Z

[DHX , D

HY ]Z = DH

[X,Y ]HZ +DV

R(X,Y )Z + Rie(X,Y )Z.

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Their coordinate expressions are

θ = Γkjml dxl ⊗ dxj ⊗

(dxm ⊗ ∂

∂xk

)Rie =

12

[Hk(Γilj)−Hl(Γikj) + ΓikrΓrlj − ΓilrΓ

rkj ]dx

k ∧ dxl ⊗(

dxj ⊗ ∂

∂xi

),

where Γijkl = VlΓijk. From the coordinate expression of Rie it follows that

Rie = −DVR,

i. e. if X, Y , Z are vector fields along τ , then Rie(X,Y )Z = −[DVZR](X,Y ).

The vertical covariant derivative of the canonical vector field along τ is the iden-tity tensor DV T = I, while the horizontal covariant derivative defines an important(1,1) tensor field associated to the connection which is called the tension t = −DHT.Its coordinate expression is

t = (Γij − Γijkvk)dxj ⊗ ∂

∂xi.

Notice that t = 0 iff the connection is linear, i. e. the functions Γijk do not dependon vl. In this case the tensor Rie coincides with the Riemann curvature tensor ofthe linear connection.

3 Induced linear connection

We remind that a covariant derivative on a vector bundle π:E → N is a C∞(N)-linear operator mapping every vector field V on N to a derivation DV of the C∞(N)-module Sec (π) of sections of the bundle E, such that it coincides precisely with Von C∞(N). That is, D satisfies the following properties:

1. DV+Wσ = DV σ + DWσ

2. DfV σ = fDV σ

3. DV (σ + ρ) = DV σ + DV ρ

4. DV (fσ) = (V f)σ + fDV σ,

for all V , W ∈ X (N), f ∈ C∞(M) and σ, ρ ∈ Sec (π). A covariant derivative on abundle is equivalent to a linear connection on such bundle (see [11, 6]).

As we mentioned in the last section, the operators DH and DV are similar to realcovariant derivatives. Taking into account that sections of the bundle τ∗τ : τ∗(TM)→TM are (canonically identified with) vector fields along τ , it is clear that the horizon-tal and vertical covariant derivatives must be related with a true covariant derivative

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on such bundle. Indeed, for every vector field W on TM we define the operator DW

by

DWZ = {PV [PH(W ), ZV ]}↓ + Tτ ◦ [PV (W ), ZH ], (∗)

where Z is a vector field along τ and PH , PV are the horizontal and verticalprojectors defined by the connection. The IR-linearity of [· , ·] implies that D satisfiesproperties (1) and (3). We now prove that D satisfies (2): if F is a function on TM ,then using [FPH(W ), ZV ] = −ZVF PH(W ) + F [PH(W ), ZV ] and [FPV (W ), ZH ] =−ZHF PV (W ) + F [PV (W ), ZH ] we have

DFWZ = −ZVF PV PH(W ) + F{PV [PHW,ZV ]}↓ + FTτ ◦ [PV (W ), ZH ]= F DWZ,

because PV PH = 0. Furthermore, [PH(W ), FZV ] = PH(W )F ZV + F [PH(W ), ZV ]and [PV (W ), FZH ] = PV (W )F ZH + F [PV (W ), ZH ], so that

DW (FZ) = {PH(W )F ZV + FPV [PHW,ZV ]}↓

+ Tτ ◦ ((PV (W )F )ZH + F [PV (W ), ZH ])= (PH(W )F + PV (W )F )Z + F (PV [PH(W ), ZV ])↓

+ FTτ + ([PV (W ), ZH ]),

and taking into account that PH + PV = I, we find that D satisfies (4). Thus, D isa covariant derivative in the pull-back bundle by τ .

As we said above, every vector field W on TM can be decomposed as a sum of avertical plus a horizontal lift of a vector field along τ , W = XH +Y V . The followingrelation holds

DXH+Y V = DHX +DV

Y .

Indeed, taking into account the equation defining DH and DV , we obtain

DXH+Y V Z = (PV [XH , ZV ])↓ + Tτ + ([Y V , ZH ])= ((DH

XZ)V )↓ + Tτ + ((DVY Z)H)

= DHXZ +DV

Y Z.

We have proved the following result.

Theorem 1 Given a connection on the tangent bundle TM to a manifold M , thereexists one linear connection D on the pull-back bundle τ∗τ : τ∗(TM) → TM suchthat

DXH+Y V = DHX +DV

Y .

for all X, Y ∈ X (τ). Such connection is explicitly given by equation (*) and will besaid to be the linear connection induced by the non-linear connection on TM .

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We remind that if φ:N → M is a smooth map and H(E) is an Ehresmannconnection on the bundle π:E → M , then (Tφ[π])−1(H(E)) is an Ehresmann con-nection on the bundle φ∗π:φ∗E → N . This connection is known as the inducedconection by φ. We remark that in the case φ = τ the induced connection is dif-ferent from the induced linear connection here defined. In particular, if the originalconnection is not linear, then the induced connection is not linear too.

If we choose local coordinates (xi, vi, wi) on τ∗(TM) such that

τ∗τ(xi, vi, wi) = (xi, vi) and τ [τ ](xi, vi, wi) = (xi, wi),

then the expressions of the horizontal lifts of the coordinate vector fields with respectto the induced linear connection are(

∂

∂xi

)H

=∂

∂xi− wkΓjik

∂

∂wj

(∂

∂vi

)H

=∂

∂vi.

Consider now the Riemann curvature tensor

Rie(V,W )Z = [DV , DW ]Z − D[V,W ]Z.

of the connection D. We will show that the tensors Rie and θ are but componentsof Rie.

Theorem 2 The curvature of the connection D is given by

Rie(XH , Y H)Z = Rie(X,Y )ZRie(XV , Y H)Z = θ(X,Y )ZRie(XV , Y V )Z = 0

for X,Y, Z ∈ X (τ).

Proof: For two horizontal vector fields XH and Y H , we have

Rie(XH , Y H)Z = [DHX , D

HY ]Z − D{[X,Y ]H}H+R(X,Y )V Z

= [DHX , D

HY ]Z −DH

[X,Y ]HZ −DV

R(X,Y )Z

= Rie(X,Y )Z.

Similarly, for XV and Y H , we obtain

Rie(XV , Y H)Z = [DVX , D

HY ]Z − D{DV

XY }H−{DHY X}V

Z

= [DVX , D

HY ]Z −DH

DVXY

Z +DV

DHY X

Z

= θ(X,Y )Z.

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Finally, for two vertical vector fields we find

Rie(XV , Y V )Z = [DVX , D

VY ]Z − D{[X,Y ]V }V Z

= [DVX , D

VY ]Z −DV

[X,Y ]VZ

= 0.

Having a linear connection, we have at our disposal a menagerie of results. Inparticular, the Bianchi identities∑

ciclic

DU Rie(V,W ) = 0,

implies some relations that Rie and θ must satisfy. They are:

θ(X,Y )Z − θ(Z, Y )X = 0DHZθ(X,Y )−DH

Y θ(X,Z)− θ(X,T (Y, Z)) +DVXRie(Y, Z) = 0∑

ciclic {DHXRie(Y,Z) + Rie(X,T (Y, Z))− θ(R(X,Y ), Z)} = 0

The proof is a matter of straightforward calculation and will be omited. This Bianchiidentities were first found in [7] by calculating the Jacobi identities for DH and DV .We also mention that in [8] we found another set of relations between the torsionand the curvature of the non-linear connection, called the Bianchi identities for thenon-linear connection.

4 Characterization of linearizable sode

In this section we will consider the following problem: Given a system of secondorder differential equations

xi = f i(x1, . . . , xn, x1, . . . , xn), i = 1, . . . , n,

we ask under what circumstances there exists a coordinate transformation

xi = ξi(x1, . . . , xn), i = 1, . . . , n,

such that in the coordinates xi the system of second-order equations turns out tobe linear:

¨xi = Aij ˙xj +Bij xj + Ci, i = 1, . . . , n,

for some contants Aij , Bij and Ci.

In geometric terms, a system of second order equations (sode) on a manifold Mis interpreted as a vector field Γ in the tangent bundle to M , such that Tvτ(Γ(v)) =v for every v ∈ TM . Every sode defines a connection on the tangent bundle

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(see [3, 2]). In coordinates, if f i are the forces defined by a sode Γ, f i = Γvi,then the coefficients of the connection are Γij = −1

2∂fi/∂vj . It follows that the

functions Γijk = ∂Γij/∂vk are symmetric in the subscripts j, k, that is, the torsion

of the connection vanishes. In fact, the vanishing of the torsion is a necessary andsufficient condition for the connection to be defined by a sode.

More directly related to the sode, there are a derivation ∇ called the dynamicalcovariant derivative and a (1,1) tensor field Φ called the Jacobi endomorphism. Theycan simultaneously be defined by the equation

[Γ,XH ] = {∇X}H + {Φ(X)}V .

and contain the whole information about the sode. In fact, on functions ∇ coin-cides with LΓ. The action of ∇ over coordinate vector fields is determined by thecoefficients of the connection in such coordinates

∇(∂

∂xj

)= Γij

∂

∂xi,

so that the dynamical covariant derivative of X = Xi∂/∂xi has the expression

∇X = (ΓXi + ΓijXj)

∂

∂xi.

In terms of the induced linear connection we have ∇X = DΓX + t(X). The coordi-nate expression of the Jacobi endomorphism is

Φ =

(−∂f

i

∂xj− ΓikΓ

kj − Γ(Γij)

)dxj ⊗ ∂

∂xi,

and it satisfies the following relation

Φ(X) = R(T,X)−DHX∇T.

Moreover, Φ is related with the curvature by mean of

(DVXΦ)(Y )− (DV

Y Φ)(X) = 3R(X,Y ).

We further have the following commutators

[∇, DVX ]Y = DV

∇XY −DHXY

[∇, DHX ]Y = DH

∇XY +DV

Φ(X)Y − [DVY Φ](X)−R(X,Y )

for X and Y vector fields along τ .

We now turn to the question of the linearizability of second order ordinary differ-ential equations. In geometric terms, a sode Γ is said to be linearizable in velocities

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(v-linearizable) if for each point m ∈M there exist local coordinates (xi) in a neigh-borhood U of m such that the coordinate expression of the forces defined by Γ is ofthe form

f i(x, v) = Aij(x)vj + bi(x)

for some functions Aij , bi ∈ C∞(U).

Theorem 3 A sode Γ is v-linearizable if and only if the connection D is flat.

Proof: Assume that Γ is v-linearizable. In coordinates in which Γ is linear in thevelocities the coefficients of the connection defined by Γ are Γij = −1

2Aij(x). Thus

Γijk = 0, from where it follows Rie = 0.Conversely, if D is flat, then in a neighbourhood of each point there exists a basis

{Xi} of X (τ) = Sec (τ∗τ) such that DWXi = 0 for every vector fieldW on TM . SinceDXV = DV

X we have that Xi are basic vector fields. Moreover, taking into accountthat DXH = DH

X and that [X,Y ] = DHXY −DH

YX we find that [Xi,Xj ] = 0. Thus,there exist local coordinates (xi) for M such that Xi = ∂/∂xi. In these coordinatesthe functions Γijk vanish, from where it follows that Γij are basic functions and thatthe expression of the forces is an affine function of the velocities.

It is worthy to note that the forces depend only on the positions iff, being lin-earizable, the tension of the connection vanishes. Also note that the existence of av-linearizable sode imposes severe restrictions to the topology of the manifold M .Indeed, since θ = 0 then we have that if X and Y are vector fields on M then DH

XYis also a basic vector field, so that it defines a linear connection D on M . Since thecurvature tensor of this linear connection is just the Rie tensor, we have that D isflat. In particular, M is locally isomorphic to an ordinary affine space (with thenatural connection in it).

A sode Γ is said to be linearizable if for each point m ∈ M there exist localcoordinates (xi) in a neighbourhood U of m such that the coordinate expression ofthe forces defined by Γ is of the form

f i(x, v) = Aijvj +Bi

jxj + Ci,

for some constants Aij , Bij and Ci.

Theorem 4 A sode Γ is linearizable if and only if the connection D is flat and theJacobi endomorphism is parallel.

Proof: If Γ is linearizable, then it is v-linearizable, and therefore D is flat. Incoordinates in which Γ is linear we have that the components of the Jacobi endo-morphism are constant Φi

j = −Bij − 1

4AikA

kj , from where DΦ = 0 readily follows.

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Conversely, assume that D is flat and DΦ = 0. Since θ = 0, we have DH ◦DV =DV ◦DH and thus DV t = −DVDHT = −DHDV T = −DHI = 0. Moreover, since Φis basic the curvature vanishes, from where we have

0 = DV Φ = −DVDH∇T = −DHDV∇T = −DHDHT = DHt.

Thus the tension is a parallel tensor field. In coordinates in which the coefficientsΓijk vanish we have that tij = Γij and Φi

j are constant. It follows that ∂f i/∂vj and∂f i/∂xj are constant too, and thus Γ is linear in these coordinates.

As a consequence of the relations between the Jacobi endomorphism, the curva-ture and the Rie tensor we have the following result.

Theorem 5 A sode Γ is linearizable if and only if θ = 0, DHΦ = 0 and DV Φ = 0.

Proof: From DV Φ = 0 we have R = 0, and thus Rie = −DVR = 0.

To finish the paper we consider some previous results in the literature. In [13]the following characterization was given: A sode Γ is linearizable iff:

1. Γ = Λ + AV0 + A1(T)V , where Λ is a spray, A0 is a vector field on M and A1

is a (1,1) tensor field on M .

2. The curvature of the connection D defined by Λ is flat.

3. DA1 and D2A0 vanishes.

In order to relate it with our results, we note that the vanishing of θ is equivalentto (1) and that the connection defined by the spray Λ is given by DXY = DH

XY forX,Y ∈ X (M). Thus D is flat iff Rie vanishes. Finally, since A1 = −2t we haveDA1 = 0 iff the tension is parallel. In such case, DA0 = −Φ − t2, from where wededuce that D2A0 = 0 if and only if the Jacobi endomorphism is parallel.

Kamran [5] has also studied the problem of linearization by using Cartan’smethod of equivalence. Kamran deals with time-dependent coordinate transfor-mations and therefore their results are not applicable to our case. On the otherhand, Chern [1] has solved the equivalence problem under spatial diffeomorphism.The invariants founded there correspond to T, ∇T, Φ, R and θ, and the covariantderivatives of ∇T, one of them being the tension, t = −DHT = DV∇T. If followsthat if R and θ vanish and moreover Φ and t are constant (i. e. parallel) then thesode is equivalent to a linear one. From our results it follows that some of thisconditions are redundant.

Finally it is worth to mention that a diferent kind of linearizability problem hasbeen treated in [4, 12]. There, the problem is to linearize a non-autonomous sode

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by coordinate transformations of the form t = τ(t, x), x = ξ(t, x). These coordinatechanges induce projective transformations on IR× TM so that the formalism usedhere is not directly applicable. We hope that further research will allow us to solvethat problem.

Acknowledgements: We wish to thank Willy Sarlet and Frans Cantrijn for illu-minating suggestions. Partial financial support from CICYT and the NATO Col-laborative Research Grants Progamme is acknowledged.

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