· JOURNAL OF GEOMETRIC MECHANICSdoi:10.3934/jgm.2013.5.85 c American Institute of Mathematical...

JOURNAL OF GEOMETRIC MECHANICS doi:10.3934/jgm.2013.5.85c©American Institute of Mathematical SciencesVolume 5, Number 1, March 2013 pp. 85–129

VECTOR FIELDS WITH DISTRIBUTIONS

AND INVARIANTS OF ODES

Bronis law Jakubczyk

Institute of MathematicsPolish Academy of Sciences

ul. Sniadeckich 8

00-956 Warszawa, Poland

Wojciech Krynski

Institute of Mathematics

Polish Academy of Sciences

ul. Sniadeckich 8

00-956 Warszawa, Poland

(Communicated by Witold Respondek)

Abstract. We study dynamic pairs (X,V) where X is a vector field on asmooth manifold M and V ⊂ TM is a vector distribution, both satisfying

certain regularity conditions. We construct basic invariants of such objects and

solve the equivalence problem. In particular, we assign to (X,V) a canonicalconnection and a canonical frame on a certain frame bundle. We compute

the curvature and torsion. The results are applied to the problem of time

scale preserving equivalence of ordinary differential equations and of Veronesewebs. The framework of dynamic pairs (X,V) is shown to include sprays,

control-affine systems, mechanical control systems, Veronese webs and other

structures.

1. Introduction. In this paper we study pairs (X,V) where X is a vector field ona smooth manifold M and V ⊂ TM is a vector distribution (a sub-bundle of thetangent bundle TM), both satisfying certain regularity conditions. Such pairs willbe called dynamic pairs. They appear to encode a large variety of geometric objectsas geodesic sprays on manifolds with affine connections, spray spaces, systems ofordinary differential equations, control systems, Veronese and Kronecker webs.

The main aim of the article is to study general geometric objects attributed tothe pair (X,V) and identify the invariants. We test the proposed approach on somesimpler classes of dynamic pairs, those defined by systems of ordinary differentialequations of order ≥ 2 and a class of dynamic pairs describing Veronese webs. Thisis done in the second part of the article, where we solve the corresponding equiva-lence problems. In future papers we intend to show applications of the invariantsconstructed here to general control systems as well as to study properties of so-lutions to systems of higher order ODEs. Applications to Kronecker webs werealready given in [32]. The present paper is an extended version our preprint [26].

2010 Mathematics Subject Classification. Primary: 34C14, 53A55; Secondary: 58A30.Key words and phrases. Vector field, distribution, ordinary differential equations, semispray,

control system, G-structure, equivalence, connection, invariants.Research supported by the Polish Ministry of Science and Higher Education, grant N201 607540.

85

http://dx.doi.org/10.3934/jgm.2013.5.85

86 BRONIS LAW JAKUBCZYK AND WOJCIECH KRYNSKI

We present two classes of objects where dynamic pairs appear naturally. Acontrol system

Σ : x = X(x) +

m∑j=1

ujYj(x)

on a manifold M , where X,Y1, . . . , Ym are smooth vector fields on M and u =(u1, . . . , um)T are controls, defines a pair (X,V), where V is the distribution spannedby Y1, . . . , Ym. If the zero control is naturally distinguished (like in passive naviga-tion) and we want to study properties of the system up to “gauge” transformationsu 7→ u = H(x)u, the pair (X,V) encodes complete information about the system.In order to get insight into general properties of such systems, knowledge of basiccoordinate invariant objects associated to its dynamic pair (X,V) is crucial. Weconstruct them for systems defining regular pairs (X,V).

Regular dynamic pairs are also canonically defined by systems of second orderdifferential equations

x = F (t, x, x), (SODE)

like Euler-Lagrange equations defined by a regular Lagrangian, or by geodesic equa-tions. In this case the manifold is the extended tangent bundle M = R×TQ (treatedas a bundle M → R), the vector field X is the total derivative given in coordinatesby

XF = ∂t +

m∑j=1

xj∂xj +

m∑j=1

F j∂xj ,

and V is the vertical distribution

VF = span∂xj | j = 1, . . . ,m.

(If F does not depend on t we may take M = TQ and suppress ∂t in the definition ofXF .) Our approach, reduced to this classical case, gives (Section 4.4) two invarianttensors called Jacobi or Riemann curvature R, and Berwald curvature B (knownsince the first half of 20th century) which are independent and complete invariants.We show that if the Berwald curvature vanishes, the system (SODE) is of mechanicaltype in the sense of [8] and [35]. Namely, it is defined by an affine connection on Qand “forces” depending in affine way on velocity. If the Riemann curvature vanishes,the system corresponds to a Veronese web.

More generally, systems of higher order differential equations,

x(k+1) = F (t, x, . . . , x(k)), (ODE)

where k ≥ 1, x ∈ Rm and F is of class C∞, also define regular dynamic pairs, wherewe take M = Jk(R,Rm) - the manifold of k-jets of parametrised curves in Rm. Thevector field X is the total derivative

XF = ∂t +

k−1∑i=0

m∑j=1

xji+1∂xji

+

m∑j=1

F j∂xjk

and V is the vertical distribution

VF = span∂xjk| j = 1, . . . ,m.

A system of curvature operators, as well as a canonical connection, are obtained inthis case as part of the invariants of dynamic pairs (X,V).

VECTOR FIELDS WITH DISTRIBUTIONS 87

We outline our approach. Let (X,V) be a pair consisting of a C∞ vector field Xand a C∞ distribution V ⊂ TM of rank m on a smooth manifold M . We attach toit a sequence of distributions defined inductively, using Lie bracket, by

V0 := V, Vi+1 := Vi + [X,Vi].

We impose natural regularity conditions on (X,V), assuming that the distributionshave maximal possible ranks. More precisely, we assume that dimM = (k+1)m+1,k,m ≥ 1, and

(R1) rkVi = (i+ 1)m, for i = 0, . . . , k,(R2) Vk ⊕ spanX = TM .

Our formalism also works, with slight modifications, if dimM = (k + 1)m and thecondition (R2) is replaced with

(R2’) Vk = TM , and X(x) 6= 0, for x ∈M .

A pair (X,V) satisfying (R1) and (R2) or (R2’) is called dynamic pair or regularpair (more precisely, k-regular pair).

In Section 2 we associate canonical objects to the pair (X,V) which are: normalgenerators of V, canonical splitting of the tangent bundle, curvature operators andcanonical connection. In the case of geodesic sprays k = 1 and our curvature opera-tor K0 is equivalent to the Riemann curvature (Section 2.3). This curvature is alsowell known in the theory of second order ODEs and appears in the generalized Ja-cobi equation for sprays (see [36], Chapter 8 and references therein). Our formalismapplied to Hamiltonian systems (Section 2.3) gives a symmetric curvature operatorK0 which is equivalent to the curvature introduced in a different way by Agrachevand Gamkrelidze [2] and used for estimation of conjugate points [3]. Curvatureoperators given by explicit formulae in Section 2.4 (cf. also [24]) seem to be neweven in the case of ODEs.

In Section 3 we solve the equivalence problem, where two pairs (X,V) and (X ′,V ′)are called (locally) equivalent if there is a (local) diffeomorphism Φ: M →M suchthat Φ∗X = X ′ and Φ∗V = V ′. Following [30] we use normal generators to showthat there is a natural class of frames in TM (called normal frames), attached inan invariant way to a regular pair (X,V). This class of frames defines a canonicalG-structure P on M (where G = GL(m)) having the following property.

Two regular pairs (X,V) and (X ′,V ′) are locally equivalent if and only if thecorresponding G-structures are isomorphic.

In order to solve the equivalence problem in a more explicit way we use theprincipal G-bundle P , defined by the G-structure, and define a canonical frame onP . Our main result in Section 3 (Theorem 3.6) says the following.

Two regular pairs (X,V) and (X ′,V ′) are locally equivalent if and only if thecorresponding canonical frames are.

This theorem gives a tool for solving the equivalence problem, since there is astandard procedure for determining if two frames are equivalent. Additionally, thecanonical frame enables us to define a principal connection on P . The connectionpulled back to TM gives the connection defined in Section 2. The components ofthe curvature and torsion of the connection on P are invariants of the equivalence.

In Section 4 we apply the general formalism to study invariants of ODEs undertime scale preserving equivalence. The corresponding geometric structure in the jetspace is determined by an affine distribution, a more subtle object then the usualCartan distribution. We show (Theorem 4.5) that ODEs correspond to regular


pairs with integrable distributions V0, . . . ,Vk. We say that such pairs (X,V) are ofequation type. The invariants defined by the curvature and torsion of such pairshave a simpler structure than in the general case.

Note that the equivalence problem for ODEs is a classical one and was mainlyattacked using contact or point transformations (see [9, 11, 5, 19, 17, 18, 22] for avery partial list of contributions). We do not address this more complicated versionof the problem, which gives less hopes for a simple and complete solution. Timescale preserving equivalence problem was less studied, even if it is more natural fromthe point of view of applications. The problem was formally solved by Chern [12](systems of order two) and [13] (systems of higher order), using Cartan’s methodof equivalence (his second paper was totally ignored in the literature). The sameinvariants were also found by Kosambi [28, 29] and discussed in Cartan [10]. Evenif it was shown that the Cartan method works, the independence and meaning ofinvariants were not fully analyzed for higher order systems. In the case of SODE,the invariants were recently used to study stability of solutions [4, 41].

In our formalism an ODE is treated as a pair (X,V) with a flag of integrabledistributions. This approach can be interpreted, roughly, as dual to Chern’s [13]. Inthe cases of systems of order two and equations of order three we provide relationsbetween the invariants and consequences of their vanishing (Sections 4.4 and 4.5).The geometry of systems of second order was also extensively analysed from thepoint of view of Lagrangian mechanics and the inverse problem of the calculus ofvariations (see [15, 14] and references therein). Natural connections on the tangentbundle, corresponding to SODE, were defined (cf. e.g. [23], [15], [33], [36]). Ourcanonical connection defined in Section 2.5, specialized to SODE, corresponds to aconnection defined in [6], Theorem 2.1. A related theory of connections for higherorder systems of ODEs can be found in [6, 34] and a recent paper [7].

In Section 5 we study mechanical control systems in the sense of Bullo andLewis [8] and Ricardo and Respondek [35]. We show that, roughly, dynamic pairsof equation type with vanishing Berwald curvature are equivalent to almost me-chanical systems and, under an additional condition, to mechanical systems. Ourequivalence, called control gauge equivalence, is weaker then the state equivalenceused by Ricardo and Respondek [35], but it is stronger then feedback equivalence.

In Section 6 we study Veronese webs, i.e. one-parameter families of foliations ofspecial type, introduced by Gelfand and Zakharevich [21] and strongly related tobi-Hamiltonian systems. We prove that Veronese webs can be treated as ODEs forwhich our curvature operators vanish. In this way we relate Veronese webs withdynamic pairs and we solve the equivalence problem (c.f. Gelfand and Zakharevich[21, 42] and Turiel [38, 39]). It appears that Veronese webs play the same role fortime-scale preserving contact transformations as paraconformal structures (calledalso GL(2)-structures) play for the general contact transformations [5, 18, 22, 31].In this context the curvature operators play the role of the Wunschmann invariantsfor contact transformations.

2. Dynamic pairs.

2.1. Dynamic pairs (X,V) and their normal frames. Let M be a smoothdifferentiable manifold of dimension n. If E is a smooth vector bundle over M thenΓ(E) will denote the set of all smooth sections of E .

Consider a C∞ dynamic pair (X,V), i.e., a vector field X on M and a distributionV ⊂ TM of rank m, satisfying the regularity conditions (R1) and (R2). Then V(x)


is an m-dimensional subspace of the tangent space TxM , for every x ∈ M and,locally, there exist m smooth vector fields V1, . . . , Vm such that

V(x) = spanV1(x), . . . , Vm(x) .If V1, . . . , Vm span V on an open subset U ⊂M then we will briefly denote them asthe row vector V = (V1, . . . , Vm) and call V a local frame of V on U .

Recall that the pair (X,V) defines a sequence of distributions

V = V0 ⊂ V1 ⊂ · · · ⊂ Vk ⊂ TM,

where

Vi(x) = spanadsXY (x) | 0 ≤ s ≤ i, Y ∈ Γ(V).Above adXY = [X,Y ], ad2

XY = [X, [X,Y ]] etc., where [X,Y ] denotes the Liebracket of vector fields. Denote by X the 1-dimensional distribution spanned by X.

Definition 2.1. A local section Y of V is called normal if

adk+1X Y = 0 mod Vk−1 ⊕X . (1)

A local frame V = (V1, . . . , Vm) of V on U ⊂ M is called normal frame of V if allVi are normal sections of V.

Note that it follows from (R1) and (R2) that the vector fields X and adiXVj(x),0 ≤ i ≤ k, 1 ≤ j ≤ m, are linearly independent and they span TM . Thus, denoting

adiXV = (adiXV1, . . . , adiXVm)

we have

adk+1X V = V H0 + (adXV )H1 + · · ·+ (adkXV )Hk mod X , (2)

where Hi are m×m matrices of functions. Then V is a normal frame iff Hk = 0.Let V = (V1, . . . , Vm) and W = (W1, . . . ,Wm) be local frames of V on U . Then

there exists a unique GL(m)-valued function G : U → GL(m) such that, in thematrix notation,

W = V G. (3)

Proposition 2.2. (a) Given a regular pair (X,V) and a frame Vx0in V(x0), there is

a normal frame V = (V1, . . . , Vm) of V in neighborhood of x0 such that V (x0) = Vx0.

(b) If V and W are two normal frames of V on U then equation (3) holds, whereG : U → GL(m) satisfies

X(G) = 0. (4)

(c) If V is a frame of V on U and Hk is defined via (2), then W = V G is a normalframe of V on U if and only if

X(G) = − 1

k + 1HkG. (5)

Proof. (a) Let V = (V1, . . . , Vm) be a local frame of V and let V (x0) = Vx0. We

will find functions G = (Gji )i,j=1,...,m such that Wi =∑mj=1G

jiVj , for i = 1, . . . ,m,

are the desired vector fields, i.e.,

adk+1X W = 0 mod Vk−1 ⊕X . (6)

In the matrix notation W = V G, thus [X,W ] = [X,V G] = [X,V ]G+ V X(G) and,

inductively, adiXW = adiX(V G) =∑ij=0

(ij

)adjX(V )Xi−j(G). This implies

adk+1X W = (adk+1

X V )G+ (k + 1)(adkXV )X(G) mod Vk−1.


It follows from (R1) and (R2) that (2) holds, thus

adk+1X V = (adkXV )H mod Vk−1 ⊕X , (7)

for a certain square matrix Hk = H = (Hji ). Since adkXW = (adkXV )G mod Vk−1,

equation (6) is equivalent to the following equation for the unknown function G:

HG+ (k + 1)X(G) = 0. (8)

This is a linear first order differential equation for G, thus it can be solved, locally,so that G(x0) = Id. If G is a solution, we have adk+1

X W = 0 mod Vk−1 ⊕ X andW (x0) = V (x0)Id = Vx0

.(b) As above, we have W = V G. Since all elements Y = Vi of V satisfy (1),

the matrix H = Hk in equation (7) is zero. Equation (6) implies that the matrixvalued function G satisfies (8), thus X(G) = 0.

(c) This follows from the proof of (a), as H = Hk and equations (5) and (8)coincide.

Note that the “initial condition” V (x0) = Vx0in statement (a) can be imposed

on any local hypersurface transversal to X in M . From statement (b) we get

Corollary 2.3. If V and W are normal frames of V on U ⊂ M then W = V Gand, for all i,

adiXW = adiX(V G) = (adiXV )G. (9)

It will be convenient to have a method of computing normal generators of Vwithout solving the differential equation for G in Proposition 2.2 (c). We will dothis in Section 2.4.

2.2. Canonical splitting and curvature operators. Corollary 2.3 says that ifboth V and W = V G are normal frames of V then adiXW = (adiXV )G for any i. Itfollows that distributions

Hi = spanV i1 , . . . , V im, with V ij := adiXVj ,

do not depend on the choice of a normal frame V = (V1, . . . , Vm), for i = 1, . . . , k.Hi will be called i-th quasi-connection, for reasons which will become clear in thesecond part of the paper. We also denote H0 = V and stress that rkHi = m.Condition (R1) implies

Vi = V ⊕H1 ⊕ · · · ⊕ Hi,for i = 1, . . . , k. Condition (R2) gives the following splitting of the tangent bundle

TM = V ⊕H1 ⊕ · · · ⊕ Hk ⊕X

called canonical splitting defined by (X,V). The splitting defines canonical projec-tions

π0 : TM → V and πi : TM → Hi, i = 1, . . . , k.

The canonical splitting can be computed using a usual (not normal) basis V =(V1, . . . , Vm) of V, see Section 2.4.

We will define the most basic invariants of dynamic pairs, called curvature oper-ators. Before doing this we introduce the operators

Ai : V → Hi, Ai = πi adiX , 0 ≤ i ≤ k.


They are vector bundle morphisms, i.e. for Y ∈ Γ(V) and x ∈ M the valueAi(Y )(x) = πi[X, [X, . . . , [X,Y ] · · · ]](x) depends on Y (x) only. The regularity con-ditions imply that all Ai are isomorphisms and, in particular, A0 = Id : V → V.Similarly

πi adX : Hk → Hi, 0 ≤ i ≤ k − 1,

are bundle morphisms, but not necessarily isomorphisms.

Definition 2.4. An i-th curvature operator Ki ∈ End(V) of a dynamic pair (X,V)is

Ki = (−1)i(Ai)−1 πi adX Ak : V → V, 0 ≤ i ≤ k − 1.

The alternating sign is chosen for consistency with classical interpretations ofcurvatures and, in particular, for simplicity of the variational equation [24].

Equivalently, Ki can be defined as follows. Let x ∈ M . In a fixed basis of thespace V(x) the operator Ki(x) is represented by m×m matrix, also denoted Ki(x).If V = (V1, . . . , Vm) is a normal frame, then matrices of the curvature operators aredefined in the basis V1(x), . . . , Vm(x) of V(x) by the following equation:

adk+1X V + (−1)k−1(adk−1

X V )Kk−1 + · · · − (adXV )K1 + V K0 = 0 mod X . (10)

Due to the transformation rule (9), when a normal frame V is transformed to thenormal frame V G, the matrices Ki are transformed via

Ki 7−→ G−1KiG. (11)

The operatorsKi(x) : V(x)→ V(x)

are invariantly assigned to the dynamic pair (X,V).If the generators V are not normal, the formula (2) holds. The linear operators

defined by the matrices Hi(x) are not invariantly assigned to the pair. However,the curvature matrices can be computed in terms of matrices Hi. In particular, wehave

Proposition 2.5. If k = 1 and ad2XV = (adXV )H1 + V H0 then

K0 = −H0 +1

2X(H1)− 1

4H2

1 (12)

and

H1 = span

adXV −

1

2V H1

. (13)

Proof. We use statement (c) in Proposition 2.2, which states that W = V G is anormal frame of V, if X(G) = − 1

2H1G. Then X(X(G)) = − 12X(H1)G + 1

4H21G.

Since ad2X(V G) = (ad2

XV )G+2(adXV )X(G)+V X2(G), using the assumed formulafor ad2

XV and eliminating X(G) and X2(G) gives

ad2X(V G) = V GG−1

(H0 −

1

2X(H1) +

1

4H2

1

)G.

This leads to the formula for K0 in the normal frame W = V G, according to (10)(we may assume that G(x) = Id, at a fixed point x). The formula for H1 followsfrom adX(V G) = (adXV )G+ V X(G) and is a special case of Proposition 2.8.

The curvature matrix K0 can be used for estimating the existence and locationof conjugate points of a control system or a system of second order ODEs, see [25].

For k > 1 formulae for the curvature matrices in an arbitrary basis of V areprovided by Proposition 2.9.


2.3. Examples. Geodesic spray. Consider a geodesic equation on a manifold Nof dimension m with local coordinates (xi). In local coordinates we have

(xi)′′ = −m∑

p,q=1

Γipq(xp)′(xq)′,

where Γipq are Christoffel symbols of a connection ∇. Note that the equation doesnot depend on the torsion of the connection and thus we will assume that ∇ issymmetric. Let J2(R, N) be the space of 2-jets of curves R → N . On J2(R, N)there are local coordinates (t, xi0, x

i1, x

i2), induced by the coordinates (xi), where

i = 1, . . . ,m. The geodesic equation is uniquely defined by the submanifold

E =

(t, xi0, x

i1, x

i2) | xi2 = −

∑p,q

Γipqxp1xq1

⊂ J2(R, N).

There is a canonical projection J2(R, N)→ TN ×R which, restricted to E, definesthe diffeomorphism E ' TN ×R. In particular (xi0) = (xi) are local coordinates onN whereas (xi1) are the corresponding linear coordinates on the fibres of TN → N .

Let

X = ∂t +∑i

xi1∂xi0−∑i,p,q

Γipqxp1xq1∂xi

1

be the total derivative. We take the vertical distribution tangent to the fibres ofTN , that is V = span∂xi

1| i = 1, . . . ,m. Clearly, conditions (R1) and (R2) are

satisfied for such a pair (X,V), with the parameter k = 1. Direct computations give

adX∂xi1

= −∂xi0

+∑j,p

2Γjipxp1∂xj

1

ad2X∂xi

1=

∑j,p

2Γjipxp1adX∂xj

1

+∑j,p,q

(2∂xq

0(Γjip)x

p1xq1 − 2

∑r

ΓjirΓrpqx

p1xq1 − ∂xi

0(Γjpq)x

p1xq1

)∂xj

1,

which determines matrices H0 and H1 in (2). Since k = 1, there is only onecurvature operator K0. From Proposition 2.5 we get

K0 =

(∑p,q,r

(∂xi

0(Γjpq)− ∂xq

0(Γjip) + ΓjirΓ

rpq − ΓripΓ

jrq

)xp1x

q1

)i,j=1,...m

.

We see that K0 is a quadratic function in the coordinates on fibres xp1, xq1 and we

recognise that the coefficients Rjiqp = ∂xi0(Γjpq)−∂xq

0(Γjip)+

∑r(Γ

jirΓ

rpq−ΓripΓ

jrq) are

components of the curvature tensor R of the connection ∇. Denoting Y = y = (xi1)and x = (xi0), we get

K0(x, y)(Z) = Rx(Z, Y )Y.

A simple calculation using Bianchi identity for R gives the converse formula

Rx(Y,Z)W =1

3(K0(x, z + w)(Y )−K0(x, y + w)(Z)

−K0(x, z)(Y )−K0(x,w)(Y ) +K0(x, y)(Z) +K0(x,w)(Z)),

where x = (t, (xi0)) and we identify the elements y = Y, z = Z,w = W . We also

have H1 = span−∂xi

0+∑j,p Γjipx

p1∂xj

1

. It follows that the quasi-connection H1


coincides, in the Ehresmann sense, with the connection ∇. More general cases willbe considered in Chapter 3.

ODE with constant coefficients. Consider the following linear system of ODEsgiven in the Halphen normal form

x(k+1) +Bk−1x(k−1) + · · ·+B1x

′ +B0x = 0.

where x = (x1, . . . , xm)T ∈ Rm and Bi ∈ Rm×m are constant matrices. Let Jk(1,m)be the space of k-jets of functions from R to Rm. The standard global coordinatefunctions on Jk(1,m) are denoted (t, xji ), where i = 0, . . . , k and j = 1, . . . ,m. Weset xi = (x1

i , . . . , xmi )T . Denote (F 1, . . . , Fm)T = −(Bk−1xk−1 + · · ·+B1x1 +B0x0)

and let

X = ∂t +

k∑i=1

m∑j=1

xji∂xji−1

+

m∑j=1

F j∂xjk

be the corresponding total derivative. Taking V = span∂xjk| j = 1, . . . ,m one

can prove that (∂x1k, . . . , ∂xm

k) is a normal frame of the pair (X,V). Moreover, a

simple induction shows that the corresponding curvature operators are given by thematrices Ki = (−1)iBi. It follows that in the case of linear equations the curvatureoperator Ki coincides (up to a sign) with Wilczynski semi-invariant Pi [40].

Hamiltonian vector field. The dynamics of a Hamiltonian vector field in thepresence of a Lagrangian foliation on a symplectic manifold was extensively inves-tigated by Agrachev and his coworkers (see [1, 2, 3] and references therein). Itssimple version can also be described in our framework, assuming regularity.

Let M = R×N , where N is a symplectic manifold of dimension 2m with sym-plectic form σ. Let H be a time dependent Hamiltonian vector field on N and, thus,a vector field on M . Take X = ∂t + H, where t is the canonical time coordinateon R, and on R×N . Let Vt ⊂ TN , t ∈ R, be a family of Lagrangian distributionson N , i.e., dimVt = m and σ(v, w) = 0 for any vectors v, w ∈ Vt. Denote by Vthe resulting distribution on M = R × N . Assume that (H,V) is regular, that isspanVt, [H,Vt]x = TxN , for any (t, x) ∈M . The pair (X,V) is a regular dynamicpair on M with k = 1. Given a normal frame V = (V1, . . . , Vm) in V, we have

[X, [X,Vi]] = [H, [H,Vi]] = −∑j

(K0)jiVj , (14)

where K0 is the curvature matrix defined according to formula (10): ad2XV = −V K0

(the coefficient at X in (10) vanishes because of the term ∂t in X). One can check,consulting [1], that the above curvature K0 coincides with the curvature introducedin a different way in [2]. The following proposition is easy to prove.

Proposition 2.6. (a) The distribution

H1 = span[H,V1], . . . , [H,Vm]

is Lagrangian with respective to σ, on each fiber t ×N .(b) The matrix g = (gij) given by

gij(x) = σ([H,Vi], Vj)(x), x ∈M = R×N, i, j = 1, . . . ,m.

is symmetric, nondegenerate, and defines a pseudo-Riemannian metric on V.(c) The matrix K0 is symmetric and defines a selfadjoint (relative to g) operatorK0 : V → V.


A canonical example of the pair as above is a time dependent vector field H onthe cotangent bundle N = T ∗N of a differentiable manifold N , where Vt is thevertical distribution of the bundle π : T ∗N → N , i.e., Vt(x) = Tπ(x)N .

Another example (see [2]) is given by a Hamiltonian function h : T ∗N → R. Thecorresponding Hamiltonian vector field H and the vertical distribution never forma regular pair with k = 1 since the dimension of M = T ∗N can not be equal to2m+ 1. However, we can have a regular pair if we restrict our considerations to alevel submanifold of the Hamiltonian. Namely, take M = h = c ⊂ T ∗N and thevector field X equal to H restricted to M , X = H|M . The distribution V on Mis defined as the vertical distribution of the cotangent bundle intersected with thetangent space to M : V(x) = T (T ∗π(x)N)∩TxM , for x ∈M . Then, if dim N = m+1,

we have dimV(x) = m, dimM = 2m+ 1 and assuming regularity of the pair (H,V)makes sense (typical examples are regular). In this case the equality in formula (14)holds, modulo H, and all statements of Proposition 14 hold true, for the canonicalsymplectic form σ on T ∗N replaced with the symplectic form σ = σ|M consideredon the quotient space TxM/spanH(x).

2.4. Explicit formulae. Given a dynamic pair (X,V), we will present explicit for-mulae for normal generators W = (W1, . . . ,Wm) of V, semiconnections H1, . . . ,Hk,operators Ai : V → Hi and curvature matrices K0, . . . ,Kk−1, assuming that a fixedlocal frame V = (V1, . . . , Vm) in V is chosen.

Let Hk be the matrix defined by formula (2) or (7). Denote by D and D thedifferential operators acting on the space m×m matrices of functions on M ,

D = X − 1

k + 1RHk

, D = X +1

k + 1LHk

(15)

where RH and LH denote the operators of right and left multiplication by H = Hk.Let Id denote the m×m identity matrix. Define the sequences of matrices

Ji = Di(Id), Ji = Di(Id), i ≥ 0, (16)

and Ji = 0 = Ji, for i < 0. In particular,

J0 = Id, J1 = − 1

k + 1Hk, J2 = − 1

k + 1X(Hk) +

1

(k + 1)2H2k .

Using statement (c) of Proposition 2.2 we can prove the following

Proposition 2.7. Given a frame V = (V1, . . . , Vm) of V, if W = V G is a normalframe then

Xi(G) = JiG, Xi(G−1) = G−1Ji, (17)

and

adiXW =

i∑j=0

(i

j

)(adjXV )Ji−jG, (18)

adiXV =

i∑j=0

(i

j

)(adjXW )G−1Ji−j . (19)

Proof. We will use the equality X(G) = −1k+1HkG, cf. (5). The first formula in (17)

holds trivially for j = 0. If it holds for j = s then it also holds for s+ 1. Namely,

Xs+1(G) = X(Ds(Id)G) = (X(Ds(Id)))G+Ds(Id)X(G)


and, as X(G) = −1k+1HkG, we get Xs+1(G) = (X− 1

k+1RHk)Ds(Id)G = Ds+1(Id)G.

The second formula in (17) can be proved in the same way, taking into account thatX(G−1) = −G−1X(G)G−1 = G−1 1

k+1Hk, by (5).

Equality (18) is a consequence of the first formula in (17), since from W = V Gand the Leibnitz rule we have

adiXW =

i∑j=0

(i

j

)(adjXV )Xi−j(G).

Analogously, equality (19) is a consequence of the second formula in (17).

The semiconnections Hi and the operators Ai : V → Hi can be computed asfollows.

Proposition 2.8. Given a frame V = (V1, . . . , Vm) of V, the operators Ai : V → Hiare given on the generators Vp by the formula AiVp = V ip , where

V ip =

i∑j=0

m∑q=1

(i

j

)(adjXVq)(Ji−j)

qp, p = 1, . . . ,m,

or in the matrix form,

V i := (AiV1, . . . , AiVm) =

i∑j=0

(i

j

)(adjXVq)Ji−j . (20)

The semiconnections Hi are then given by Hi = spanV i1 , . . . , V im, i = 1, . . . , k.

Proof. Fix a point x in the domain of definition of the frame V . By Proposition 2.2we can choose a matrix G, defined in a neigbourhood of x and such that G(x) = Id,so that W = V G is a normal frame of V. Then W (x) = V (x) and Ai(x)Vp(x) =

Ai(x)Wp(x) = (adiXWp)(x) = V ip (x), where in the second equality we use the

definition of Hi and the fact that W is a normal frame and in the third one we useformula (18).

Curvature matrices in arbitrary frame V = (V1, . . . , Vm) can also be computedexplicitly (cf. [24] for the case m = 1).

Proposition 2.9. If k ≥ 1 and

adk+1X V = (adkXV )Hk + · · ·+ (adkXV )H1 + V H0 + fX

then

Ki = (−1)i+1k∑j=i

(j

i

)Jj−i

((k + 1

j

)Jk+1−j +Hj

), (21)

where Js = Ds(Id) and Js = Ds(Id) are the matrices (16).

Proof. Assume that W = V G is a normal frame in V. We fix the point x0 ofconsideration and let G(x0) = Id (i.e., W (x0) = V (x0), cf. Proposition 2.2 (a)).Denote

W i = adiXW, V i = adiXV, i ≥ 0.

We should compute matrices Ki in the expansion W k+1 =∑ki=0(−1)i+1W iKi+gX,

cf. (10). Note that formula (18) implies

W k+1 =

k+1∑i=0

(k + 1

i

)V iJk+1−iG = V k+1 +

k∑j=0

(k + 1

i

)V iJk+1−iG. (22)


This equation and the expansion V k+1 =∑ki=0 V

iHi + fX give

W k+1 =

k∑i=0

V i(Hi +

(k + 1

i

)Jk+1−iG

)+ fX.

Using formula (19) for V i = adiXV we get

W k+1 =

k∑i=0

i∑j=0

(i

j

)W jG−1Ji−j

(Hi +

(k + 1

i

)Jk+1−iG

)+ fX

=

k∑j=0

k∑i=j

(i

j

)W jG−1Ji−j

(Hi +

(k + 1

i

)Jk+1−iG

)+ fX.

Since we took G(x0) = Id at the point of consideration, at this point we obtain

W k+1 =∑ki=0(−1)i+1W iKi + fX, with Ki given by the formula (2.9).

2.5. Canonical connection. In this section we will show that the splitting

TM = H0 ⊕ · · ·Hk ⊕Xtogether with the associated projections πi : TM → Hi and the isomorphisms

Ai : V → Hi,where H0 = V and Hi are quasi-connections defined by a regular pair (X,V), canbe used to construct linear connections on M . For local sections Y ∈ Γ(Hi) andZ ∈ Γ(Hj) of Hi and Hj we define

∇ijY Z = πj([Y,Z]), for i 6= j.

Then ∇ijY Z is tensorial with respect to Y and satisfies ∇ijY (fZ) = Y (f)Z + f∇ijY Z,for any smooth function f , thus it is a partial connection differentiating sections ofthe bundle Hj along vector fields in Hi.

We would like to glue different ∇ij together to get a linear connection ∇can onM . We require that ∇can preserves the splitting i.e., for any Y ∈ Γ(TM),

∇canY : Γ(Hi)→ Γ(Hi) and ∇canY : Γ(X )→ Γ(X ). (23)

Additionally, we insist that the resulting connection should be consistent with iso-morphisms Ai in the following sense

A−1i ∇

canY Ai(Z) = ∇canY Z, ∀ Y ∈ Γ(TM), ∀ Z ∈ Γ(V), (24)

which means that Ai are parallel with respect to ∇can. The meaning of condition(24) will become evident in Section 3.3.

For i, j = 0, . . . , k we define

∇canY Z = Aj∇i0Y A−1j (Z), ∀ Y ∈ Γ(Hi), ∀ Z ∈ Γ(Hj), i > 0, (25)

∇canY Z = AjA−11 ∇01

Y A1A−1j (Z), ∀ Y ∈ Γ(H0), ∀ Z ∈ Γ(Hj). (26)

We also define

∇canY Z = πi([Y, Z]), ∀ Y ∈ Γ(X ), ∀ Z ∈ Γ(Hi), (27)

∇canY X = 0, ∀ Y ∈ Γ(TM). (28)

These formulae define a unique connection on M . The connection will be calledcanonical connection. Clearly, ∇can satisfies conditions (23) and (24). (If Y ∈ Γ(X ),condition (24) follows from properties of connections and the fact that ∇canX Z = 0for any normal vector field Z ∈ V.)


This connection can also be characterized by certain conditions on its torsion.Let us recall that torsion tensor of a connection ∇ is defined as follows

T (∇)(Y1, Y2) = ∇Y1Y2 −∇Y2Y1 − [Y1, Y2],

and it is (2,1)-tensor. In our case, the splitting TM = H0⊕· · ·Hk⊕X decomposesany tensor into components taking values in different components of the splitting.In particular, we denote

T (∇)ijl = T (∇)|Hi⊕Hj⊕(Hl)∗ , T (∇)il = T (∇)|Hi⊕X⊕(Hl)∗ .

Further, using isomorphisms Ai : V → Hi we can treat the components T (∇)ijl and

T (∇)il as tensors on V. Precisely, for Y,Z ∈ Γ(V), we define

T (∇)ijl (Y,Z) = A−1l T (∇)ijl (Ai(Y ), Aj(Z))

T (∇)il(Y ) = A−1l T (∇)il(Ai(Y ), X)

Now, we are able to characterize the canonical connection ∇can defined by (25)-(28) in terms of its torsion.

Theorem 2.10. Let (X,V) be a regular pair on a manifold M . Then ∇can is theunique connection on M which preserves the canonical splitting, satisfies (24), (28)and

T (∇)kk = 0, T (∇)011 = 0, T (∇)0i

0 = 0, i = 1, . . . , k. (29)

Remark 2.11. The situation here can be compared to the situation in Riemanniangeometry. In our case we have a frame bundle with the structure group GL(m),whereas in the Riemannian case the structure group is O(n). The compatibilityconditions (23) and (24) for the connection are analogs of metricity of Levi-Civitaconnection. In the present case we can normalize only a part of the torsion, whereasin the Riemannian case all torsion can be eliminated.

The coefficients (Christoffel symbols) of ∇can can be explicitly computed.

Proposition 2.12. Given a local basis V = (V1, . . . , Vm) of V, let V 0p = Vp and

V ip = AiV0p be the vector fields described explicitly in Proposition 2.8. The con-

nection ∇can is uniquely determined by Christoffel symbols Γirpq, Γrq defined by therelations

∇V ipV 0q =

∑r

ΓirpqV0r , ∇XV 0

q =∑r

ΓrqV0r , (30)

where p, q, r ∈ 1, . . . ,m and i = 0, . . . , k. Namely, ∇V ipX = 0 and, for j =

1, . . . , k,

∇V ipV jq =

∑r

ΓirpqVjr , ∇XV jq =

∑r

ΓrqVjr .

The symbols Γirpq, Γrq are equal to the coefficients

Γ0rpq = b1rpq1, Γirpq = −birqp0, i > 0, Γrq = Hr

q (31)

of the commutator relations

[X,V kj ] =∑s

Hsj V

ks mod Vk−1 ⊕X , (32)

[V 0p , V

lq ] =

∑i,j

bljpqiVij mod X . (33)


Proof. The first statement is a consequence of the properties (23) and (24) of ∇can.The first two equalities in (31) follow from the definition (33) of the coefficients

bljpqi and from the definition (25), (26) of ∇can (with j = 0). The third equality

in (31) follows from the fact that (32) can be written as ∇canX V kj =∑sH

sj V

ks

which is equivalent to ∇canX V 0j =

∑sH

sj V

0s . The latter follows from the equalities

V kj = AkV0j and the commutation property ∇canAk = Ak∇can.

Before we prove Theorem 2.10, we will discuss other possible choices for canonicallinear connections assigned to a dynamical pair (X,V). We first note that thecomposition Aj2 A−1

j1is a well defined isomorphism of Hj1 and Hj2 . Therefore, for

arbitrary i, j1 ∈ 0, . . . , k and for j2 6= i the formula

(Y,Z) 7→ Aj1A−1j2∇ij2Y Aj2A

−1j1

(Z), Y ∈ Γ(Hi), Z ∈ Γ(Hj1),

defines a new partial connection on the bundle Hj1 in the direction of Hi and the

definition is also valid for i = j1. So defined partial connection ∇Y Z satisfies (24).Now we are able to define other connections invariantly assigned to (X,V). They

will be parameterized by mappings p : 0, 1, . . . , k 7→ 0, 1, . . . , k having no fixedpoints. We fix such p and define

∇can(p)Y Z = AjA

−1p(i)∇

ip(i)Y Ap(i)A

−1j (Z), ∀ Y ∈ Γ(Hi), ∀ Z ∈ Γ(Hj). (34)

This formula allows to differentiate an arbitrary section of H0 ⊕ · · · ⊕ Hk in anarbitrary direction in H0⊕· · ·⊕Hk. The definition of differentiation in the directionof the vector field X does not depend on p,

∇can(p)X Z = πi([X,Z]), ∀ Z ∈ Γ(Hi). (35)

We also define

∇can(p)Y X = 0, ∀ Y ∈ Γ(TM). (36)

Again, it will be explained in Section 3.3 why these definitions are natural from thepoint of view of the geometry of (X,V). At this point we state the following resultwhich follows directly from the definitions and concludes the above constructions.

Theorem 2.13. If (X,V) is a regular pair on a manifold M then for any p theequations (34)-(36) define a linear connection on M such that:(a) ∇can(p) preserves the splitting TM = H0 ⊕ · · ·Hk ⊕X , i.e.

∇can(p)Y : Γ(Hi)→ Γ(Hi), ∇can(p)

Y : Γ(X )→ Γ(X ), ∀ Y ∈ Γ(TM). (37)

(b) The operators Ai : V → Hi are parallel with respect to ∇can(p), i.e.

∇can(p)Y Ai(Z) = Ai(∇can(p)

Y Z), ∀ Y ∈ Γ(TM), ∀ Z ∈ Γ(V). (38)

(c) The vector field X is parallel with respect to ∇can(p), i.e.

∇can(p)Y X = 0, ∀ Y ∈ Γ(TM). (39)

(d) If V = (V1, . . . , Vm) is a normal frame of V then ∇can(p)X Vi = 0 for any i.

The connections ∇can(p) are not all possible connections on M satisfying state-ments (a)-(d) of the theorem. For instance one can also take a linear combination oftwo connections from the family. The freedom of possible choices will be discussedin a systematic manner in Section 3.4.


Remark 2.14. Note that the canonical connection ∇can is ∇can(p) with

p(0) = 1, p(1) = 0, p(2) = 0, . . . , p(k) = 0. (40)

∇can is canonical in the sense that if two dynamic pairs (X,V) and (X ′,V ′) areequivalent (by means of a diffeomorphism) then the corresponding connections∇canand (∇′)can are also equivalent. The other connections ∇can(p) have the same prop-erty. The canonical connection ∇can was chosen for further use only for simplicity.

Proof of Theorem 2.10. First of all we are going to prove by direct computationthat the canonical connection ∇can satisfies (29) indeed. We will use the definitionof ∇can. We fix a normal frame (V1, . . . , Vm) of V. Then

T (∇can)kk(Vp) = A−1k πk(∇canAk(Vp)X −∇

canX Ak(Vp)− [Ak(Vp), X]) = 0

because ∇canX Ak(Vp) = Ak(∇canX Vp) = 0 by statements (b) and (d) of Theorem 2.13,

∇canAk(Vp)X = 0 by the definition of ∇can and πk[Ak(Vp), X] = −πk(adk+1X Vp) = 0 by

the definition of normal frame. Similarly

T (∇can)011 (Vp, Vq) = A−1

1 π1(∇canVpA1(Vq)−∇canA1(Vq)Vp − [Vp, A1(Vq)]) = 0

because ∇canVpA1(Vq) = π1[Vp, A1(Vq)] by the definition of ∇can, and ∇canA1(Vq)Vp ∈ V

i.e. π1(∇canA1(Vq)Vp) = 0 by statement (a) of Theorem 2.13. Analogously

T (∇can)0i0 (Vp, Vq) = π0(∇canVp

Ai(Vq)−∇canAi(Vq)Vp − [Vp, Ai(Vq)]) = 0

because ∇canAi(Vq)Vp = −π0[Vp, Ai(Vq)]] by definition of ∇can, and ∇canVpAi(Vq) ∈ Hi

i.e. π0(∇canVpAi(Vq)) = 0 for i > 0.

Now we will prove that if ∇ is another connection on M satisfying (23), (24)and (29) then ∇ = ∇can. It is sufficient to show that ∇Y and ∇canY coincide on V,because both ∇ and ∇can uniquely extend from V = H0 to all Hi, i > 0, in such away that (24) is satisfied. (Note that condition (28) implies ∇YX = 0 for any Y .)In the considerations we use a fixed frame V = (V1, . . . , Vm) of V.

From the fact that T (∇)0i0 (Vp, Vq) = π0(∇Vp

Ai(Vq)−∇Ai(Vq)Vp−[Vp, Ai(Vq)]) = 0

and from ∇VqAi(Vp) ∈ Hi it follows that ∇Ai(Vq)Vp = −π0[Vp, Ai(Vq)]]. Hence

∇Ai(Vq)Vp = ∇canAi(Vq)Vp, i = 1, . . . , k.

From the fact that T (∇)011 (Vp, Vq) = A−1

1 π1(∇VpA1(Vq)−∇A1(Vq)Vp−[Vp, A1(Vq)]) =

0 and from the assumptions ∇A1(Vq)Vp ∈ V, ∇VqA1(Vp) ∈ H1 it follows that

∇VpA1(Vq) = π1[Vp, A1(Vq)]. Applying A−1

1 to both sides we get

∇VpVq = ∇canVpVq.

Finally, the equality T (∇)kk(Vp) = A−1k πk(∇Ak(Vp)X −∇XAk(Vp)− [Ak(Vp), X]) =

0 and the assumption ∇Ak(Vp)X = 0 imply that ∇XAk(Vp) = πk[X,Ak(Vp)] =

∇canX Ak(Vp). Hence ∇XAk(Vp) = ∇canX Ak(Vp) and applying A−1k to both sides

gives∇XVp = ∇canX Vp.

Summarizing, for arbitrary section Y ∈ Γ(TM) and any p = 1, . . . ,m we have∇Y Vp = ∇canY Vp which finishes the proof.


3. Equivalence problem and invariants.

3.1. Canonical bundle. In order to identify further invariants of dynamic pairs,and solve the equivalence problem, we introduce a sub-bundle FN = FN (X,V) ofthe tangent frame bundle, called canonical or normal frame bundle of regular pair(X,V).

Definition 3.1. Fix x ∈M . A frame Fx in TxM of the form

Fx = (V 0, V 1, . . . , V k, X(x)), where V i = (V i1 , . . . , Vim),

is called normal at x if there is a local normal frame V = (V1, . . . , Vm) in V such

that V ij = (adiXVj)(x). Equivalently, Fx is called normal if there is a local frame Vof V, possibly not normal, such that

V 0 = V (x), V i = AiV (x) = (AiV1(x), . . . , AiVm(x)), i = 1, . . . , k, (41)

where the operators Ai = πi adiX : V → Hi are the vector bundle isomorphismsfrom Section 2.2. Both definitions coincide since they coincide for a local normalframe V and, given the pair (X,V), the second one depends on the value V (x),only. A local normal frame in TM is a smooth local field of normal frames x 7→ Fx.

Denote by FN (x) the set of all normal frames in TxM . The set

FN = FN (X,V) :=⋃x∈MFN (x)

forms a bundle over M , called normal frame bundle corresponding to the pair (X,V)or canonical bundle of (X,V). This is a smooth sub-bundle,

FN ⊂ F ,of the bundle π : F →M of all frames on M .

There is a natural right action of GL(m) on FN given by

(Fx, G) 7−→ RGFx := (V 0G,V 1G, . . . , V kG,X(x))

where G ∈ GL(m) and V iG =(∑m

j=1Gj1V

ij , . . . ,

∑mj=1G

jmV

ij

). This action is

briefly denoted

RGFx = FxG, where G = diag(G,G, . . . , G, 1).

With respect to this action FN is a principal GL(m)-bundle. This means thatGL(m) acts transitively and freely on each fiber FN (x). To check this pick twonormal frames F(x) and F′(x) at x, given by local normal frames V and V ′ of V.Then, by Proposition 2.2, there is a matrix valued function x 7→ G(x) such that

V = V ′G, where X(G) = 0. Thus adXV = (adXV′)G and, generally, adiXV =

(adiXV′)G. Thus, the corresponding local normal frames F and F′ are related by

the block diagonal matrix

G = diag(G, . . . , G, 1), F = F′G.If x ∈ M is fixed, the group of block diagonal matrices G(x) is isomorphic to thegroup GL(m), with the isomorphism given by G(x) 7→ G(x). We conclude that

Proposition 3.2. FN = FN (X,V) is a principal GL(m)-bundle and a sub-bundleof the bundle F of frames on M , i.e., a GL(m)-structure. Two regular pairs (X,V)and (X ′,V ′) are locally equivalent if and only if the corresponding canonical bundlesFN (X,V) and FN (X ′,V ′) are isomorphic as GL(m)-structures.


Proof. The second statement follows directly from the definition of FN (X,V) andfrom equivariance of the Lie bracket with respect to diffeomorphisms. Note thatany isomorphism of FN (X,V) and FN (X ′,V ′) is given by a diffeomorphism Φ ofM such that Φ∗FN (X,V) = FN (X ′,V ′) which implies Φ∗V = V ′.

The connections ∇can and ∇can(p) (see Section 2.5) are compatible with GL(m)-structure FN in the following sense. By definition, a linear connection ∇ on M iscompatible with GL(m)-structure FN if the corresponding parallel transport trans-forms a normal frame onto a normal frame. It means that for any smooth curve γ(t)(say, a trajectory of a vector field Y ) the corresponding parallel transport Pt0,tFγ(t0)

preserves the canonical splitting, the operators Ai, and the vector field X. Equiv-alently, given a normal frame V = (V1, . . . , Vm) of V along γ and the correspond-

ing normal frame F = (V 0, . . . , V k, X) along γ in TM (where V i = adiXV ), thenPt0,tFγ(t0) = Fγ(t)G(t), where G(t) = diagG(t), . . . , G(t), 1 for some smooth curve

G(t) in GL(m). This is equivalent to the fact that, if x 7→ (V 0(x), . . . , V m(x), X(x))is a local section of FN , then for an arbitrary vector field Y on M we have

∇Y ((V 0(x), . . . , V m(x), X(x))) = (V 0(x), . . . , V m(x), X(x))g(x)

with a matrix g(x) ∈ g = diag(g, . . . , g, 0) ∈ gl(n) | g ∈ gl(m) ' gl(m), where g isthe Lie algebra of the structural group diag(G, . . . , G, 1) ∈ GL(n) | G ∈ GL(m) 'GL(m). In other words ∇Y has values in the representation of gl(m) associated tothe frame bundle FN . It follows that a connection ∇ is compatible with FN (X,V)if and only if it satisfies the conditions

∇Y : Γ(Hi)→ Γ(Hi), ∇Y : Γ(X )→ Γ(X ), (42)

∇YAi(Z) = Ai(∇Y Z), ∇YX = 0, (43)

which hold for any Y ∈ Γ(TM) and Z ∈ Γ(V). Hence, statements (a)-(c) ofTheorem 2.13 can be rephrased as follows.

Corollary 3.3. If (X,V) is a regular pair on a manifold M then the connection∇can is compatible with the GL(m)-structure FN on M . The same holds for con-nections ∇can(p), for any mapping p : 0, . . . , k → 0, . . . , k without fixed points.

3.2. Compatible connections and associated frames. Our further aim nowis to assign to any (X,V), in an invariant way, a frame on its canonical bundleFN = FN (X,V). Then, two pairs (X,V) and (X ′,V ′) will be equivalent if and onlyif the corresponding frames are equivalent. In this way we will reduce the problemof equivalence of dynamic pairs to the problem of equivalence of frames, which hasa well known solution. We will provide explicit formulae for the frame in terms ofthe original data (X,V) which will make our solution effective.

Note that any distribution D on FN of rank n = dimM which is transversal tofibers defines a unique frame on FN . Namely, let F be a point in FN which is ann-tuple of vectors F = (V 0, V 1, . . . , V k, X) on M , where V i = (V i1 , . . . , V

im). Then

there exists a unique tuple of vectors lifted to D(F),

(Vij ,X | j = 1, . . . ,m, i = 0, . . . , k),

defined by the conditions that Vij ,X ∈ D(F) and

π∗(Vij) = V ij , π∗(X) = X. (44)

Varying F ∈ FN we obtain in this way global vector fields on FN . We will brieflydenote Vi = (Vi

1, . . . ,Vim) and V = (V0, . . . ,Vk).


In addition, on FN there are defined fundamental vector fields which are tangentto the fibres of FN and which come from the action of the structural group GL(m).The fundamental vector fields with the Lie bracket form a Lie algebra isomorphicto the Lie algebra g of the structural group, where

g = diag(g, g, . . . , g, 0) | g ∈ gl(m) ⊂ gl(n)

is naturally isomorphic to gl(m). Vector fields corresponding to matrices est with1 at the position (s, t) = (row, column) and 0 elsewhere will be denoted Gt

s andthe collection of all such vector fields will be shortly denoted G = (Gt

s)s,t=1,...,m.Clearly the tuple (V,X,G) is a frame on FN defined uniquely by D.

Let us describe a frame on FN in local coordinates. If U ⊂M is an open subsetand U 3 x 7→ F(x) = (V 0(x), . . . , V k(x), X(x)) is a fixed local section of FN thenany point F′ ∈ FN can be encoded by its projection x = π(F′) ∈ U and an elementG ∈ GL(m) such that

F′ = RGF(x) = (V 0(x)G,V 1(x)G, . . . , V k(x)G,X(x)).

In this way we get a local trivialisation FN = U × GL(m). The trivialisationpossesses a natural coordinate system on fibers of FN : (Gts)s,t=1,...,m where (t, s) =(row, column). The fundamental vector fields Gt

s, in these coordinates, are givenby

Gts =

∑r

Grs∂Grt. (45)

Additionally, it follows from (44) that the lifted vector fields X and Vij can be

written in coordinates as

X = X +∑s,t

αstGts, Vi

j =∑p

GpjVip +

∑s,t

βisjtGts, (46)

for some functions αst and βisjt on FN .It is a basic observation that there is one-to-one correspondence between linear

connections ∇ on M which are compatible with frame bundle structure FN (seeSection 3.1) and horizontal distributions D on FN which are GL(m)-invariant, i.e.

D(RGF) = (RG)∗D(F), (47)

for any F ∈ FN and any G ∈ GL(m). Such distributions on FN are called principalconnections on FN . For a given ∇ the corresponding D is spanned by vector fieldson FN tangent to curves defined by parallel transport of normal frames on M . Viceversa, the parallel transport defined by integral curves of D defines a unique ∇.

The Christoffel symbols of ∇ determine the coefficients αst and βisjt of the associ-ated canonical frame (46) on FN , which spans D, as follows.

Proposition 3.4. Assume that ∇ is a linear connection on a manifold M compat-ible with FN (X,V). Let x 7→ F(x) = (V 0(x), . . . , V k(x), X(x)) be a section of FN ,

where V i = (V i1 , . . . , Vim). Define coefficients Γpriq and Γpq by (30), i.e.

∇XV 0q =

∑r

ΓrqV0r , ∇V i

pV 0q =

∑r

ΓirpqV0r .

The corresponding principal connection D on FN is spanned by

X = X−∑s,t,q,r

ΓrqGqt (G

−1)srGts, Vi

j =∑p

Gpj (Vip−∑s,t

∑q,r

ΓirpqGqt (G

−1)srGts). (48)


Proof. It follows from Proposition 2.12 that Γpriq and Γpq define ∇ uniquely. To

show (48) we should find the vertical components of the vector fields X and Vij ,

tangent to D. They can be computed by finding the derivatives X(G) and Vij(G),

where G are treated as m2 coordinate functions on the locally trivialized bundleFN 'M ×GL(m). Write the parallel transport equation

∇γ(t)RGF = 0 (49)

for a normal frame RGF = (V 0(x)G, . . . , V k(x)G,X(x)) where γ is a trajectory ofX. Then (49) is equivalent to k+1 equations ∇X(V i)G+V iX(G) = 0. For i = 0 we

get 0 =∑q(∇X(V 0

q )Gqp + V 0q X(Gqp)) =

∑q(∑r ΓrqV

0r G

qp + V 0

q X(Gqp)). Comparing

the coefficients at V 0q gives

∑r ΓqrG

rp+X(Gqp) = 0 thus X = X−

∑p,q(∑r ΓqrG

rp)∂Gq

p.

Since ∂Gqp

= (G−1)sqGps , this gives the first formula in (48). The proof of the second

formula is analogous.

3.3. Canonical frame and equivalence. To state our next result we need anotion of canonical frame.

Definition 3.5. Let (X,V) be a regular pair and let Dcan denote the principalconnection on the bundle FN corresponding to the canonical linear connection∇can.The frame (V,X,G) on FN defined by Dcan is called canonical frame of (X,V).

The frame (V,X,G) defines functions T stjpqi , Tsjpi , S

stpq, S

sp, R

stupqw, R

supw by the fol-

lowing relations

[Vsp,V

tq] =

∑i,j

T stjpqiVij + SstpqX +

∑u,w

RstupqwGwu , (50)

[X,Vsp] =

∑i,j

T sjpi Vij + SspX +

∑u,w

RsupwGwu . (51)

The functions are called structural functions of the frame (V,X,G). They consti-tute a set of basic invariants of the frame.

The main result of this part of the paper is the following characterization of ∇canin terms of the structural functions of the corresponding canonical frame on FN .The result provides a solution to the equivalence problem and characterizes regularpairs with maximal symmetry group. The structural functions of the canonicalframe give basic invariants of the original dynamic pair.

Theorem 3.6. Assume that (X,V) is a regular pair on a manifold M .(a) Let Dcan be a principal connection on FN defined by the canonical linear con-nection ∇can on M and let (V,X,G) be the corresponding frame on FN . Then(V,X,G) is the unique frame on FN which satisfies the following conditions:

[X,Vkp ] = 0 mod V0, . . . ,Vk−1,G,X, (52)

[V0p,V

1q ] = 0 mod V0,V2, . . . ,Vk,G,X, (53)

[V0p,V

iq] = 0 mod V1, . . . ,Vk,G,X, i = 1, . . . , k, (54)

or equivalently T kqpk = 0, T 01jpq1 = 0 and T 0ij

pq0 = 0. It also satisfies

[X,Vip] = Vi+1

p mod G, i = 0, . . . , k − 1, (55)

and

[Gts,X] = 0, [Gt

s,Vij ] = δtjV

is. (56)


(b) Two pairs (X,V) and (X ′,V ′) on M are diffeomorphic if and only if the corre-sponding frames on FN (X,V) and FN (X ′,V ′) are diffeomorphic.(c) The symmetry group of (X,V) is at most (k+ 1)m+m2 + 1-dimensional, and itis maximal if and only if (X,V) is locally equivalent to a pair defined by the system

x(k+1) +Kk−1x(k−1) + · · ·+K1x

′ +K0x = 0, x(t) ∈ Rm,

where the matrices are of the form Ki = diag(ki, . . . , ki) with constant ki ∈ R.

Proof. In order to prove statement (a) we shall show that conditions (52)-(54) areequivalent to (29). In fact it is evident from the point of view of the theory ofconnections (c.f. [37]) because the structural functions defined by Lie brackets[Vi

p,Vjq] and [X,Vj

q] can be easily expressed in terms of the torsion T (∇can) as

will be explained in Section 3.4. Actually, structural functions T kqpk , T 01jpq1 , and

T 0ijpq0 give an alternative description of torsion tensors T (∇)kk, T (∇)01

1 and T (∇)0i0 ,

respectively.This result also follows directly from Proposition 3.4 if one computes explicitly

the brackets. In this way we can also get (56) and (55).(b) Our construction is invariant with respect to the action of diffeomorphisms,

thus (X,V) and (X ′,V ′) are equivalent if and only if the corresponding frames are.(c) The first part of statement (c) follows directly from statements (a) and (b).

Indeed, by (b) all symmetries of (X,V) give rise to symmetries of the canonicalframe and on the other hand the dimension of the symmetry group of a frame isbounded from above by the dimension of the ambient manifold (see [27], Chapter1, Theorem 3.2). In our case this dimension is dimFN = (k + 1)m+m2 + 1.

The symmetry group of (X,V) is maximal if and only if all structural functionsof the frame (V,X,G) are constant. The only possibly non-constant structuralfunctions appear for [X,Vi

j ] and [Vip,V

lq]. Proposition 3.4 implies that all structural

functions are homogeneous in vertical coordinates Gst . For the Lie bracket [Vip,V

lq]

they are homogeneous of order one (functions next to Gst and Vi

j), or two (a functionnext to X). Thus, in order to be constant they have to vanish, i.e. in the mostsymmetric case

[Vip,V

lq] = 0.

For the Lie bracket [X,Vij ], i < k, the similar homogeneity argument implies

[X,Vi] = Vi+1.

and for [X,Vk] we get

[X,Vk] = −V0K0 + · · ·+ (−1)kVk−1Kk−1

where, in coordinates, Ki = G−1KiG, as can be directly checked consulting Propo-sition 3.4 and assuming that V = (V1, . . . , Vm) is a normal frame of V. Thus, in the

most symmetric case Ki have to be diagonal and constant so that G−1KiG = Ki

are constant, independent on G.In this way we determined all possible structural functions of maximally sym-

metric regular pairs. This finishes the proof since the structural functions coincidewith the structural functions of the pair (X,V) corresponding to the system

x(k+1) +Kk−1x(k−1) + · · ·+K1x

′ +K0x = 0,


with diagonal and constant Ki. (The canonical frame of the pair (X,V), which

corresponds to the linear system, is given by Gst , X = X and Vi

j = GsjadiX∂xs0,

since vector fields adiX∂xs0

are constant and all their Lie brackets vanish.)

It follows from Propositions 2.12 and 3.4 that the ensuing explicit formulae forthe canonical frame hold.

Corollary 3.7. Let (X,V) be a regular dynamic pair on a manifold M and letx 7→ F(x) = (V 0(x), . . . , V k(x), X(x)) be a section of FN , i.e., V i = (V i1 , . . . , V

im)

and V ij = AiV0j . Assume that

[X,V kj ] =∑s

Hsj V

ks mod Vk−1 ⊕X (57)

and[V 0p , V

lq ] =

∑i,j

bljpqiVij mod X . (58)

Then the canonical vector fields on FN spanning Dcan are given by

X = X −∑s,t

HrqG

qt (G

−1)srGts, Vi

j =∑p

GpjVip −

∑s,t

∑p,q,r

ΓirpqGpjG

qt (G

−1)srGts,

(59)where

Γ0rpq = b1rpq1, Γirpq = −birqp0, i = 1, . . . , k. (60)

Proof. It follows from Proposition 3.4 because from the definition of ∇can one getsthat: Γ0r

pq = b1rpq1, Γirpq = −birqp0 and Γrq = Hrq , see Proposition 2.12.

Example. For illustration we continue the example of geodesic equation fromSection 2.3 (see Section 4.4 for a general case). We will use the above corollaries.In order to solve the equivalence problem one has to construct V0, V1 and G onFN = R × TN × GL(m) (we locally trivialize the canonical bundle). The vectorfields in G are standard, given by (45).

Take Vj = V 0j = ∂xj

1and compute the corresponding components V 1

j = A1Vj

of the normal frame in TM . From the formulae for adXVj and ad2XVj computed

earlier we see that the matrix H1 is (H1)ji = 2∑p Γjipx

p1. This and adXVj =

−∂xj0

+ 2∑i,p Γijpx

p1∂xi

1give

V 1j = A1Vj = −∂xj

0+∑p,i

Γijpxp1∂xi

1.

We have

[X,V 1j ] = −

∑i,u

Γijuxu1V

0j mod V1, [V 0

p , V1q ] =

∑j

ΓjpqV0j

(here Γjpq denote the coefficients in the original geodesic equation). Hence, Corollary3.7 gives

X = X +∑

p,q,r,s,t,u

(G−1)ptΓqpux

u1G

sqG

rs∂Gr

t

V0i =

∑j

Gji∂xj1,

V1i =

∑j

Gji

(−∂xj

0+∑p,q

Γpjqxq1∂xp

1+∑p,q,r

ΓpjrGrq∂Gp

q

).


It is straightforward to verify that

[V0i ,V

0j ] = 0, [V0

i ,V1j ] = 0, [X,V0

j ] = V1j , [X,V1

j ] = −Ki0jV

1i mod G,

with K0 = G−1K0G and K0 computed earlier, and

[V1i ,V

1j ] =

∑s,t,p,q

GsiGtjR

pstqx

q1∂xp

1+

∑s,t,p,q,r

GsiGtjG

rqR

pstq∂Gp

r,

where

Rpstq = ∂xt0(Γpsq)− ∂xs

0(Γptq) +

∑r

(ΓptrΓ

rsq − ΓpsrΓ

rtq

)are components of the curvature tensor of ∇. Due to formula (56) we see that thereare no more nontrivial structural functions on FN .

3.4. Torsion and curvature. We can describe the principal connection given byTheorem 3.6 by a connection form. The corresponding curvature and torsion of theconnection can be used as an alternative description of the invariants of the pair(X,V).

Let (θji , α, ωst ) be the coframe on FN dual to the frame (Vi

j ,X,Gts) defined by

a principal connection D. Denote φ = (θji , α) and ω = (ωst ). Then ω is a 1-formon FN with values in the Lie algebra g ' gl(m), called the connection form. Itdefines the connection D by D(F) = kerω(F) for F ∈ FN . The 1-form φ on FNwith values in Rn is called the canonical soldering form, where n = (k + 1)m + 1is the dimension of the manifold M . The following Cartan structural equations aresatisfied

dφ+ ω ∧ φ = Θ, dω + ω ∧ ω = Ω

and define torsion Θ and curvature Ω of the principal connection D, both being2-forms with values in Rn and g, respectively.

Remark 3.8. Let ∇ be the linear connection on M associated to a principal con-nection D on the bundle π : FN →M . Recall that torsion and curvature tensors of∇ are

T (∇)(Y1, Y2) = ∇Y1Y2 −∇Y1

Y2 − [Y1, Y2],

R(∇)(Y1, Y2) = ∇Y1∇Y2−∇Y2

∇Y1−∇[Y1,Y2].

For arbitrary sections Y1,Y2,Y3 ∈ Γ(D) we have

π∗(θ|D)−1(Θ(Y1,Y2)) = T (∇)(π∗(Y1), π∗(Y2)), (61)

π∗(θ|D)−1(Ω(Y1,Y2)θ(Y3)) = R(∇)(π∗(Y1), π∗(Y2))π∗(Y3), (62)

where in above θ|D is treated as a linear isomorphism of D and Rn. These formulaeshow that T (∇) and R(∇) are completely determined by Θ and Ω, and vice versa(the latter holds since Θ and Ω are horizontal, that is any of them vanishes if oneof its arguments is vertical).

Remark 3.9. The torsion and the curvature can be also described in terms of thestructural functions. Namely, since Θ has values in Rn, where n = (k + 1)m + 1,

we can decompose Θ = (Θji , Θ)j=1,...,m

i=0,...,k . Similarly, we can write Ω = (Ωst )s,t=1,...,m.Then

Θji =

∑p<qs,t

T stjpqi θqt ∧ θps +

∑l,r

T ljri θrl ∧ α, Θ =

∑p<qi,j

Sijpqθqj ∧ θ

pi +

∑i,p

Sipθpi ∧ α


and

Ωst =∑p<qi,j

Rijspqtθqj ∧ θ

pi +

∑i,p

Risptθpi ∧ α.

The coincidence of the coefficients in these formulae and the structural functionsintroduced in Section 3.3 follows from the general formula dβ(Y,Z) = −β([Y, Z]) fora 1-form β belonging to a coframe and Y , Z being frame vector fields in D = kerω.

Note that the structural functions satisfy additional relations which follow fromthe Bianchi identities. We will not use them in full generality in the present paper,but only restrict to the simplest cases, important for our applications. Namely, inthe next section we will consider cases k = 1 and k = 2 with additional integrabilityconditions. We will work in terms of canonical frame rather than connection anduse the identities in the form of Jacobi identity for the vector fields in the canonicalframe.

The freedom of choosing torsion normalisation conditions in Theorem 3.6 can beexplained following Sternberg [37]. Note that both Θ and Ω vanish if one of theirarguments is in the vertical distribution spanG. Thus, at a fixed point F ∈ FN , wemay identify them with elements of hom(D(F)∧D(F),Rn) and hom(D(F)∧D(F), g),respectively. Moreover, using the isomorphism φ|D(F) : D(F) → Rn we obtain thatΘ ∈ hom(Rn ∧Rn,Rn) and Ω ∈ hom(Rn ∧Rn, g). If we fix a point in F ∈ FN thenit follows that the set of sub-spaces of TFFN which are transversal to the fibre is anaffine space modeled on the linear space hom(Rn, g). If two connections differ at Fby an element η ∈ hom(Rn, g) then their torsions at F differ by

δη ∈ hom(Rn ∧ Rn,Rn).

where δ : hom(Rn, g)→ hom(Rn ∧ Rn,Rn) is Spencer operator

δη(Y1, Y2) = η(Y1)Y2 − η(Y2)Y1.

The approach of Sternberg [37], page 318, says that in order to choose a canonicalconnection one should fix a subspace N ⊂ hom(Rn ∧Rn,Rn) such that N ⊕ Im δ =hom(Rn ∧ Rn,Rn) and then consider connections with torsion in N . Such a con-nection is unique provided that ker δ = 0. This is the case in our situation:

Lemma 3.10. The kernel of the Spencer operator is trivial, ker δ = 0.

Proof. Let η ∈ hom(Rn, g) and δη = 0. There exists a ∈ hom(Rn, gl(m)) such thatη = diag(a, . . . , a, 0) and η(Y1)Y2 − η(Y2)Y1 = 0 for any two vectors Y1, Y2 ∈ Rn.Both Y1 and Y2 are column vectors of length (k + 1)m + 1, where k ≥ 1. TakeY1 = (y1, 0, 0, . . . , 0)T and Y2 = (0, y2, 0, . . . , 0)T for some y1, y2 ∈ Rm. Then weget a(y1)y2 = 0 (and also a(y2)y1 = 0) for arbitrary y1 and y2. Thus a = 0 andconsequently η = 0.

An upshot of Lemma 3.10 is that in order to solve the problem of equivalence fordynamic pairs one should fix, once and for all, a subspace N ⊂ hom(Rn ∧ Rn,Rn)transversal to Im δ and then assign to a dynamic pair a unique connection on FNwith torsion having values in N . Choosing normalization conditions (52), (53), (54)was a choice of the subspace N ⊂ hom(Rn ∧Rn,Rn). Of course, there is a freedomin choosing another transversal subspace N and the only criterion for choosing oneseems simplicity of the resulting invariants.


4. Ordinary differential equations. Consider a system of m ordinary differen-tial equations of order k + 1 ≥ 2,

x(k+1) = F (t, x, x(1), . . . , x(k)), (F )

where x = (x1, . . . , xm) ∈ Rm and F = (F 1, . . . , Fm) is a smooth map Rn → Rm,n = 1 + (k + 1)m. Two systems (F ) and (F ′) of this form will be called equivalent(alternatively, time-scale preserving equivalent or affine-contact equivalent), if thereexists a smooth diffeomorphism Rm+1 → Rm+1 of the form

t 7−→ t′ = t+ c, x 7−→ x′ = Φ(t, x), (63)

with c ∈ R, which maps the set of solutions to (F ) onto the set of solutions of (F ′).In the present section we will focus on the problem whether two systems of the

form (F ) are time-scale preserving equivalent and on determining invariants. Westart with providing a geometric background to the definition of equivalence.

4.1. Jet space and its affine distribution. Let Jk(1,m) denote the space of kjets of smooth functions R→ Rm. The space Jk(1,m) is endowed with the naturalcoordinate system (t, y) := (t, x0, . . . , xk), where xi = (x1

i , . . . , xmi ). Recall that any

parametrised curve x : I → Rm, with I ⊂ R an open interval, has its k-jet extensionjkx : I → Jk(1,m) defined by (jkx)(t) = (t, x(t), x(1)(t), . . . , x(k)(t)). On each suchcurve we identify xi(t) = x(i)(t).

For any given (t, y) ∈ Jk(1,m) there is a smooth curve x : I → Rm such that(jkx)(t) = (t, y). The vector tangent to this curve at (t, y) is of the form

v = ∂t +

k−1∑i=0

m∑j=1

xji+1∂xji

+

m∑j=1

uj∂xjk,

where uj are arbitrary numbers and, again, we identify xi = x(i)(t). All suchvectors form an affine subspace of the tangent space to M = Jk(1,m) at the point(t, y) ∈M . This subspace is denoted Ak(t, y) ⊂ Tt,yM and we have

Ak(t, y) = Dk(t, y) + span∂x1k, . . . , ∂xm

k,

where Dk denotes the vector field

Dk(t, y) = ∂t +

k−1∑i=0

m∑j=1

xji+1∂xji.

We will call Ak canonical affine distribution on the space Jk(1,m) of k-jets ofparametrised curves in Rm. We may write

Ak = Dk + Vk,

where Vk denotes the involutive distribution Vk = span∂x1k, . . . , ∂xm

k.

Proposition 4.1. A diffeomorphism Ψ : Jk(1,m)→ Jk(1,m) which preserves Akis the k-jet extension of a diffeomorphism (63), which means that it is of the form

(t, x0, . . . , xk) 7−→ (t+ c,Φ(t, x0), (DkΦ)(t, x0, x1), . . . , (Dk−1k Φ)(t, x0, . . . , xk)).

(64)In particular Ψ preserves the 1-form dt. Vice versa, any Ψ as in (64) preserves Ak.


Proof. Suppose, Ψ : Jk(1,m)→ Jk(1,m) preserves Ak, i.e., Ψ∗Ak = Ak. Denotingwith the same same symbol Ak the set (the sheaf) of vector fields belonging to thedistribution Ak we have

[Ak,Ak] = V1k ,

[Ak, [Ak,Ak]] = V2k ,

...

[Ak, [· · · [Ak,Ak]]] = Vkk ,where [·, ·] denotes the Lie bracket and V1

k , . . . ,Vkk denote the involutive distributions

Vik = span∂xjs| s = k − i, k − i+ 1, . . . , k, j = 1, . . . ,m.

Since Ψ preserves Ak, it also preserves these distributions and the corresponding fo-liations. In particular, it preserves Vk−1

k and Vkk , which means that t is transformedinto t′ and (t, x) is transformed into (t′, x′). The 1-form α = dt is determined by theconditions α(Vkk ) = 0 and α(Y ) = 1, for Y ∈ Ak, thus it is also preserved by Ψ. Thisimplies that t is mapped into t+ c, c ∈ R and, therefore, (t′, x′) = (t+ c,Φ(t, x)).

Since through any point in Jk(1,m) there passes a k-jet extension of a curve inRm, and for any curve s 7→ γ(s) = (s, x(s)) we have

x′i(Ψ(γ(s))) =

(d

ds

)iΨ(γ(s)) = (Di

kΦ)(γ(s)),

it follows that x′i = (DikΦ)(t, x) and, thus, Ψ is of the desired form.

The converse implication is straightforward.

In geometric theory of ODEs one uses a Cartan distribution Ck, which is thevector distribution spanned by the more subtle object, the affine distribution Ak.Ck consists of vectors tangent to k-jet extensions of unparametrised curves and

Ck = spanDk,Vk, Ak = Y ∈ Ck : dt(Y ) = 1.

4.2. Dynamic pair of (F ) and equivalence. The system (F ) can be equivalentlydefined by a submanifold EF ⊂ Jk+1(1,m),

EF = (t, x0, . . . , xk+1) ∈ Jk+1(1,m) | xk+1 − F (t, x0, . . . , xk) = 0.

The functions t, x0, . . . , xk restricted to EF define a system of coordinates on EF ,since the projection π : Jk+1(1,m)→ Jk(1,m) restricted to EF is a diffeomorphism,π|EF

: EF → Jk(1,m).The canonical affine distribution Ak+1 = Dk+1 + Vk+1 on M = Jk+1(1,m),

intersected with the tangent space to the submanifold EF , defines a unique vector

XF (t, y) = Ak+1(t, y) ∩ T(t,y)EF , for any (t, y) ∈ EF .

This follows from the fact that Vk+1 and TEF are mutually transversal subspacesin TJk+1(1,m), at any point in EF . In this way (F ) defines a vector field XF onEF . XF is called total derivative corresponding to (F ). In coordinates, it is givenby

XF = ∂t +

k−1∑i=0

m∑j=1

xji+1∂xji

+

m∑j=1

F j∂xjk. (65)

Consider the Lie square [Ak+1,Ak+1], which is a vector distribution on the man-ifold Jk+1(1,m). In coordinates, [Ak+1,Ak+1] = span∂xi

k, ∂xj

k+1|i, j = 1, . . . ,m.


We define the distribution on EF by intersecting [Ak+1,Ak+1] with the tangentbundle TEF ,

VF = TEF ∩ [Ak+1,Ak+1]

and, in coordinates,VF = span∂x1

k, . . . , ∂xm

k. (66)

It is easy to check that the pair (XF ,VF ) is regular, i.e., it satisfies conditions (R1)and (R2) on M = EF . We will call it the dynamic pair of system (F ).

Consider two equations (F ) and (F ′).

Proposition 4.2. The following statements are equivalent.(a) Equations (F ) and (F ′) are time-scale preserving equivalent.(b) There is a diffeomorphism of Jk+1(1,m) which preserves the canonical affinedistribution Ak+1 and transforms EF onto EF ′ .(c) There is a diffeomorphism of Jk+1(1,m) preserving the Cartan distribution Ck+1

and the 1-form dt, and transforming EF onto EF ′ .(d) The dynamic pairs (XF ,VF ) and (XF ′ ,VF ′) are diffeomorphic.

Proof. Equivalence of (a), (b) and (c) follows from Proposition 4.1 and the remarkfollowing it. From the definitions XF = TEF ∩ Ak+1, VF = TEF ∩ [Ak+1,Ak+1]we see that (b) implies (d).

In order to show (d)⇒ (b) assume that there is a diffeomorphism ψ : EF → EF ′

which transforms the pair (XF ,VF ) into (XF ′ ,VF ′). Since EF and Jk(1,m) arediffeomorphic via the natural projection π : Jk+1(1,m) → Jk(1,m) restricted to

EF , and so are EF ′ and Jk(1,m), there is a diffeomorphism ψ : Jk(1,m)→ Jk(1,m)corresponding to ψ : EF → EF ′ . Moreover, after projections both pairs (XF ,VF )and (XF ′ ,VF ′) have, in natural coordinates in Jk(1,m), the form (65) and (66).

Since ψ transforms the projected pair into the projected pair, it follows from (65)

and (66) that they both span the same affine distribution Ak = Dk + Vk. Thus ψpreserves the canonical affine distribution Ak in Jk(1,m). We then deduce from

Proposition 4.1 that ψ is of the form (64). Let Ψ be the 1-prolongation of ψ, whichmeans that it is of the form (64), with k replaced by k + 1. Then Ψ automaticallypreserves Ak+1 and it is easy to see that it transforms EF to EF ′ .

The proposition can also be deduced from the classical Lie-Backlund theorem,cf. [20], or from Theorem 1 in [17], with additional condition Ψ∗(dt) = dt.

Equations (F ) and (F ′) satisfying one of the above conditions will simply becalled equivalent. Taking into account conditions (b) and (c) one could also callthem affine-contact equivalent or time-scale preserving contact equivalent.

Condition (d) implies that we can use Theorem 3.6 in order to solve the equiva-lence problem for systems (F ). We can assign to (F ) a canonical connection and acanonical frame on the normal frame bundle of the pair (XF ,VF ). We obtain

Theorem 4.3. The following conditions are equivalent.(a) Equations (F ) and (F ′) are equivalent.(b) The dynamic pairs (XF ,VF ) on EF and (XF ′ ,VF ′) on EF ′ are diffeomorphic.(c) The canonical frames (V,X,G) and (V′,X′,G′) in Theorem 3.6, correspondingto dynamic pairs (XF ,VF ) and (XF ′ ,VF ′) and living on the normal frame bundlesπ : FN → Jk(1,m) and π : F ′N → Jk(1,m), are diffeomorphic.

We also deduce that any system (F) has at most (k+ 1)m+m2 + 1-dimensionalgroup of time-scale preserving contact symmetries and it has maximal dimension if


and only if it is equivalent to a linear system with constant and diagonal coefficients.In this way, the problem of time-scale preserving equivalence of systems of ODEsis reduced to the geometry of pairs (X,V).

4.3. Dynamic pairs of ODEs. Not all dynamic pairs (X,V) correspond to sys-tems of ODEs. In order to characterize such pairs we introduce

Definition 4.4. Let X be a smooth vector field and V be a smooth distributionof constant rank m on a manifold M . The pair (X,V) is of equation type if thereexists a system (F ) and a diffeomorphism Φ: M → EF such that Φ∗(X) = XF andΦ∗(V) = VF . The pair (X,V) is locally of equation type if for any x ∈ M thereexists a neighbourhood U 3 x such that (X|U ,V|U ) is of equation type.

Theorem 4.5. A pair (X,V) is locally of equation type if and only if it satisfiesconditions (R1), (R2) and, additionally,

(R3) Vi are integrable for i = 0, . . . , k,(R4) adXVk = Vk.

Moreover, condition (R3) is equivalent to:(R3’) Vi are integrable for i = k − 1 and i = k.

Proof. It is straightforward to check that conditions (R1)-(R4) are satisfied for anarbitrary equation (F ) and the corresponding (XF ,VF ).

In order to prove the theorem in the opposite direction let us notice that (R1)and (R3) imply that Vk defines a corank one foliation on M . Thus we can choosea local coordinate t on M such that leaves of Vk are given by equations: t = const.Additionally, it follows from (R3) that we can choose remaining coordinates such

that Vi = t = c, xj0 = cj0, · · · , xjk−i−1 = cjk−i−1 | j = 1, . . . ,m, for i = 0, . . . , k− 1,

where c and cjs are constants. We have

X = f∂t +

k∑i=0

m∑j=1

f ji ∂xji

for certain functions f and fi = (f1i , . . . , f

mi ).

Note that (R4) implies that f is constant on leaves of Vk. If not, then the Liebracket of X and some vector field tangent to Vk would be transversal to Vk, andhence it would violate condition (R4). Thus, we can reparametrise t so that f ≡ 1.

Similarly, let us notice that fi depend on t and x0, . . . , xi only. Otherwise, theLie bracket of X and a vector field in Vk would stick out of Vk. We will modifycoordinates xji in such a way that X is of the form XF for some system (F ). Firstly,we set

y0 = x0 and y1 = f0(t, y0, x1).

Using (R1) we will see that (t, y0, y1, x2, . . . , xk) can be taken as new coordinates.Indeed, (R1) implies that the matrix (∂xs

1f t0)s,t=1,...,m has maximal possible rank

m, because rkV1 − rkV0 = m. We continue the reasoning and inductively define

yi = fi−1(t, y0, . . . , yi−1, xi).

At each step we get new coordinate system (t, y0, . . . , yi, xi+1, . . . , xk). Finally weobtain

X = ∂t +

k−1∑i=0

m∑j=1

yji+1∂yji+

m∑j=1

F j∂yjk.

where F j(t, y0, . . . , yk) = f jk(t, y0, . . . , yk) define the desired system of ODEs.


In order to complete the proof it is sufficient to prove that (R3’) implies (R3). Weproceed by induction. Assume that Vi+1, . . . ,Vk are integrable, where i < k − 1.Let Y1 and Y2 be two sections of Vi ⊂ Vi+1. Then, by assumption, [Y1, Y2] is asection of Vi+1. Moreover, Jacobi identity implies that [X, [Y1, Y2]] is also a sectionof Vi+1, since [X,Y1] and [X,Y2] are sections of Vi+1 by the definition of Vi+1. Itfollows, that [Y1, Y2] is a section of Vi. If not, then by condition (R1), the bracket[X, [Y1, Y2]] would be a non-trivial section of Vi+2 mod Vi+1.

Theorem 4.5 implies that the canonical frame in Theorem 3.6 satisfies

[Vip,V

jq] = 0 mod V0, . . . ,Vr,G, where r = maxi, j. (67)

This fact implies

Corollary 4.6. The canonical frame of Theorem 3.6, corresponding to dynamicpair (XF ,VF ) of system (F), satisfies the following conditions

[V0p,V

0q ] = 0, (68)

[V0p,V

1q ] = 0 mod G, (69)

[V0p,V

iq] = 0 mod V0, . . . ,Vi−1,G, i = 2, . . . , k, (70)

[X,Vip] = Vi+1

p mod G, i = 0, . . . , k − 1, (71)

[X,Vkp ] = 0 mod V0, . . . ,Vk−1,G. (72)

It is also uniquely determined by the first three of them.

Proof. Conditions (69), (71) and (72) follow from the definitions of canonical frames(Section 3.3). In order to prove the remaining ones we will use identities (56),without mentioning. From (67) we have

[V0p,V

0q ] = T 00r

pq0 V0r +R00t

pqsGst .

Taking Lie bracket of both sides with X and using [X,Gst ] = 0, [X,V0

r ] = V1r

mod G, gives

[[X,V0p],V

0q ] + [V0

p, [X,V0q ]] = T 00r

pq0 V1r mod V0,G.

Using again [X,V0r ] = V1

r mod G on the left-hand side and the identity [V0i ,V

1j ] =

0 modulo V0, G (satisfied for the canonical frame) we see that the left-hand sideequals to zero, modulo V0,G. Thus T 00r

pq0 = 0.We repeat the same procedure, taking this time Lie bracket of the above identity

with V1r (now T 00r

pq0 = 0). Applying the Jacobi identity on the left-hand side and

using the identity [V0i ,V

1j ] = 0 modulo V0, G we find that this side vanishes

modulo V0, G. The right-hand side equals to R00tpqs[V

1r ,G

st ] = −R00t

pqrV1t modulo

V0, G, thus R00tpqr = 0 and (68) is proved.

In order to prove (70) note that this condition is satisfied for i = 1, by thedefinitions of the canonical frame (Theorem 3.6). Suppose now that (70) is satisfiedfor some i ≥ 1 and take the Lie bracket of both sides with X. We see from (71) andcondition (67) that is is also satisfied for i+ 1.

Theorem 4.5 implies that the distribution Vk is integrable. Let S be a leafof the corresponding foliation (a hypersurface) and let us choose a normal frameFx = (V 0, . . . , V k, X(x)) at each point x ∈ S. Then (V 0, . . . , V k) constitutes aframe of manifold S and X is transversal to S, by (R2). We will call (V 0, . . . , V k)


a normal frame of a pair (V, X) on S. Such a frame together with the vector fieldX and the curvature matrices K0, . . . ,Kk−1 determine V in a neighbourhood of S.

Corollary 4.7. Assume that two dynamic pairs (X,V) and (X ′,V ′) are of equation

type, X = X ′, and there exists a common leaf S of distributions Vk and V ′k with acommon normal frame (V 0, . . . , V k). Additionally, assume that there exist a normalframe of V and a normal frame of V ′ which coincide on S and are such that theassociated matrices of curvature operators coincide in a neighbourhood of S. ThenV = V ′ in a neighbourhood of S.

Proof. Let V and V ′ be normal frames of V and V ′, respectively, such that theassociated curvature operators coincide on a neighbourhood of S and V = V ′ on S.We can assume that V = V ′ = V 0 on S (if not, we take V := V 0G and V ′ := V 0G′

where G(x), G′(x) ∈ GL(m) are transition matrices from V 0(x) to V (x) and V ′(x)).Now, we know that both V and V ′ satisfy equation (10) with the same coefficients

Ki. Moreover, adiXV = adiXV′ = V i on S for i = 0, . . . , k. Thus, the uniqueness

theorem for ODEs implies that V = V ′ on a neighbourhood of S. ConsequentlyV = spanV = spanV ′ = V ′ on a neighbourhood of S.

4.4. Systems of order 2. Consider a system (F) of second order ODEs on Rm,

x′′ = F (t, x, x′).

Instead of it, we can consider the corresponding dynamic pair (XF ,VF ) or, equiv-alently, a general dynamic pair (X,V) satisfying (R1)-(R4), with k = 1. In thiscase we can describe all structural functions of the corresponding canonical frameon normal frame bundle FN . We have the following

Theorem 4.8. Let (F ) be a system of second order ODEs. The canonical framethe dynamic pair (XF ,VF ) of (F ) satisfies the following structural equations

[X,V0p] = V1

p,

[X,V1p] = −Kr

0pV0r + R1t

psGst ,

[V0p,V

0q ] = 0,

[V0p,V

1q ] = R01t

pqsGst ,

[V1p,V

1q ] = T 11r

pq0 V0r +R11t

pqsGst .

The invariants R1, T 110 and R11 are determined by K0, namely

T 11spq0 =

1

3

(V0p(K

s0q)−V0

q(Ks0p)), (73)

R11spqr =

1

3

(V0rV

0p(K

s0q)−V0

rV0q(K

s0p)), (74)

R1spq =

1

2T 11spq0 −

1

2

(V0p(K

s0q) + V0

q(Ks0p)), (75)

and they satisfy ∑cyclp,q,r

R11tpqr = 0, (76)

∑cyclp,q,r

V1p(T

11tqr ) = 0 (77)


(∑cycl denotes the cyclic sum). Moreover, R01t

pqs is symmetric in lower indices and

V0p(R

01tqrs) = V0

q(R01tprs), (78)

V0r(R

11tpqs) = V1

p(R01trqs)−V1

q(R01trps), (79)

X(R01tpqs) = R11t

pqs + V0p(R

1tqs), (80)

X(R11tpqs) = −Kr

0pR01trqs + Kr

0qR01trps + V1

p(R1tqs)−V1

q(R1tps), (81)∑

cyclp,q,r

(V1p(R

11tqrs)− T 11u

qr R01tups

)= 0. (82)

If K0 vanishes then [X,V1p] = 0, [V1

p,V1q ] = 0 and the only nonzero invariant R01

satisfies the relations

X(R01tpqs) = 0 and Vi

p(R01tqrs) = Vi

q(R01tprs), i = 0, 1.

Note that K0 = G−1K0G, in coordinates on FN .

Proof. A priori, due to Corollary 4.6, the structural equations of the canonical framehave the form:

[X,V0p] = V1

p + R0tpsG

st ,

[X,V1p] = −Kr

0pV0r + R1t

psGst ,

[V0p,V

0q ] = 0,

[V0p,V

1q ] = R01t

pqsGst ,

[V1p,V

1q ] = T 11r

pq0 V0r + T 11r

pq1 V1r +R11t

pqsGst .

In order to identify the relations between the structural functions we will use theJacobi identities for the following combinations of the vector fields (the remainingare either trivial or follow from the considered ones): (X,V0

p,V0q), (X,V0

p,V1q),

(X,V1p,V

1q), (V0

p,V0q ,V

1r), (V0

p,V1q ,V

1r) and (V1

p,V1q ,V

1r).

First, let us consider the third equation and take the Lie bracket of both sideswith X. We have [X, [V0

p,V0q ]] = 0 and, applying Jacobi identity, the first and

fourth equations and the identities (56), we obtain

−R01tqpsG

st +R01t

pqsGst + R0t

pqV0t − R0t

qpV0t −V0

q(R0tps)G

st + V0

p(R0tqs)G

st = 0,

thus

R0tpq = R0t

qp, (83)

R01tpqs = R01t

qps + V0q(R

0tps)−V0

p(R0tqs). (84)

Next, let us Lie bracket both sides of the fourth structural equation with X. Onthe left-hand side we have [X, [V0

p,V1q ]] and, applying Jacobi identity, the first, the

second and the third equations, and the identities (56) we get:

T 11rpq0 V0

r + T 11rpq1 V1

r +R11tpqsG

st −V1

q(R0tps)G

st + R0t

pqV1t

−V0p(K

r0q)V

0r + V0

p(R1tqs)G

st − R1t

qpV0t .

On the right-hand side we obtain X(R01tpqs)G

st . Therefore,

R1rqp = T 11r

pq0 −V0p(K

r0q), (85)

R0rpq = −T 11r

pq1 , (86)

X(R01tpqs) = R11t

pqs + V0p(R

1tqs)−V1

q(R0tps). (87)


In particular, we get that R0rpq is anti-symmetric in p and q, because T 11r

pq1 is. But

(83) reads that R0tpq is symmetric in p and q. Thus

R0rpq = T 11r

pq1 = 0. (88)

This proves that the structural equations are as stated in the theorem.Additionally, equation R0r

pq = 0 together with (87) proves the relation (80) andsimplifies (84) to

R01tpqs = R01t

qps. (89)

Now, let us Lie bracket both sides of the last structural equation with X. Onthe left-hand side we have [X, [V1

p,V1q ]] and, applying Jacobi identity, the second

and the fourth structural equations, and (56), we obtain

−Kr0pR

01trqsG

st + V1

q(Kr0p)V

0r −V1

q(R1tps)G

st + R1t

pqV1t

+Kr0qR

01trpsG

st −V1

p(Kr0q)V

0r + V1

p(R1tqs)G

st − R1t

qpV1t .

On the right-hand side, taking into account the first structural equation and (88),we get:

T 11rpq0 V1

r + X(T 11rpq0 )V0

r + X(R11tpqs)G

st .

Thus we have

X(T 11rpq0 ) = V1

q(Kr0p)−V1

p(Kr0q) (90)

T 11rpq0 = R1t

pq − R1tqp (91)

and (81) (which was to be proved). Combining (85) and (91) we can express T 11rpq0

and R1tpq in terms of V0

p(Kr0q). Precisely, taking into account that T 11r

pq0 = −T 11rqp0 ,

we find

T 11rpq0 =

1

3(V0

p(Kr0q)−V0

q(Kr0p)), R1r

pq = 2T 11rpq0 −V0

p(Kr0q),

which gives (73) and (75) in the formulation of the theorem.Consider next the third structural equation and bracket it with V1

r . We have[V1

r , [V0p,V

0q ]] = 0 and, after applying Jacobi identity and the fourth structural

equation,

−R01tprqV

0t + V0

q(R01tprs)G

st +R01t

qrpV0t −V0

p(R01tqrs)G

st = 0.

This implies (78) and the relation R01tprq = R01t

qrp which, together with (89), implies

that R01tpqr is symmetric with respect to indices p, q, and r.

Let us now bracket the last structural equation with the vector field V0r . On the

left-hand side we have [V0r , [V

1p,V

1q ]] and, applying Jacobi identity,

−V1q(R

01trps)G

st +R01t

rpqV1t + V1

p(R01trqs)G

st −R01t

rqpV1t .

On the right-hand side, taking into account the third structural equation and T 11rpq1 =

0, we obtain:V0r(T

11tpq0)V0

t + V0r(R

11tpqs)G

st −R11t

pqrV0t .

Thus we get the relation (79) and R11tpqr = V0

r(T11tpq0). The latter equation and (73)

give the desired formula (74) for R11tpqr.

Finally, we consider the Jacobi identity∑

[V1p, [V

1q ,V

1r ]] = 0, where we take the

cyclic sum over p, q, r. Taking into account the last structural equation we get

0 =∑

cyclp,q,r

(V1p(T

11sqr )V0

s + V1p(R

11tqrs)G

st − T 11u

qr R01tupsG

st −R11t

qrs[Gst ,V

1p])


which, taking into account [Gst ,V

1p] = δspV

1t , implies vanishing of the cyclic sums∑

R11tpqr = 0,

∑V1p(T

11tqr ) = 0,

∑(V1p(R

11tqrs)− T 11u

qr R01tups

)= 0,

i.e., identities (76), (77) and (82). This ends the proof of the theorem as, if K0 = 0,all the invariants vanish except of R01t

pqr.

Let us find the structural functions in terms of function F defining the equation,in natural coordinates (t, xj , yj) on the space of 1-jets. We have

XF = ∂t +

m∑j=1

yj∂xj +

m∑j=1

F j(t, x, y)∂yj

and VF = spanV1, . . . , Vm, where Vj = ∂yj . Let V = (V1, . . . , Vm). We compute

ad2XFV = (adXF

V )H1 + V H0, where

H1 =(−∂ysF t

)s,t=1,...,m

, H0 =(∂xsF t −XF (∂ysF

t))s,t=1,...,m

.

Therefore, by Proposition 2.5, we get

K0 =

(−∂xsF t +

1

2X(∂ysF

t)− 1

4

m∑r=1

∂ysFr∂yrF

t

)s,t=1,...,m

and H1 = spanV 11 , . . . , V

1m where

V 1j = −∂xj − 1

2

m∑s=1

∂yjFs∂ys .

Thus, in the case of SODE, our H1 is equivalent to a non-linear connection usedin [6, 14, 15, 34] (in Finsler Geometry the corresponding linear connection ∇can isusually called Berwald connection).

Since Vj = ∂yj , we have

[Vp, V1q ] = −1

2

m∑s=1

∂yp∂yqFsVs

and using Corollary 3.7 we can write

V0j =

m∑s=1

Gsj∂ys ,

V1j = −

m∑s=1

Gsj∂xs − 1

2

m∑s,t=1

Gtj∂ytFs∂ys −

1

2

m∑r,s,t=1

GujGwt (G−1)sr∂yu∂ywF

rGts.

Then

[V0p,V

1q ] =

m∑s,t=1

R01spqtG

ts =

m∑u,w,s,t=1

GupGwq R

suwtG

ts,

[V1p,V

1q ] =

m∑u,w,s,t=1

GupGwq T

tuw(G−1)stV

0s mod G


where

Rsuwt = −1

2∂yu∂yw∂ytF

s,

T tuw =1

2

(∂xu∂ywF

t − ∂xw∂yuFt)

+1

4

m∑v=1

(∂yuF

v∂yv∂ywFt − ∂ywF v∂yv∂yuF t

).

By Theorem 4.8 we know that T is expressed in terms of K0. As a conclusion weget that all invariants of a system (F ) are expressed by K0, R and their derivatives.This strengthens a result of [12] (problem (B)).

Corollary 4.9. A system of second order ODEs is equivalent to the trivial systemx′′ = 0 if and only if K0 vanishes and F is a polynomial of degree at most 2 in x′.

Proof. The trivial system has vanishing invariants K0 and R01. Vanishing of R01

means that Rsuwt = 0 which is equivalent to the fact that F is polynomial of degreeat most 2 of y.

In the case of geodesic equation for a Finsler metric the tensor defined by− 1

2∂yu∂yw∂ytFs is called Berwald curvature (cf. [36]). In our setting it appears

as a component of curvature: R01suwt. Vanishing of Berwald curvature of geodesic

equations is necessary an sufficient condition for Finsler metric to be Riemannian.The structural functions Rijsuwt can be equivalently interpreted as components

of curvature tensor R(∇can) of the canonical linear connection ∇can on M , asexplained in Section 3.4 (see equation (62)). In particular the coefficients R01s

uwt

correspond toR(∇can)01 = R(∇can)|V⊕H1 .

If we use the isomorphism A1 : V → H1, as we did in Section 2.5 for torsion tensor,then we can interpret R(∇can)01 as a (3,1)-tensor on V.

Definition 4.10. The (3,1)-tensor on V defined by

B(Y1, Y2, Y3) = R(∇can)01(Y1, A1(Y2))Y3,

where Y1, Y2, Y3 ∈ Γ(V), will be called Berwald curvature of a dynamic pair (X,V).

B is well defined for an arbitrary pair, not necessarily of equation type. Thefollowing explicit formula for B follows from the definition of ∇can,

B(Y1, Y2, Y3) = A−11 π1([Y1, A1 π0([A1(Y2), Y3])])

− π0([A1(Y2), A−11 π1([Y1, A1(Y3)])])

+ A−11 π1([π0([A1(Y2), Y1]), A1(Y3)])

− π0([π1([Y1, A1(Y2)]), Y3]).

From the formulae preceding Corollary 4.9 we get

Corollary 4.11. In the case of SODE Berwald curvature B of the correspond-ing dynamic pair is symmetric (3,1)-tensor and coincides with the usual Berwaldcurvature,

B(∂yu , ∂yw , ∂yt) = −1

2∂yu∂yw∂ytF

s.

In Section 5 we will show that vanishing of B is crucial in characterization ofmechanical control systems.


Remark 4.12. In the case of autonomous SODE our connection ∇can is equivalentto a linear connection appearing in [6] (see also [7], [15], [33] and [34]). Consequently,T 11

0 , R01 and R11 correspond to the torsion and curvature tensors defined in thecited papers. Besides, K0 coincides with the Jacobi endomorphism in this case.The Berwald curvature is also defined in [33] in terms of (X,V), with integrable V,and is used for characterizing SODE which are second order in the velocities. Thislatter issue is also crucial for characterizing mechanical systems, cf. Section 5 and[35].

4.5. Equations of order 3. Let (F ) be an equation of the third order

x′′′ = F (t, x, x′, x′′).

As before we consider time-scale preserving contact transformations and we wantto solve the equivalence problem for (F ).

Theorem 4.13. Let (F ) be a third order ODE. The canonical frame of the dynamicpair (XF ,VF ) satisfies the following structural equations

[X,V0] = V1 − LG,

[X,V1] = V2,

[X,V2] = −K0V0 +K1V

1 + R2G,

[V0,V1] = 2V0(L)G,

[V0,V2] = LV1 +R02G,

[V1,V2] = T 120 V0 − 2X(L)V1 + 2LV2 +R12G.

where L = T 021 = R0 = GL and L, K0, K1 are functions of global coordinates

(t, x, y, p) on J2(1, 1) (G is the fiber coordinate), and

R2 = −1

2V0(K0) +

1

2V1(K1)−X2(L)− LK1,

T 120 =

1

2V0(K0) +

1

2V1(K1)−X2(L)− LK1

R02 = V1(L) + 2V0X(L)− 2(L)2, R12 = V0(T 120 ).

Moreover X(L) = 13V0(K1) and

V0(R12) + 2LR02 − 2X(L) + 2V2V0(L) = V1(R02).

Proof. The proof, based on Jacobi identity applied to the canonical frame, is anal-ogous to the proof of Theorem 4.8. We leave it to the reader.

Corollary 4.14. All structural functions of the canonical frame are combinationsof L, K0, K1 and their derivatives, where

L = −1

3∂2pF,

K1 = ∂yF −X(∂pF ) +1

3(∂pF )2,

K0 = ∂xF −X(∂yF ) +1

3∂yF∂pF +

2

3X2(∂pF )− 2

3X(∂pF )∂pF +

2

27(∂pF )3.

In order to prove the corollary we will compute structural functions in termsof function F . We will use the following formulae, being direct consequences ofPropositions 2.9 and 2.8.


Proposition 4.15. If

ad3XV = H2ad2

XV +H1adXV +H0V

then

K1 = H1 −X(H2) +1

3H2

2 , (92)

K0 = −(H0 −

1

3X2(H2)− 2

9H2X(H2) +

2

27H3

2 +1

3H2H1 +

2

9X(H2)H2

)and

H1 = span

adXV −

1

3V H2

, (93)

H2 = span

ad2XV −

2

3(adXV )H2 + V

(1

9H2

2 −1

3X(H2)

).

Let now (t, x, y, p) denote global coordinates on J2(1, 1) (y corresponds to x′ andp corresponds to x′′). Then

XF = ∂t + y∂x + p∂y + F (t, x, y, p)∂p

and VF = spanV , where V = ∂p. We check that

ad3XFV = −∂pFad2

XFV − (2X(∂pF )−∂yF )adXF

V + (X(∂yF )−X2(∂pF )−∂xF )V.

Therefore, from Proposition 4.15 we easily derive the formulae for K0 and K1 inCorollary 4.14. Moreover, H1 = spanV 1 and H2 = spanV 2, where

V 1 = −∂y −2

3∂pF∂p,

V 2 = ∂x +1

3∂pF∂y +

(∂yF +

4

9(∂pF )2 − 2

3X(∂pF )

)∂p.

In order to construct the canonical frame we compute

[V, V 1] = −2

3∂2pFV,

[V, V 2] = −1

3∂2pFV

1 +

(1

3∂y∂pF −

2

3X(∂2

pF )

)V,

and equation Corollary 3.7 implies

V0 = GV,

V1 = GV 1 −G2

3∂2pFG,

V2 = GV 2 +G

(1

3∂y∂pF −

2

3X(∂2

pF )

)G.

Then

[V0,V2] = −G1

3∂2pFV1 mod G.

This gives the structural function T 021 = −G 1

3∂2pF and proves the first formula in

Corollary 4.14. In addition we get

Corollary 4.16. A third order ODE is time-scale preserving contact equivalent tothe trivial equation x′′′ = 0 if and only if it is affine in x′′ and K0 = K1 = 0.


Remark 4.17. Unlike in the case of SODE, our curvature operators do not coincidewith components R(i) of the Jacobi endomorphism for higher order ODEs (see[7] for the definition of Jacobi endomorphism). The reason is that our approachspecialized to differential equations is dual to the approach of Chern, as mentionedin the introduction. We differentiate the Lie algebra of vector fields rather thanthe algebra of differential forms on a manifold. We will explain the duality in moredetails in a future paper.

In the case of third order equations, we have the following formulae which expressJacobi endomorphism in terms of the curvature operators

R(1) = −1

2K1, R(0) = −1

2K0 −

1

2XF (K1).

One can also express the Wunschmann invariant W in terms of K0 and K1. Namelyone has

W = K0 +1

2XF (K1)

(see [16] for an analogous formula involving the Jacobi endomorphism). The similarformulae hold for equations of higher order and generalizations of the Wunschmanninvariant found by Bryant, Dunajski and Tod and others [5, 18]. Actually, theformulae go back to Wilczynski [40] and construction of his contact invariants oflinear ODEs in terms of semi-invariants Pi (cf. [31], remark on page 528).

5. Mechanical control systems. We will now show that vanishing of the Berwaldtensor B, appearing in Section 4.4 as one of two basic invariants of second orderODEs (SODE), corresponds to the fact that SODE describes a mechanical systemin the sense of Bullo and Lewis [8]. More precisely, we will show that a control-affinesystem Σ, defined below, corresponds to a fully actuated mechanical control systemin the sense of [8] and [35] iff the dynamic pair defined by Σ is of equation type(with k = 1) and its tensor B vanishes. We will also characterize more general, notfully actuated, mechanical control systems.

Consider a control system

Σ : x = X(x) +

m∑j=1

ujYj(x),

where the state x is in a manifold M and the control u = (u1, . . . , um) is in Rm.The vector fields X,Y1, . . . , Ym are assumed smooth.

We say that Σ is control gauge equivalent or simply gauge equivalent to anothersystem Σ′ of the same form if there is a diffeomorphism Φ: M →M ′ such that

Φ∗F = F ′, and Φ∗G = G′,where G is the distribution G(x) = spanY1(x), . . . , Ym(x) and G′ is analogously de-fined by Σ′. This is equivalent to existence of a nonsingular matrix (aij) of functions

on M such that Yj =∑i aijY′i .

Motivated by definitions in [8] and [35] we will call Σ mechanical control systemif it is defined by the tuple (Q,∇, d, g0, g1, . . . , gm), where Q is a manifold, ∇ is aconnection on Q having Christoffel symbols Γijk, d : TQ→ TQ is a linear endomor-phism and g0, . . . , gm are vector fields on Q. All of ∇, d and g0, g1, . . . , gm are timedependent, in general. This means that Σ is defined by the second order ODEs

∇q q = g0(t, q) + d(t, q)q +

m∑i=1

ujgj(t, q)


or, equivalently, by the system of first order equations given in local coordinates by

t = 1

Σmech : qi = vi

vi = −Γijk(t, q)vjvk + dij(t, q)vj + gi0(t, q) +

m∑r=1

urgir(t, q).

Note that the drift term X here has a special form and is quadratic in velocity, whilethe vector fields Y1, . . . , Ym have zero components in qi. Such a system is calledfully actuated if m = dimQ and g1, . . . , gm are linearly independent everywhere.

The system Σ will be called almost mechanical control system if it satisfies allrequirements for mechanical control system with the exception that the vector fieldsg1, . . . , gm can be velocity dependent, gj(t, q, v). As the coefficients of the gaugetransformation depend on x = (t, q, v), any fully actuated almost mechanical controlsystem is gauge equivalent to a mechanical control system.

Proposition 5.1. A control system Σ is locally gauge equivalent to a fully actuatedmechanical control system if and only if the following conditions hold.(i) The pair (X,G) is a 1-regular dynamic pair on M , i.e. rkG = m, rk (G +[X,G]) = 2m and

TM = (G + [X,G])⊕ spanX.(ii) (X,G) is of equation type, i.e., the distributions G and G+[X,G] are integrable.(iii) The Berwald curvature B of (X,G) vanishes.

Proof. If (i) and (ii) are satisfied then (X,G) is of equation type, i.e., M is locallyequivalent to the jet space J1(R,Rm) and G can be taken the distribution tangentto the fibers of the bundle J1(R,Rm)→ J0(R,Rm), while the vector field X is thetotal derivative corresponding to some second order system of ODEs, i.e., it is ofthe form X = ∂t+vi∂qi +F i(t, q, v)∂vi . Vanishing of the Berwald curvature implies

that F i are quadratic with respect to v1, . . . , vm (see Corollary 4.11). Thus it hasthe same form as the drift term in Σmech. Since G spans the distribution tangentto the fibers of J1(R,Rm)→ J0(R,Rm), we have G = span∂v1 , . . . , ∂vm and thusG is the same as in the fully actuated system Σmech.

For proving the converse implication it is enough to check that conditions (i),(ii) and (iii) hold for system Σmech. This straightforward exercise is left to thereader.

If we do not require of Σmech to be fully actuated then the required conditionsare more subtle.

Theorem 5.2. A control system Σ is locally control gauge equivalent to a mechan-ical control system (resp. almost mechanical control system) if and only if thereexists a distribution V such that conditions (i)-(v) (resp. (i)-(iv)) stated below hold.(i) The pair (X,V) is a 1-regular dynamic pair.(ii) (X,V) is of equation type, i.e., the distributions V and V+[X,V] are integrable.(iii) The Berwald curvature B of (X,V) vanishes.(iv) G ⊂ V.(v) [V, [X,G]] ⊂ V + [X,G].

Proof. It is straightforward to check that conditions (i)-(iv) (resp. (i)-(v)) are sat-isfied for almost mechanical (resp. mechanical) control systems. Clearly, they areinvariant under gauge equivalence. This proves the “only if” part of the theorem.


We will prove the “if” part. If conditions (i), (ii) and (iii) hold then it followsfrom the proof of Proposition 5.1 that the pair (X,V) is of equation type, thecorresponding second order differential equation is quadratic in velocity and thedistribution V is spanned by ∂v1 , . . . , ∂vm . The last fact and condition (iv) implythat the vector fields Yj are of the form Yj =

∑i gij(t, q, v)∂vi . This means that the

system Σ is locally gauge equivalent to an almost mechanical control system.In order to show equivalence to a mechanical control system, under condition

(v), it remains to show the followingFact. Given a dynamic pair (X,V) of a second order ODE and a distribution

G ⊂ V, we can change a basis in G (gauge transformation) so that new gjj depend

on (t, q), only, if and only if condition (v) holds.In local coordinates we can write X = ∂t+ vi∂qi +F i∂vi , V = span∂vj (we use

the summation convention). Denote ∂r := ∂vr . Then

[X,Yj ] = −∑i

gij∂qi mod V,

[∂r, [X,Yj ]] = −∑i

(∂rgij)∂qi mod V.

If gij do not depend on v then ∂rgij = 0 and [∂r, [X,Yj ]] ∈ V, which implies (v).

To prove the converse implication denote gj = (g1j , . . . , g

mj ). Then condition (v)

implies that [∂r, [X,Yj ]] ∈ span[X,Y1], . . . , [X,Ym]+V. From the above formulaefor the Lie brackets it follows then that there are functions bsrj such that

∂rgj =∑s

bsrj gs.

We look for a gauge transformation Zi =∑j a

jiYj such that the coefficients hji of

new vector fields Zi do not depend on v. This will hold if ∂rhji = 0. Since hi = aji gj ,

with the summation convention, we have

∂rhi = (∂raji )gj + aji∂r gj = (∂ra

ji )gj + asi b

jrsgj .

Thus our requirement is fulfilled if we take the gauge transformation matrix (aji )such that

∂raji = −bjrsasi , where ∂r = ∂vr .

Such a transformation exists, locally. Namely, we first solve the above system ofdifferential equations for r = 1, obtaining new gji independent of v1, then for r = 2,

and so on. After m steps we obtain coefficients gji independent of v.

Remark 5.3. Note that for any system Σ locally gauge equivalent to a mechanicalcontrol system the involutivity condition(vi) [G,G] ⊂ Gholds, since the vector fields Yj of Σmech commute if their coefficients are indepen-dent of v. Ricardo and Respondek [35] characterized mechanical control systemsusing state equivalence. In their case the vertical distribution V can be explicitlyconstructed assuming that the system is geodesically accessible.


6. Veronese webs. We apply our results to get local classification of Veronesewebs of corank 1. Such webs were introduced by Gelfand and Zakharevich [21]in connection to bi-hamiltonian systems. It was conjectured in [21], and provedby Turiel in [38], that Veronese webs determine bi-hamiltonian structures. Normalforms of Veronese webs were provided in [39] (see also [42]). Below we show thatthe framework of dynamic pairs includes Veronese webs (and thus, by results of[21, 38], it includes bi-hamiltonian structures).

We will skip some details as they can be found in a separate article [32]. Thearticle also covers more advanced applications of dynamical pairs to Kronecker webs.

Let

R 3 t 7→ Ftbe a family of corank 1 foliations on a manifold S of dimension k+ 1. Assume thatωt are smooth one-forms annihilating Ft. We say that a family Ft is a Veroneseweb if there exist pointwise linearly independent smooth one-forms

α0, . . . , αk

such that for every x ∈ S

ωt(x) = tkα0(x) + tk−1α1(x) + · · ·+ tαk−1(x) + αk(x).

If we add a one-form at infinity ω∞ = αk then, for every x ∈ S, we get a Veronesecurve in the projectivisation of the cotangent space T ∗xS:

RP 1 3 (s : t) 7−→ R

(k∑i=0

sitk−iαi(x)

)∈ P (T ∗xS). (94)

This curve has a canonical parameter defined by the map t 7→ Ft.We say that two Veronese webs Ft on a manifold S and F ′t on a manifold

S′ are equivalent if there exists a diffeomorphism Φ: S → S′ such that Φ(Ft) = F ′tfor any t ∈ R.

6.1. Dynamic pairs of Veronese webs and equivalence. Let

RP 1 3 (s : t) 7→ R

(k∑i=0

sitk−iYi(x)

)∈ P (TxS)

be the Veronese curve in the projective space P (TxS) dual to the curve (94). Bydefinition, this is a curve Zt(x) in TxS such that

span

Zt(x),

d

dtZt(x), . . . ,

dk−1

dtk−1Zt(x)

= TxFt = kerωt(x), (95)

where

Zt = tkY0 + tk−1Y1 + · · ·+ tYk−1 + Yk (96)

and Y0, . . . , Yk are pointwise linearly independent vector fields on S.Denote

MF =⋃x∈S

P

(k∑i=0

sitk−iαi(x) | (s : t) ∈ RP 1

)⊂ P (T ∗S).

ThenMF is k+2 dimensional manifold. Topologically it is a circle bundle pr : MF →S. In fact MF ' S × S1. Note that the fibres of MF have a canonical affineparameter given by t. If x ∈ S and t ∈ R then (x, t) is a point in MF . On MF


there is a canonical vertical (i.e. tangent to fibres) vector field, denoted XF . Incoordinates

XF = ∂t.

Moreover, MF itself is equipped with a canonical foliation with leaves given by theequations (t : s) = const. This foliation can be treated as a horizontal connectionon the bundle MF → S. Therefore, in particular, we can lift the vector Zt(x) to

a unique vector Zt(x) at the point (x, t) ∈ MF . In this way we obtain a global

vector field (x, t) 7→ Z(x, t) ∈ T(x,t)MF defined on MF . We introduce the rank 1distribution:

VF (x, t) = spanZ(x, t).

Lemma 6.1. The pair (XF ,VF ) on MF satisfies (R1), (R2) and is of equationtype.

Proof. We begin with the observation that adiXFVF is spanned by Y 0, Y 1, . . . , Y i,

where Y i is the lift of ∂itZt to MF . By (96) Y 0, . . . , Y k are independent at anypoint of MF and thus (R1) and (R2) are satisfied. To finish the proof it is sufficient

to prove that adk−1XFVF and adkXFVF are integrable (see condition (R3’) of Theorem

4.5). Integrability of adk−1XFVF immediately follows from the definitions of XF , VF

and from (95). Namely, pr(adk−1XFVF (x, t)) = TxFt. On the other hand adkXFVF is

the distribution tangent to foliation t = const on MF .

We would like to know which dynamic pairs of equation type define Veronesewebs.

Definition 6.2. Let X, V be a smooth vector field and a smooth line field on amanifold M . We say that (X,V) is of Veronese type, if there exists a Veronese webFt on a manifold S such that (X,V) is diffeomorphic to the pair (XF ,VF ) on themanifold MF . We say that (X,V) is locally of Veronese type, if for any x ∈M thereexists a neighbourhood U 3 x and a Veronese web Ft on a manifold S such that(X|U ,V|U ) is diffeomorphic to the pair (XF |V ,VF |V ) for an open subset V ⊂MF .

Theorem 6.3. Let (F ) be an equation of order k+ 1. The corresponding dynamicpair (XF ,VF ) on M = EF ' Jk(1, 1) is locally of Veronese type if and only if allcurvature operators K0, . . . ,Kk−1 vanish.

Proof. First, note that if (X,V) is of Veronese type then in local coordinates onMF we have X = ∂t and V(t, x) is spanned by

Z(t, x) = tkY0(x) + tk−1Y1(x) + · · ·+ tYk−1(x) + Yk(x),

see formula (96). Since adk+1∂t

Z = 0, it follows that Z is a normal generator of Vand all curvature operators vanish.

On the other hand, if (X,V) is of equation type and all its curvature operators

vanish, then we can choose a section V of V such that adk+1X V = 0. Let us choose

an open subset U ⊂ M with local coordinates such that X = ∂t on U (we canalways locally trivialise X). Then along any integral curve of X contained in U weget the formula

V (t) = tkV0 + tk−1V1 + · · ·+ tVk−1 + Vk,

where V0, . . . , Vk are constant vectors along an integral curve of X. Indeed, theequation adk+1

X V = 0 means that along an integral curve of X the vector field V is

a solution to the equation dk+1Vdtk+1 = 0 and thus V is polynomial in t.


Take U so that the set of trajectories of X in U forms a Hausdorff manifold anddefine S to be the quotient space S = U/X. This means that a point x ∈ S isan integral line of X with parameter t belonging to some segment Ix ⊂ R. If weproject V(t) = spanV (t) to S for every t ∈ Ix we get a segment of Veronese curvein P (TxS). Since a Veronese curve is uniquely determined by a finite number of itspoints, we can uniquely extend the segment of Veronese curve to the full Veronesecurve. The dual Veronese curve in P (T ∗S) defines the desired Veronese web.

Theorem 3.6 applied to the pair (XF ,VF ) give the following:

Theorem 6.4. The following conditions are equivalent.(a) Veronese webs Ft and F ′t are equivalent.(b) The dynamic pairs (XF ,VF ) and (XF ′ ,VF ′) are diffeomorphic by a diffeomor-phism preserving t.(c) The canonical frames (X,V,G) and (X′,V′,G′) on the normal frame bundlesπ : FN → MF and π : F ′N → MF ′ , corresponding to dynamical pairs (XF ,VF ) and(XF ′ ,VF ′) via Theorem 3.6, are diffeomorphic by a diffeomorphism preserving t.

Proof. Assume first that Ft and F ′t are equivalent Veronese webs and theequivalence is established by Φ: S → S′. Let Ψ: MF → MF ′ be the lift of Φdefined in an obvious way. By definition of equivalence of webs we get Φ(Ft) = F ′tand hence Ψ preserves t. Moreover, Ψ maps fibres of MF → S onto fibres ofMF ′ → S′ and we get that Ψ∗XF = XF ′ . It is also a direct consequence of thedefinitions that Ψ∗VF = VF ′ because Φ∗(kerωt) = kerω′t, for any t, which implies

that Φ∗(spanZt) = spanZ ′t and, consequently, Ψ∗(spanZ) = spanZ ′.On the other hand, if Ψ: MF → MF ′ establish equivalence of dynamic pairs

(XF ,VF ) and (XF ′ ,VF ′) then Ψ∗XF = XF ′ and thus it transports fibres of MF →S onto fibres of MF ′ → S′. Hence, Ψ defines a mapping Φ: S → S′. If Ψ∗VF = VF ′then also Ψ∗adk−1

XFVF = adk−1

XF′VF ′ . The projection of a leaf of adk−1

XFVF is a leaf

of the foliation Ft, for some t, thus Φ maps leaves of Ft onto leaves of F ′t. Ifadditionally Ψ preserves t we get that Φ(Ft) = F ′t for any t. This proves (a)⇔ (b).

(b) ⇔ (c) follows directly from Theorem 3.6.

Corollary 6.5. A Veronese web has at most k+2-dimensional group of symmetries.The group has maximal dimension if and only if the web is flat, i.e., it is given by

the kernel of the 1-forms ωt =∑ki=0 t

k−idxi, in some coordinates on S.

Proof. Note that Theorem 3.6 imply that the group of symmetries of a web is atmost k + 3 dimensional. However, statement (c) of Theorem 6.4 says that not allsymmetries of the canonical frame of a dynamic pair (XF ,VF ) define symmetriesof Ft. Namely a symmetry must keep t invariant. Therefore we get that thedimension of the symmetry group is bounded from above by k + 2. Moreover,it follows that if the symmetry group has maximal possible dimension then thestructural functions of the canonical frame (V,G) have to be constant on each leafFN |t=const ⊂ FN . Therefore the part of the curvature and the torsion of thecanonical connection which involves V vanish. Then, using Jacobi identity appliedto [X, [Vi,Vj ]] we get that the part of the curvature involving X and Vi also vanish.Moreover, taking into account that Ki ≡ 0 for an arbitrary Veronese web (Lemma6.1) we get that the Veronese web is flat.


6.2. Veronese webs on a plane. Let F = Ft be a Veronese web on R2 definedby a family of 1-dimensional distributions

spantY0 + Y1,

where Y0, Y1 are smooth vector fields on R2. Theorems 4.8, 4.5 and 6.3 imply

Theorem 6.6. The canonical frame on the bundle FN , corresponding to the dy-namic pair (XF ,VF ), satisfies:

[X,V0] = V1, [X,V1] = 0, [V0,V1] = RG

for a certain function R such that X(R) = 0.

Since V0 and V1 are homogeneous of order one with respect to the fiber coor-dinate in the normal bundle FN → MF , and G are homogeneous of order zero,it follows that R = G2R, in coordinates, where R is a function on MF . However,X(R) = 0 implies that R is in fact well defined on R2. Let us fix a point x ∈ R2 andlet x = (x, 0) ∈ MF (x). Let us also choose a point ν in the bundle FN (x). Thenwe can introduce a coordinate system on R2 in the following way:

(x0, x1) 7→ pr exp(x0V0) exp(x1V

1)(ν), (97)

where pr : FN → R2 is the projection composed of the projections FN →MF → R2.If we change ν 7→ νG for some G ∈ GL(1) ' R∗, the coordinates are multiplied bya real number. Therefore we get a canonical local system of coordinates on R2 withthe origin in x, given up to multiplication by a constant. In this coordinates we canexpress function R and get the function on R2 intrinsically assigned to the web.

Corollary 6.7. There is one-to-one correspondence between germs of Veronesewebs at 0 ∈ R2 and germs of functions R : R2 → R at 0 given up to the transforma-tions

R(x0, x2) 7→ G2R(Gx0, Gx1), G 6= 0.

Proof. We shall show how to recover the web from the function R. First we considerR2 × GL(1) with coordinates x0, x1 and G (this space is the level set t = 0of the canonical bundle FN ). We define the function R(x0, x1, G) = GR(x0, x1)on R2 × GL(1). It follows from the definition of the canonical coordinate system(formula (97)) and the relation [G,Vi] = Vi that we can assume V0 = G∂x0 andV1 = Ga∂x0 +Gb∂x1 +Gc∂G for some functions a, b, c in variables x0, x1. Thus theequation [V0,V1] = RG in Theorem 6.6 implies the following:

∂x0a− c = 0, ∂x0

b = 0, G∂x0c = R.

On the plane x0 = 0 we have a = c = 0 and b = 1 (again we use the definition (97)of the coordinate system). Hence we are able to recover a, b and c in a unique way.The web on the (x0, x1)-plane is spanned by: pr∗(V

0(x0, x1, 1) + tV1(x0, x1, 1))where pr : R2 ×GL(1)→ R2 is the projection on the first factor.

6.3. Veronese webs on R3. Let F = Ft be a Veronese web on R3 given by thekernel of ωt, where

ωt = t2α0 + tα1 + α2

and α0, α1, α2 are smooth one-forms on R3. Theorems 4.13, 4.5 and 6.3 imply:


Theorem 6.8. The canonical frame on the bundle FN , corresponding to the dy-namic pair (XF ,VF ), satisfies the relations:

[X,V0] = V1 − TG, [X,V1] = V2, [X,V2] = 0,

[V0,V1] = 2V0(T )G,

[V0,V2] = TV1 + (V1(T )− 2T 2)G,

[V1,V2] = 2TV2,

for a certain function T such that

X(T ) = 0 and V0V2(T ) + V2V0(T ) = V1V1(T )− 4TV1(T ) + 2T 3.

Let us denote

V0 = V0, V1 = [X,V] = V1 − TG, V2 = V2.

Then (V0, V1,X,G) is a new frame on FN and the distribution D = spanV,Xis a new principal connection. The structural equations for the frame (V,X,G),which we call second canonical frame, take the following elegant form.

Theorem 6.9. The second canonical frame satisfies:

[X, V0] = V1, [X, V1] = V2, [X, V2] = 0,

[V0, V1] = T V0 + V0(T )G,

[V0, V2] = T V1 + V1(T )G,

[V1, V2] = T V2 + V2(T )G,

for a function T such that

X(T ) = 0 and V0V2(T ) + V2V0(T ) = V1V1(T ). (98)

Proof. Follows from Theorem 6.8 by direct computations.

Remark 6.10. The functions T in Theorems 6.8 and 6.9 coincide.

In coordinates T = GT , where T is a function on MF . However, similarly to thecase of Veronese web on the plane, it follows from X(T ) = 0 that T is well definedon R3. Let us fixed a point x ∈ R3 and take x = (x, 0) ∈MF (x). Let us also choosea point ν in FN (x). We introduce the following coordinate system on R3:

(x0, x2, x1) 7→ pr exp(x1V1) exp(x2V

2) exp(x0V0)(ν),

where pr : FN → R3 is the projection. Note that at the beginning we go along V0

then along V2 and finally along V1.We are able to compute any derivative of the form ∂a+b+cT /∂xa0∂x

b2∂x

c1 at the

origin of our coordinate system. Indeed, this is equivalent to (V0)a(V2)b(V1)c(T ).Moreover, it follows from the structural equations and (98) that any derivative of

the form (V0)a(V2)b(V1)c(T ) with arbitrary c ∈ N can be written as a sum of

derivatives (V0)a(V2)b(V1)c(T ) where c = 1 or c = 0. Therefore if we know T and

S = V1(T ) on the plane x1 = 0 then we are able to recover all possible derivatives

∂a+b+cT /∂xa0∂xb2∂x

c1 at the origin of our coordinate system. Hence, if all data are

analytic then we can recover T on R3.The coordinate system is unique up to the choice of the point ν ∈ FN (x). If we

change ν then every coordinate function is multiplied by G ∈ GL(1) ' R∗. We get


Corollary 6.11. In the analytic category there is one-to-one correspondence be-tween germs of Veronese webs at 0 ∈ R3 and germs at 0 of two functions T and Sin two variables: x0 and x2. The functions are given up to the following transfor-mations:

T (x0, x2) 7→ GT (Gx0, Gx2), S(x0, x2) 7→ GS(Gx0, Gx2).

Proof. The proof is similar to the proof of Corollary 6.7 and we skip it.

Acknowledgments. We are grateful to Boris Doubrov who recently informed uson the paper of Chern [13] and provided us with a copy of it. We also acknowledgehelpful suggestions of the reviewers concerning the bibliography.

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