Vector fields with distributions and invariants of ODEs

We study pairs (X,V) where X is a vector field on a smooth 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. For a given pair (X,V) we construct a canonical connection on a certain frame bundle. The results are applied to the problem of time-scale preserving equivalence of ordinary differential equations. The framework of pairs (X,V) is shown to include sprays, Hamiltonian systems, Veronese webs and other structures.


Introduction
In this paper we study pairs (X, V) where X is a vector field on a manifold M and V ⊂ T M is a vector distribution (a sub-bundle of the tangent bundle T M ), both satisfying certain regularity conditions. Such pairs appear to encode a large variety of geometric objects as geodesic sprays on Riemannian (pseudo-Riemannian) manifolds and, more generally, manifolds with affine connections, spray spaces, control systems, systems of ordinary differential equations, Veronese and Kronecker webs.
The aim of the article is twofold. The first aim is to study general geometric objects attributed to the pair (X, V) and identify the invariants. The second aim is to test the proposed approach on systems of ordinary differential equations of order ≥ 2, where pairs (X, V) appear as a tool. We use time scale preserving equivalence. This is done in the second part of the article, where we solve the corresponding equivalence problem. The approach was partially developed in the thesis [19].
The equivalence problem for ODEs is a classical one and was mainly attacked using contact or point transformations (see Cartan [5], Chern [6], Bryant [3], Fels [13], Doubrov, Komrakov and Morimoto [11], Dunajski and Tod [12], Godlinski and Nurowski [16] for a very partial list of contributions). We do not address this version of the problem, which is more complicated and gives less hopes for a simple and complete solution (i.e., identifying complete sets of independent invariants).
Time scale preserving equivalence problem was less studied, even if it is more natural from the point of view of applications. The problem was formally solved by Chern [7] (systems of order two) and [8] (systems of higher order), using Cartan's method of equivalence (his second paper was totally ignored in the literature). Our approach (Section 3) treats ODE's as pairs (X, V) with a flag of integrable distributions and can be thought, roughly, as dual to Chern's [8] (more details on dual approaches and formulas for invariants will be provided in a future paper). We additionally give relations between invariants of systems of order two (Section 3.4).
The setting considered here includes control systems and variational equations appearing there (cf. e.g. [21,4,2]). It also includes equivalence problems of special geometric structures, one of them being a Veronese web (Gelfand and Zakharevich [14,26], Touriel [25]). We show that a Veronese web can be coded as a pair (X, V) and its invariants can be found as a special case of our invariants (Section 4).
The simplest invariants, the curvature operators which we construct in Section 2.2, generalise the curvatures used in [17] for an analysis of the variational equation.
In the special case of geodesic spray on the tangent bundle of a Riemann or Finsler manifold, or a general spray (cf. Shen [22]), there is only one curvature operator K 0 , equivalent to the Riemann curvature (Section 2.3). It is the curvature which appears in the generalised Jacobi equation (see [22], Chapter 8). Our formalism applied to Hamiltonian systems (Section 2.3) gives a symmetric curvature operator K 0 which is equivalent to the curvature introduced in a different way by Agrachev and Gamkrelidze [2] and used for estimation of conjugate points.
We outline our approach. The subject of the study is a pair (X, V) of a vector field X and a distribution V ⊂ T M on a smooth manifold M . We attach to it a sequence of distributions defined inductively, using Lie bracket, by We impose natural regularity conditions on (X, V), assuming that the distributions have maximal possible ranks. More precisely, we assume that dim M = (k +1)m+1, k, m ≥ 1, and (R1) rk V i = (i + 1)m, for i = 0, . . . , k, Our formalism also works with modified assumptions, where dim M = (k + 1)m and (R2) is replaced with (R2') V k = T M , and X(x) = 0, for x ∈ M .
A pair (X, V) satisfying (R1) and (R2) is called dynamic pair or regular pair. We show that there is a natural class of frames in T M (called normal frames), attached in an invariant way to a regular pair (X, V). This class of frames defines a canonical G-structure P on M , where G = Gl(m). The G-structure has the following property.
Two regular pairs (X, V) and (X , V ) are locally equivalent if and only if the corresponding G-structures are isomorphic.
Above, two pairs (X, V) and (X , V ) are called (locally) equivalent if there is a (local) diffeomorphism Φ : M → M such that Φ * X = X and Φ * V = V .
In order to solve the equivalence problem in a more explicit way we use the principal G-bundle P , defined by the G-structure, and define a canonical frame on P . Our main results in Section 2 (Theorems 2.9 and 2.11) say the following.
Two regular pairs (X, V) and (X , V ) are locally equivalent if and only if the corresponding canonical frames are.
These theorems give a tool for solving the equivalence problem, since there is a standard procedure for determining if two frames are equivalent. Additionally, the canonical frames enable us to define a principal connection on P . The connection can be used for the analysis of geometric properties of the pair (X, V).
Important geometric objects attached to (X, V) are curvature operators, defined in Section 2.2 (see also [19] and [17]). In the classical case of semi-sprays defined by differential equations of order two there is only one such operator.
In the second part (Section 3) we study systems of ordinary differential equations x (k+1) = F (t, x, . . . , x (k) )), where x ∈ R m and F is of class C ∞ . Then we take M = J k (R, R m ) -the manifold of k-jets of parametrised curves in R m . The vector field X is the total derivative A vertical distribution V = V F on E F can be defined as the restriction to E F of the kernel of the canonical projection J k+1 (1, m) → J k−1 (1, m). In coordinates V F = span{∂ x j k | j = 1, . . . , m}.
At the end (Section 4) we study Veronese webs, i.e. one-parameter families of foliations of special type, introduced by Gelfand and Zakharevich [14] and strongly related to bi-hamiltonian systems. We prove that Veronese webs can be treated as ODEs for which our curvature operators vanish. In this way we relate Veronese webs with dynamic pairs and we solve the equivalence problem. It appears that Veronese webs play the same role for time-scale preserving contact transformations as paraconformal structures (called also Gl(2)-structures) play for the general contact transformations [3,12,16,20].
Other examples of pairs (X, V) that can be interesting for applications are: (1) Given a Riemannian or a pseudo-Riemannian manifold N , we take X to be the geodesic spray on M = R × T N and V the vertical distribution on the bundle R × T N → N .
(2) For a vector field X on a fibre bundle E → B, we may take M = E and Vthe vertical distribution tangent to fibres.
(3) Given a control systemẋ = f (x)+ i u i g i (x) on a manifold M , we take X = f and V = span{g 1 , . . . , g m }.
2 Vector fields with distributions 2.1 Dynamic pairs (X, V) and their normal frames Let M be a smooth differentiable manifold of dimension n. Consider a C ∞ dynamic pair (X, V), i.e., a vector field X on M and a distribution V ⊂ T M of rank m, satisfying the regularity conditions (R1) and (R2). Then V(x) is an m-dimensional subspace of the tangent space T x M , for every x ∈ M and, locally, there exist m smooth vector fields V 1 , . . . , V m such that If V 1 , . . . , V m span V on an open subset U ⊂ M then we will briefly denote them as the row vector V = (V 1 , . . . , V m ) and call V a local frame of V on U .
Recall that the pair (X, V) defines a sequence of distributions denotes the Lie bracket of vector fields. Denote by X the 1-dimensional distribution spanned by X.
Note that it follows from (R1) and (R2) that the vector fields X and ad i X V j (x), 0 ≤ i ≤ k, 1 ≤ j ≤ m, are linearly independent and they span T M . Thus, denoting where H i are m × m matrices of functions. Then V is a normal frame iff H k = 0. Let V = (V 1 , . . . , V m ) and W = (W 1 , . . . , W m ) be local frames of V on U . Then there exists a unique Gl(m)-valued function G : U → Gl(m) such that, in the matrix notation, Proposition 2.2 (a) Given a regular pair (X, V) and a frame V . . , m, are the desired vector fields, i.e., In the matrix notation It follows from (R1) and (R2) that (2) holds, thus for a certain square matrix (6) is equivalent to the following equation for the unknown function G: This is a linear first order differential equation for G, thus it can be solved, locally, so that G(x 0 ) = Id. If G is a solution, we have ad k+1 (b) As above, we have W = V G. Since all elements Y = V i of V satisfy (1), the matrix H = H k in equation (7) is zero. Equation (6) implies that the matrix valued function G satisfies (8), thus X(G) = 0.
(c) This follows from the proof of (a), as H = H k and equations (5) and (8) coincide.
Note that the "initial condition" V (x 0 ) = V x 0 in statement (a) can be imposed on any local hypersurface transversal to X in M . From statement (b) we get

Curvature operators
We will define the most basic invariants of dynamic pairs, called curvature operators. Proposition 2.2 says that if both V and W = V G are normal frames of V then X(G) = 0. Hence, ad i X W = (ad i X V )G for any i. It follows that distributions do not depend on the choice of a normal frame V = (V 1 , . . . , V m ), for i = 1, . . . , k. H i will be called i-th quasi-connection, for reasons which will become clear in the second part of the paper. We also denote H 0 = V and stress that rk H i = m.
The last relation defines projections Notice that the operators are vector bundle isomorphisms, similarly as In particular, A 0 = Id : V → V.
The alternating sign is chosen for consistency with classical interpretations of curvatures and, in particular, for simplicity of the variational equation [17]. Equivalently, K i can be defined as follows. Let x ∈ M . In a fixed basis of the space V(x) the operator K i (x) is represented by m × m matrix, also denoted K i (x). If V = (V 1 , . . . , V m ) is a normal frame, then matrices of the curvature operators are defined in the basis V 1 (x), . . . , V m (x) of V(x) by the following equation: Due to the transformation rule (9), when normal generators V is transformed to normal generators V G, the matrices K i are transformed via The operators 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 H i (x) are not invariantly assigned to the pair. However, the curvature matrices can be computed in terms of matrices H i . In particular, we have and Proof. We use statement (c) in Proposition 2.2, which states that , using the assumed formula for ad 2 X V and eliminating X(G) and X 2 (G) gives This leads to the formula for K 0 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 the connection H 1 follows from ad X (V G) = (ad X V )G + V X(G).

Examples
where K 0 is the curvature matrix defined according to formula (10): ad 2 X V = V K 0 (the coefficient at X in (10) vanishes because of the term ∂ t in X). One can check, consulting [1], that the above curvature K 0 coincides with the curvature introduced in a completely different way in [2]. The following proposition is easy to prove.
is Lagrangian with respective to σ, on each fiber {t} × N . (b) The matrix g = (g ij ) given by is symmetric, nondegenerate, and defines a pseudo-Riemannian metric on V.
(c) The matrix K 0 is symmetric and defines a selfadjoint (relative to g) operator Proof. Since V is Lagrangian, we have σ(V i , V j ) = 0. Lie differentiating this equality with respect to H gives and proves the symmetry of g in (b), since σ is antisymmetric. Differentiating twice gives . The side terms are zero which follows from (14) and the fact that V is Lagrangian.
]) = 0, which shows (a). Differentiating three times yields and, applying (14) and the summation convention, where in the second equality we use the fact that σ(H((K 0 ) s i )V s , V j ) = 0, as V is Lagrangian. Using antisymmetry of σ and the definition of g ij we get

Remark.
A canonical example of the pair as above is a time dependent vector field H on the cotangent bundle N = T * Ñ of a differentiable manifoldÑ , where V t is the vertical distribution of the bundle π : T * Ñ →Ñ , i.e., V t (x) = T π(x)Ñ .
Another example (see [2]) is given by a Hamiltonian function h : T * Ñ → R. The corresponding Hamiltonian vector field H and the vertical distribution never form a regular pair with k = 1 since the dimension of M = T * Ñ can not be equal to 2m + 1. However, we can have a regular pair if we restrict our considerations to a level submanifold of the Hamiltonian. Namely, take M = {h = c} ⊂ T * Ñ and the vector field X equal to H restricted to M , X = H| M . The distribution V on M is defined as the vertical distribution of the cotangent bundle intersected with the tangent space to M : Then, if dimÑ = m + 1, we have dim V(x) = m, dim M = 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 canonical symplectic form σ on T * Ñ replaced with the symplectic formσ = σ| M considered on the quotient space T x M/span{H(x)}.
Geodesic spray. Consider a geodesic equation on a manifold N of dimension m with local coordinates (x i ). In local coordinates we have where Γ i pq are Christoffel symbols of a connection ∇. Note that the equation does not depend on the torsion of the connection and thus we will assume that ∇ is symmetric. Let J 2 (R, N ) be the space of 2-jets of functions R → N . On J 2 (R, N ) there are local coordinates (t, , induced by the coordinates (x i ), where i = 1, . . . , m. The geodesic equation is uniquely defined by the submanifold There is a canonical projection J 2 (R, N ) → T N × R which, restricted to E, defines the diffeomorphism E T N × R. In particular (x i 0 ) = (x i ) are local coordinates on N whereas (x i 1 ) are the corresponding linear coordinates on the fibres of T N → N .
be the total derivative. We take the vertical distribution tangent to the fibres of T N , that is V = span{∂ x i 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 Since k = 1, there is only one curvature operator K 0 . From Proposition 2.5 we get We see that K 0 is a quadratic function in the coordinates on fibres x p 1 , x q 1 and we recognise that the coefficients A simple calculation using Bianchi identity for R gives the converse formula where x = (t, (x i 0 )) and we identify the elements y = Y, z = Z, w = W . We also have It follows that the quasi-connection H 1 coincides, in the Ehresmann sense, with the connection ∇. More general cases will be considered in Chapter 3.
ODE with constant coefficients. Consider the following linear system of ODEs be the space of k-jets of functions from R to R m . The standard global coordinate functions on J k (1, m) are denoted (t, x j i ), where i = 0, . . . , k and j = 1, . . . , m. We be the corresponding total derivative, where we abbreviate is a normal frame of the pair (X, V). Moreover, a simple induction gives that the corresponding curvature operators are given by the matrices K i = (−1) i A i .

Canonical bundle
In order to identify further invariants of dynamic pairs, and solve the equivalence problem, we introduce a sub-bundle F N (X, V) of the tangent frame bundle, called canonical or normal frame bundle of (X, V). Assume (X, V) satisfy (R1) and (R2).
where the operators A i = π i • ad i X : V → H i are the vector bundle isomorphisms from Section 2.2. Both definitions coincide since they coincide for a local normal frame V and, given the pair (X, V), the second one depends on the value V (x), only. A local normal frame in T M is a smooth local field of normal frames x → F x .
Denote by F N (x) the set of all normal frames in T x M . The set 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, There is a natural right action of Gl(m) on F N given by With respect to this action F N is a principal Gl(m)-bundle. This means that Gl(m) acts transitively and freely on each fiber F N (x). To check this pick two normal frames F (x) and F (x) at x, given by local normal frames V and V of V.
Thus, the corresponding local normal frames F and F are related by the block diagonal matrix If x ∈ M is fixed, the group of block diagonal matrices G(x) is isomorphic to the group Gl(m), with the isomorphism given by G(x) → G(x). We conclude that Proof. The second statement follows directly from the definition of F N (X, V) and from equivariance of the Lie bracket with respect to diffeomorphisms. Note that

Canonical frame and connection, equivalence
Assume that a pair (X, V) satisfies conditions (R1) and (R2). To any such pair we will assign, in an invariant way, a frame on the canonical bundle. Moreover, we will show that the normal frame bundle π : F N → M with the structural group where n = (k + 1)m + 1, possesses a canonical principal connection understood as a smooth distribution D on F N , which is transversal to the fibres and which satisfies for any F ∈ F N and any A ∈ Gl(m). Here R A is the transformation of F N induced by the right action: . Then there exists a unique tuple of vectors lifted to D, i.e., such that V i j , X ∈ D(F ) and We will briefly denote . In addition, on F N there are defined fundamental vector fields which are tangent to the fibres of F N and which come from the action of the structural group Gl(m). The fundamental vector fields with the Lie bracket form a Lie algebra isomorphic to the Lie algebra g of the structural group, where is naturally isomorphic to gl(m). Vector fields corresponding to matrices e s t with 1 at the position (s, t) = (row, column) and 0 elsewhere will be denoted G t s and the collection of all such vector fields will be shortly denoted G = (G t s ) s,t=1,...,m . Clearly the tuple (F, G) is a frame on F N defined uniquely by D.
Vice versa, any frame (F, G) on F N , where G consists of the fundamental vector fields, defines a unique Ehresmann connection D = span{X, V i j : 0 ≤ i ≤ k, 1 ≤ j ≤ m} on F N . In our case both the connection and the frame will be Gl(m) invariant.
We will show that one can choose D in a canonical way. Then two pairs (X, V) and (X , V ) will be equivalent if and only if the corresponding connections are equivalent (or if and only if the corresponding frames are equivalent).
Let (θ j i , α, ω s t ) be the coframe on F N dual to the frame (V i j , X, G t s ). We put . These functions are called structural coefficients of the frame. We will see in the next section that they are coefficients of the torsion and curvature of the connection D. Some of them can be normalised so that the connection and frame are unique.
Namely, the main result of this part of the paper says the following: Theorem 2.9 Assume that (X, V) satisfies (R1) and (R2), with a given k ≥ 1.
(a) There exists a unique principal connection on F N such that the corresponding frame (V, X, G) satisfies the following conditions: It then also satisfieŝ The symmetry group of (X, V) is at most (k + 1)m + m 2 + 1-dimensional, and it is maximal if and only if (X, V) is locally equivalent to a pair defined by the system where x ∈ R m and matrices K i are diagonal and constant.

Lemma 2.10
There is a unique vector field X on F N satisfying (17) and π * X = X.
Proof. Let us fix an open subset U ⊂ M and a section U x → F x ∈ F N of normal frames, F x = (V 0 (x), . . . , V k (x), X(x)). Then a point ν ∈ F N is encoded by its projection x = π(ν) ∈ U and an element G = G(ν) ∈ Gl(m) such that In this way we get a local trivialisation F N = U × Gl(m). Using this trivialisation we introduce natural coordinates on fibers of F N (the coordinates depend on the initial choice of the section x → F x but our construction will not). The vertical vector fields G t s , in these coordinates, are given by where G r s (ν) is the (r, s) = (row, column) coordinate of the above defined matrix G = G(ν). The lifted vector fields X and V i j (see (16)) can be written in coordinates as for some functions α s t and β is jt . By direct checking we see that This relation and (29) give q are coefficients of the matrixĤ = G −1 HG. Consequently, we haveT ks jk = α s j +Ĥ s j and condition (17) is satisfied if and only if α s t = −H s j . In this way the first part of the lemma is proved and we get the explicit formula The second part follows directly from the first part if we choose the section of normal Then H i j = 0 and X = X. Thus any integral curve in F N of X is a section of a local normal frame in T M coming from a local normal frame V G (the matrix G is kept constant along such a curve).
Proof of Theorem 2.9. (a) We will use the local section U x → F x ∈ F N of normal frames, F x = (V 0 (x), . . . , V k (x), X(x)), and local coordinates defined in the proof of Lemma 2.10.
The lifted vector fields can be written Assume that We will show that β is jt are uniquely determined by conditions (18) and (19), namely Taking into account that we will see that the resulting frame on F N consists of vertical vector fields G t s and of can be identified with right invariant vector fields on Gl(m) corresponding to matrices e r q , with 1 at the position (q, r) and 0 elsewhere. To prove (34) note that from (32) and (33) we obtain Thus (18) is equivalent to the equations for p, q, j = 1, . . . , m, and (19) is equivalent to the equations: for p, q, j = 1, . . . , m and l = 1, . . . , k. It follows from these equations that (β ls pq ) are uniquely determined by (18) and (19) and are given by (34).
It is evident from (36) that so constructed frame is invariant under the right action of Gl(m) on F N . This means that the distribution D = span{X, V i j }, where i = 0, . . . , k and j = 1, . . . , m, defines a principal connection on F N . Thus statement (a) of the theorem is proved. Note that condition (25) is a direct consequence of (36), thus so are conditions (20) and (21) .
(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 statement (a), since the dimension of the symmetry group of a frame is bounded from above by the dimension of the ambient manifold (see [18], Chapter 1, Theorem 3.2). In our case this dimension is dim F N = (k + 1)m + m 2 + 1.
The symmetry group of (X, V) is maximal if and only if the structural functions of the frame (G s t , X, V i j ) are constant. We have (26) and, by definition, Lie brackets of vertical vector fields G t s behave as generators of gl(m), thus have constant structural functions. Therefore, in order to finish the proof we have to check when the structural functions of [X, V i j ] and of [V i p , V l q ] are constant as functions on F N . The coefficients β i j in (34) are homogeneous of order one with respect to G. It implies that the structural functions of the bracket [V i p , V l q ] are homogeneous of order either one (functions next to G s t and V i j ), or two (a function next to X). Thus, in order to be constant they have to vanish. Condition (25), already proved, says: Moreover, By homogeneity argument, we see that in the most symmetric case coefficients next to X and G vanish, whereas K i have to be diagonal and constant so that In conclusion we see that structural functions of a pair (X, V) with maximal symmetry group coincide with the structural functions of the pair (X, V) corresponding to the system with diagonal and constant K i . (The canonical frame of the pair (X, V), which corresponds to the linear system, is given by G s t , X = X and V i j = G s j ad i X ∂ x s 0 , since vector fields ad i X ∂ x s 0 are constant and all their Lie brackets vanish.) There is also another way of defining the canonical frame on the bundle F N , with condition (19) replaced by a condition on the structural functionsR is pt = ω s t ([X, V i p ]). It will be more convenient in certain situations.
Theorem 2.11 Assume that (X, V) satisfies (R1) and (R2). There exists a unique principal connection on F N such that the corresponding frame (V, X, G) satisfies (17), (18) and the following condition: Two pairs (X, V) and (X , V ) are diffeomorphic if and only if the corresponding frames are diffeomorphic.
Note that condition (39), taking into account the form (32) of V i j , is equivalent to Proof. The proof is similar to the proof of Theorem 2.9. We get from (17) and Lemma 2.10, in coordinates used in its proof, that X = X − s,tĤ s t G t s . Then we use equation (37) with l = 1 and we see that condition (18) is equivalent to (38), and to (34) with i = 0. In this way we normalise V 0 = (V 0 1 , . . . , V 0 m ) so that (32) whereĜ q r = t G q t ∂ G r t and the coefficients b 1r pq1 are determined from the relation where V 0 (x), . . . , V k (x), X(x) is a section of the normal frame bundle F N . Then (40) defines V i uniquely. The resulting frame defines a principal connection. Note that we can take V i j = ad i X V j , where (V 1 , . . . , V m ) is a local normal frame of V. In this case we can apply formula (31) withĤ = 0, thus X = X. We get which follows from X = X, (40) and (41).

Definition 2.12
The frame of Theorem 2.9 will be called first canonical frame, denoted V, and the frame of Theorem 2.11 the second canonical frame, denotedṼ. By definition, the vector fields X and G in both canonical frames coincide.
The following direct corollaries of the proofs of Theorems 2.9, 2.11 and Definition 2.7 will be useful.

Corollary 2.13
The coefficients (35), needed for determining the first normal frame (36), and the coefficients (43), needed for determining the second normal frame (41), can be computed using any local frame V of V k by defining the section of F N via Corollary 2.14 If k = 1, the operator Example. For illustration we continue the example of geodesic equation from Section 2.3 (see Section 3.4 for a general case). We will use the above corollaries. In order to solve the equivalence problem one has to construct V 0 , V 1 and G on F N = R × T N × Gl(m) (we locally trivialise the canonical bundle). The vector fields in G are standard, given by (27). Take and compute the corresponding components V 1 j = A 1 V j of the normal frame. From the formulas for ad X V j and ad 2 X V j computed earlier we see that the matrix H 1 is (H 1 ) j i = 2 p Γ j ip x p 1 . This and ad We have 1 and the coefficients defined in the proof of Theorem 2.9 via formula (33) are b 1j pq1 = 0 and b 1j pq0 = Γ j pq . Hence, formulas (36) and (35) give It is straightforward to verify that are components of the curvature tensor of ∇. Due to formula (26) we see that there are no more nontrivial structural functions on F N .

Torsion and curvature
We can describe the principal connection given by Theorem 2.9 (or Theorem 2.11) by a connection form. The corresponding curvature and torsion of the connection can be used as an alternative description of the invariants of the pair (X, V). Denote φ = (θ j i , α) and ω = (ω s t ). Then ω is a 1-form on F N with values in the Lie algebra g gl(m), called the connection form. It defines the connection D by D(F ) = ker ω(F ). The 1-form φ on F N with values in R n is called the canonical soldering form, where n = (k + 1)m + 1 is the dimension of the manifold M . The following Cartan structural equations are satisfied dφ + ω ∧ φ = Θ, dω + ω ∧ ω = Ω and define torsion Θ and curvature Ω of the connection, both being 2-forms with values in R n and g, respectively.
The structural functions from Section 2.5 can be described in terms of the torsion and curvature. Namely, since Θ has values in R n , where n = (k + 1)m + 1, we can decompose Θ = (Θ j i ,Θ) j=1,...,m i=0,...,k . Similarly, we can write Ω = (Ω s t ) s,t=1,...,m . Then and The coincidence of the coefficients in these formulas and the structural functions introduced in Section 2.5 follows from the general formula dβ(Y, Z) = −β([Y, Z]) for a 1-form β belonging to a coframe and Y , Z being frame vector fields in D = ker ω.
Note that the structural functions satisfy relations which follow from the Bianchi identities DΘ = Ω ∧ θ and DΩ = 0, where D denotes the covariant derivative. We will not use them in full generality in the present paper, but only restrict to the simplest cases, important for our applications. Namely, in the next section we will consider cases k = 1 and k = 2 with additional integrability conditions. We will work in terms of canonical frame rather than connection and use the identities in the form of Jacobi identity for the vector fields in the canonical frame.
The freedom of choosing torsion normalisation conditions in Theorem 2.9 can be explained following Sternberg [23]. Note that both Θ and Ω vanish if one of their arguments is in the vertical distribution span G. Thus at a fixed point F ∈ F N they can be considered as elements of hom(D(F ) ∧ D(F ), R n ) and hom(D(F ) ∧ D(F ), g), respectively. Moreover, using the isomorphism φ| D(F ) : D(F ) → R n we obtain that Θ ∈ hom(R n ∧ R n , R n ) and Ω ∈ hom(R n ∧ R n , g). If we fix a point in F ∈ F N then it follows that the set of sub-spaces of T F F N which are transversal to the fibre is an affine space modeled on the linear space hom(R n , g). If two connections differ at F by an element η ∈ hom(R n , g) then their torsions at F differ by δη ∈ hom(R n ∧ R n , R n ).
An upshot of Lemma 2.15 is that in order to solve the problem of equivalence for dynamic pairs one should fix, once and for all, a subspace N ⊂ hom(R n , g) transversal to Im δ and then assign to a dynamic pair a unique connection on F N with torsion having values in N . Choosing normalisation conditions (17), (18), (19) was a choice of the subspace N ⊂ hom(R n , g). Of course, there is a freedom in choosing another transversal subspace N and the only criterion for choosing one seems simplicity of the resulting invariants.

Ordinary differential equations
Consider a system of m ordinary differential equations of order k + 1 ≥ 2, where x = (x 1 , . . . , x m ) ∈ R m and F = (F 1 , . . . , F m ) is a smooth map R n → R m , 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 there exists a smooth diffeomorphism R m+1 → R m+1 of the form 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. We start with providing a geometric background to the definition of equivalence.

Jet space and its affine distribution
Let J k (1, m) denote the space of k jets of smooth functions R → R m . The space J k (1, m) is endowed with the natural coordinate system (t, y) := (t, x 0 , . . . , x k ), where x i = (x 1 i , . . . , x m i ). Recall that any parametrised curve x : I → R m , with I ⊂ R an open interval, has its k-jet extension j k x : I → J k (1, m) defined by (j k x)(t) = (t, x(t), x (1) (t), . . . , x (k) (t)). On each such curve we identify x i (t) = x (i) (t).
For any given (t, y) ∈ J k (1, m) there is a smooth curve x : I → R m such that (j k x)(t) = (t, y). The vector tangent to this curve at (t, y) is of the form where u j are arbitrary numbers and, again, we identify x i = x (i) (t). All such vectors form an affine subspace of the tangent space to M = J k (1, m) at the point (t, y) ∈ M . This subspace is denoted A k (t, y) ⊂ T t,y M and we have We will call A k canonical affine distribution on the space J k (1, m) of k-jets of parametrised curves in R m . We may write x 0 , . . . , x k )). (45) In particular Ψ preserves the 1-form dt. Vice versa, any Ψ as in (45) preserves A k .
Proof. Suppose, Ψ : J k (1, m) → J k (1, m) preserves A k , i.e., Ψ * A k = A k . Denoting with the same same symbol A k the set (the sheaf) of vector fields belonging to the distribution A k we have . . , k, j = 1, . . . , m}. Since Ψ preserves A k , it also preserves these distributions and the corresponding foliations. In particular, it preserves V k−1 k and V k k , which means that t is transformed into t and (t, x) is transformed into (t , x ). The 1-form α = dt is determined by the conditions α(V k k ) = 0 and α(Y ) = 1, for Y ∈ A k , thus it is also preserved by Ψ. This implies that t is mapped into t + c, c ∈ R and, therefore, (t , x ) = (t + c, Φ(t, x)).
Since through any point in J k (1, m) there passes a k-jet extension of a curve in R m , and for any curve s → γ(s) = (s, x(s)) we have it follows that x i = (D i Φ)(t, x) and, thus, Ψ is of the desired form. The converse implication is straightforward.
Remark. In geometric theory of ODEs one uses a Cartan distribution C k , which is the vector distribution spanned by the more subtle object, the affine distribution A k . C k gives all vectors tangent to k-jet extensions of unparametrised curves. We have C k = span{D k , V k } and A k = {Y ∈ C k : dt(Y ) = 1}.

Dynamic pair of (F) and equivalence
The system (F ) can be equivalently defined by a submanifold E F ⊂ J k+1 (1, m), The functions t, x 0 , . . . , x k restricted to E F define a system of coordinates on E F , since the projection π : , intersected with the tangent space to the submanifold E F , defines a unique vector This follows from the fact that V k+1 and T E F are mutually transversal subspaces in T J k+1 (1, m), at any point in E F . In this way (F) defines a vector field X F on E F . X F is called total derivative corresponding to (F) and, in coordinates, it is given by Consider the Lie square [A k+1 , A k+1 ], which is a vector distribution on J k+1 (1, m) and, in coordinates, |i, j = 1, . . . , m}. We define the distribution on E F by intersecting [A k+1 , A k+1 ] with the tangent bundle T E F , In coordinates, It is easy to check that the pair (X F , V F ) is regular, i.e., it satisfies conditions (R1) and (R2) on M = E F . We will call it the dynamic pair of system (F). Consider two equations (F ) and (F ). Proof. Equivalence of (a), (b) and (c) follows from Proposition 3.1 and the remark following it. From the definitions In order to show (d) ⇒ (b) assume that there is a diffeomorphism ψ : E F → E F which transforms the pair (X F , V F ) into (X F , V F ). Since E F and J k (1, m) are diffeomorphic via the natural projection π : J k+1 (1, m) → J k (1, m) restricted to E F , and so are E F and J k (1, m), there is a diffeomorphismψ : J k (1, m) → J k (1, m) corresponding to ψ : E F → E F . Moreover, after projections both pairs (X F , V F ) and (X F , V F ) have, in natural coordinates in J k (1, m), the form (46) and (47). Sinceψ transforms the projected pair into the projected pair, it follows from (46) and (47) that they both span the same affine distribution A k = D k + V k . Thuŝ ψ preserves the canonical affine distribution A k in J k (1, m). We then deduce from Proposition 3.1 thatψ is of the form (45). Let Ψ be the 1-prolongation ofψ, which means that it is of the form (45), with k replaced by k + 1. Then Ψ automatically preserves A k+1 and it is easy to see that it transforms E F to E F .
Equations (F) and (F') satisfying one of the above conditions will simply be called equivalent. Taking into account conditions (b) and (c) one could also call them affine-contact equivalent or time-scale preserving contact equivalent.
Condition (d) implies that we can use Theorems 2.9 or 2.11 in order to solve the equivalence problem for systems (F). We can assign to (F) a canonical connection and a canonical frame on the normal frame bundle of the pair (X F , V F ). We obtain (c) The canonical frames (V, X, G) and (V , X , G ) in Theorem 2.9 (resp. Theorem 2.11), corresponding to dynamic pairs (X F , V F ) and (X F , V F ) and living on the normal frame bundles π : F N → J k (1, m) and π : F N → J k (1, m), are diffeomorphic.
We also deduce that any system (F) has at most (k + 1)m + m 2 + 1-dimensional group 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 ODEs is reduced to the geometry of pairs (X, V).

Dynamic pairs of ODEs
Not all dynamic pairs (X, V) correspond to systems of ODEs. In order to characterise such pairs we introduce  Proof. It is straightforward to check that conditions (R1)-(R4) are satisfied for an arbitrary equation (F ) and the corresponding (X F , V F ). In order to prove the theorem in the opposite direction let us notice that (R1) and (R3) imply that V k defines a corank one foliation on M . Thus we can choose a local coordinate t on M such that leaves of V k are given by equations: t = const. Additionally, it follows from (R3) that we can choose remaining coordinates such . . , m}, for i = 0, . . . , k − 1, where c and c j s are constants. We have for certain functions f and f i = (f 1 i , . . . , f m i ). Note that (R4) implies that f is constant on leaves of V k . If not, then the Lie bracket of X and some vector field tangent to V k would be transversal to V k , and hence it would violate condition (R4). Thus, we can reparametrise t so that f ≡ 1.
Similarly, let us notice that f i depend on t and x 0 , . . . , x i only. Otherwise, the Lie bracket of X and a vector field in V k would stick out of V k . We will modify coordinates x j i in such a way that X is of the form X F for some system (F ). Firstly, we set y 0 = x 0 and y 1 = f 0 (t, y 0 , x 1 ).
At each step we get new coordinate system (t, y 0 , . . . , y i , x i+1 , . . . , x k ). Finally we obtain where F j (t, y 0 , . . . , y k ) = f j k (t, y 0 , . . . , y k ) define the desired system of ODEs. In order to complete the proof it is sufficient to prove that (R3') implies (R3). We proceed by induction. Assume that V i+1 , . . . , V k are integrable, where i < k − 1. Let Y 1 and Y 2 be two sections of V i ⊂ V i+1 . Then, by assumption, [Y 1 , Y 2 ] is a section of V i+1 . Moreover, Jacobi identity implies that [X, [Y 1 , Y 2 ]] is also a section of V i+1 , since [X, Y 1 ] and [X, Y 2 ] are sections of V i+1 by the definition of V i+1 . It follows, that [Y 1 , Y 2 ] is a section of V i . If not, then by condition (R1), the bracket [X, [Y 1 , Y 2 ]] would be a non-trivial section of V i+2 mod V i+1 . Theorem 3.5 implies that the distribution V k is integrable. Let S be a leaf of the corresponding foliation (a hypersurface) and let us choose a normal frame F x = (V 0 , . . . , V k , X(x)) at each point x ∈ S. Then (V 0 , . . . , V k ) constitutes a frame 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 field X and the curvature matrices K 0 , . . . , K k−1 determine V in a neighbourhood of S. Corollary 3.7 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 V k and V k with a common normal frame (V 0 , . . . , V k ). Additionally, assume that there exist a normal frame of V and a normal frame of V which coincide on S and are such that the associated matrices of curvature operators coincide in a neighbourhood of S. Then V = V in a neighbourhood of S.
Proof. Let V and V be normal frames of V and V , respectively, such that the associated 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 0 G and V := V 0 G 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 K i . Moreover, ad i X V = ad i X V = V i on S for i = 0, . . . , k. Thus, the uniqueness theorem for ODEs implies that V = V on a neighbourhood of S. Consequently V = span{V } = span{V } = V on a neighbourhood of S.

Systems of order 2
Consider a system of second order ODEs on R m , Instead of (F ), we can consider the corresponding dynamic pairs (X F , V F ) or, equivalently, general dynamic pairs (X, V) satisfying (R1)-(R4), with k = 1.
Theorem 3.8 The first and the second canonical frames on F N , corresponding to dynamic pair (X F , V F ) of (F ), coincide and they satisfy: The invariantsR 1 , T 11 0 and R 11 are determined byK 0 , namely and they satisfy cycl{p,q,r} cycl{p,q,r} ( cycl denotes the cyclic sum). Moreover, R 01t pqs is symmetric in lower indices and cycl{p,q,r} IfK 0 vanishes then [X, V 1 p ] = 0, [V 1 p , V 1 q ] = 0 and the only nonzero invariant R 01 satisfies the relations Next, let us Lie bracket both sides of the fourth structural equation with X. On the left-hand side we have [X, [V 0 p , V 1 q ]] and, applying Jacobi identity, the first, the second and the third equations, and the identities (26) we get: On the right-hand side we obtain X(R 01t pqs )G s t . Therefore, In particular, we get thatR 0r pq is anti-symmetric in p and q, because T 11r pq1 is. But (66) reads thatR 0t pq is symmetric in p and q. Thuŝ This proves that the structural equations are as stated in the theorem. Moreover, from these equations we see that they satisfy the axioms of both, the first and the second canonical frames, thus these frames coincide by their uniqueness. Additionally, equationR 0r pq = 0 together with (70) proves the relation (63) and simplifies (67) to R 01t pqs = R 01t qps .
Now, let us Lie bracket both sides of the last structural equation with X. On the left-hand side we have [X, [V 1 p , V 1 q ]] and, applying Jacobi identity, the second and the fourth structural equations, and (26), we obtain On the right-hand side, taking into account the first structural equation and (71), we get: and (64) (which was to be proved). Combining (68) and (74) we can express T 11r pq0 andR 1t pq in terms of V 0 p (K r 0q ). Precisely, taking into account that T 11r pq0 = −T 11r qp0 , we find which gives (56) and (58) in the formulation of the theorem.
Consider next the third structural equation and bracket it with V 1 r . We have [V 1 r , [V 0 p , V 0 q ]] = 0 and, after applying Jacobi identity and the fourth structural equation, qrs )G s t = 0. This implies (61) and the relation R 01t prq = R 01t qrp which, together with (72), implies that R 01t pqr is symmetric with respect to indices p, q, and r. Let us now bracket the last structural equation with the vector field V 0 r . On the left-hand side we have and, applying Jacobi identity, On the right-hand side, taking into account the third structural equation and T 11r pq1 = 0, we obtain: i.e., identities (59), (60) and (65). This ends the proof of the theorem as, ifK 0 = 0, all the invariants vanish except of R 01t pqr .
Let us find the structural functions in terms of function F defining an equation and coordinates (t, x j , y j ) on the space of 1-jets, where y j corresponds to the first derivative of x j . We have Therefore, by Proposition 2.5, we get We have and using (36) we can write Then By Theorem 3.8 we know thatT is expressed in terms of K 0 . As a conclusion we get that all invariants of a system (F ) are expressed by K 0 ,R and their derivatives. This strengthens a result of [7] (problem (B)). We get the following characterisation of trivial systems.
Corollary 3.9 A system of second order ODEs is equivalent to the trivial system x = 0 if and only if K 0 vanishes and F is a polynomial of degree at most 2 in x .
Proof.R s uwt = − 1 2 ∂ y u ∂ y w ∂ y t F s = 0 means that F is polynomial of degree at most 2 as a function in x .
Remark. Let us notice that if (F ) is a geodesic equation for a Finsler metric then the quantity ∂ y u ∂ y w ∂ y t F s is called Berwald curvature (cf. [22]). In our setting it appears as a component of curvature: R 01s uwt . Vanishing of Berwald curvature is necessary an sufficient condition for Finsler metric to be Riemannian.

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 want to solve the equivalence problem for (F ).
Theorem 3.10 Let (F ) be a third order ODE. The first canonical frame on F N satisfies: where L = T 02 1 =R 0 = GL andL, K 0 , K 1 are functions of global coordinates (t, x, y, p) on J 2 (1, 1) (G is the fiber coordinate), and Moreover X(L) = 1 3 V 0 (K 1 ) and Proof. The proof, based on Jacobi identity applied to the canonical frame, is analogous to the proof of Theorem 3.8.

Corollary 3.11
All structural functions of the canonical frame are combinations of L, K 0 , K 1 and their derivatives, wherẽ In order to prove the corollary we will compute structural functions in terms of function F . We start with the following analog of Proposition 2.5. and equation (34) implies Then This gives the structural function T 02 1 = −G 1 3 ∂ 2 p F and proves the first formula in Corollary 3.11. In addition we get

Veronese Webs
We apply our results to get local classification of Veronese webs of corank 1. Such webs were introduced by Gelfand and Zakharevich [14] in connection to bi-hamiltonian systems. It was conjectured in [14], and proved by Turiel in [24], that Veronese webs determine bi-hamiltonian structures. Normal forms of Veronese webs were provided in [25] (see also [26]). Below we show that the framework of dynamic pairs includes Veronese webs (and thus, by results of [14,24], it includes bi-hamiltonian structures).
Let R t → F t be a family of corank 1 foliations on a manifold S of dimension k + 1. Assume that ω t are smooth one-forms annihilating F t . We say that a family {F t } is a Veronese web if there exist pointwise linearly independent smooth one-forms If we add a one-form at infinity ω ∞ = α k then, for every x ∈ S, we get a Veronese curve in the projectivisation of the cotangent space T * x S: This curve has a canonical parameter defined by the map t → F t . We say that two Veronese webs {F t } on a manifold S and {F t } on a manifold S are equivalent if there exists a diffeomorphism Φ : S → S such that Φ(F t ) = F t for any t ∈ R.

Dynamic pairs of Veronese webs and equivalence
Let be the Veronese curve in the projective space P (T x S) dual to the curve (77). By definition, this is a curve Z t (x) in T x S such that where and Y 0 , . . . , Y k are pointwise linearly independent vector fields on S. Denote Then M F is k + 2 dimensional manifold, more precisely a circle bundle pr : M F → S. Note that the fibres of M F have a canonical parameter given by t. If x ∈ S and t ∈ R then (x, t) is a point in M F . On M F there is a canonical vertical (i.e. tangent to fibres) vector field, denoted X F . In coordinates Moreover, M F itself is equipped with a canonical foliation with leaves given by the equations {t = const}. This foliation can be treated as a horizontal connection on the bundle M F → S. Therefore, in particular, we can lift the vector Z t (x) to a unique vectorẐ t (x) at the point (x, t) ∈ M F . In this way we obtain a global vector field (x, t) →Ẑ(x, t) ∈ T (x,t) M F defined on M F . We introduce the rank 1 distribution: Proof. We begin with the observation that ad i X F V F is spanned byŶ 0 ,Ŷ 1 , . . . ,Ŷ i , whereŶ i is the lift of ∂ i t Z t to M F . By (79)Ŷ 0 , . . . ,Ŷ k are independent at any point of M F and thus (R1) and (R2) are satisfied. To finish the proof it is sufficient to prove that ad k−1 X F V F and ad k X F V F are integrable (see condition (R3') of Theorem 3.5). Integrability of ad k−1 X F V F immediately follows from the definitions of X F , V F and from (78). Namely, pr(ad k−1 On the other hand ad k X F V F is the distribution tangent to foliation {t = const} on M F .
We would like to know which dynamic pairs of equation type define Veronese webs.
Definition 4.2 Let X, V be a smooth vector field and a smooth line field on a manifold M . We say that (X, V) is of Veronese type, if there exists a Veronese web {F t } on a manifold S such that (X, V) is diffeomorphic to the pair (X F , V F ) on the manifold M F . We say that (X, V) is locally of Veronese type, if for any x ∈ M there exists a neighbourhood U x and a Veronese web {F t } on a manifold S such that Proof. First, note that if (X, V) is of Veronese type then in local coordinates on M F we have X = ∂ t and V(t, x) is spanned bŷ see formula (79). Since ad k+1 ∂tẐ = 0, it follows thatẐ is a normal generator of V and 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 ad k+1 X V = 0. Let us choose an open subset U ⊂ M with local coordinates such that X = ∂ t on U (we can always locally trivialise X). Then along any integral curve of X contained in U we get the formula where V 0 , . . . , V k are constant vectors along an integral curve of X. Indeed, the equation ad k+1 X V = 0 means that along an integral curve of X the vector field V is a solution to the equation d k+1 V dt k+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 and define S to be the quotient space S = U/X. This means that a point x ∈ S is an integral line of X with parameter t belonging to some segment I x ⊂ R. If we project V(t) = span{V (t)} to S for every t ∈ I x we get a segment of Veronese curve in P (T x S). Since a Veronese curve is uniquely determined by a finite number of its points, we can uniquely extend the segment of Veronese curve to the full Veronese curve. The dual Veronese curve in P (T * S) defines the desired Veronese web. Theorems 2.9 and 2.11 applied to the pair (X F , V F ) give the following: (c) The canonical frames (X, V, G) and (X , V , G ) on the normal frame bundles π : F N → M F and π : F N → M F , corresponding to dynamical pairs (X F , V F ) and (X F , V F ) via Theorem 2.9 (resp. Theorem 2.11), are diffeomorphic by a diffeomorphism preserving t.
Proof. Assume first that {F t } and {F t } are equivalent Veronese webs and the equivalence is established by Φ : S → S . Let Ψ : M F → M F be the lift of Φ defined in an obvious way. By definition of equivalence of webs we get Φ(F t ) = F t and hence Ψ preserves t. Moreover, Ψ maps fibres of M F → S onto fibres of M F → S and we get that Ψ * X F = X F . It is also a direct consequence of the definitions that Ψ * V F = V F because Φ * (ker ω t ) = ker ω t , for any t, which implies that Φ * (span{Z t }) = span{Z t } and, consequently, Ψ * (span{Ẑ}) = span{Ẑ }.
On the other hand, if Ψ : M F → M F establish equivalence of dynamic pairs (X F , V F ) and (X F , V F ) then Ψ * X F = X F and thus it transports fibres of M F → S onto fibres of M F → S . Hence, Ψ defines a mapping Φ : The projection of a leaf of ad k−1 X F V F is a leaf of the foliation F t , for some t, thus Φ maps leaves of {F t } onto leaves of {F t }. If additionally Ψ preserves t we get that Φ(F t ) = F t for any t. This proves (a) ⇔ (b).
Corollary 4.5 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 ω t = k i=0 t k−i dx i , in some coordinates on S.
Proof. Note that Theorem 2.9 imply that the group of symmetries of a web is at most k + 3 dimensional. However, statement (c) of Theorem 4.4 says that not all symmetries of the canonical frame of a dynamic pair (X F , V F ) define symmetries of {F t }. Namely a symmetry must keep t invariant. Therefore we get that the dimension of the symmetry group is bounded from above by k+2. Moreover, it follows that if the symmetry group has maximal possible dimension then the structural functions of the canonical frame (V, G) have to be constant on each leaf F N | {t=const} ⊂ F N . Therefore the part of the curvature and the torsion of the canonical connection which involves V vanish. Then, using Jacobi identity applied to [X, [V i , V j ]] we get that the part of the curvature involving X and V i also vanish. Moreover, taking into account that K i ≡ 0 for an arbitrary Veronese web (Lemma 4.1) we get that the Veronese web is flat.

Veronese webs on a plane
Let F = {F t } be a Veronese web on R 2 defined by a family of 1-dimensional distri- where Y 0 , Y 1 are smooth vector fields on R 2 . Theorems 3.8, 3.5 and 4.3 imply Theorem 4.6 The first canonical frame on the bundle F N , corresponding to the dynamic pair (X F , V F ), satisfies: for a certain function R such that X(R) = 0.
Remark. Equation [X, V 0 ] = V 1 implies that the first and the second canonical frames, given by Theorems 2.9 and 2.11, coincide for Veronese webs on a plane.
Since V 0 and V 1 are homogeneous of order one with respect to the fiber coordinate in the normal bundle F N → M F , and G are homogeneous of order zero, it follows that R = G 2R , in coordinates, whereR is a function on M F . However, X(R) = 0 implies thatR is in fact well defined on R 2 . Let us fix a point x ∈ R 2 and letx = (x, 0) ∈ M F (x). Let us also choose a point ν in the bundle F N (x). Then we can introduce a coordinate system on R 2 in the following way: where pr : F N → R 2 is the projection composed of the projections F N → M F → R 2 .
If we change ν → νG for some G ∈ Gl(1) R * , the coordinates are multiplied by a real number. Therefore we get a canonical local system of coordinates on R 2 with the origin in x, given up to multiplication by a constant. In this coordinates we can express functionR and get the function on R 2 intrinsically assigned to the web.

Corollary 4.7
There is one-to-one correspondence between germs of Veronese webs at 0 ∈ R 2 and germs of functionsR : R 2 → R at 0 given up to the transformations R(x 0 , x 2 ) → G 2R (Gx 0 , Gx 1 ), G = 0.
Proof. We shall show how to recover the web from the functionR. First we consider R 2 × Gl(1) with coordinates x 0 , x 1 and G (this space is the level set {t = 0} of the canonical bundle F N ). On R 2 ×Gl(1) we define R(x 0 , x 1 , G) = GR(x 0 , x 1 ). It follows from the definition of the canonical coordinate system (formula (80)) and the relation [G, V i ] = V i that we can assume V 0 = G∂ x 0 and V 1 = Ga∂ x 0 + Gb∂ x 1 + Gc∂ G for some functions a, b, c in variables x 0 , x 1 . Thus the equation [V 0 , V 1 ] = RG in Theorem 4.6 implies the following: On the plane {x 0 = 0} we have a = c = 0 and b = 1 (again we use the definition (80) of the coordinate system). Hence we are able to recover a, b and c in a unique way. The web on the (x 0 , x 1 )-plane is spanned by: pr * (V 0 (x 0 , x 1 , 1) + tV 1 (x 0 , x 1 , 1)) where pr : R 2 × Gl(1) → R 2 is the projection on the first factor.
Remark. It can be easily verified that the following substitution relates the frames of Theorem 4.8 and Theorem 4.9: Moreover, the functions T in Theorems 4.8 and 4.9 coincide.
In coordinates T = GT , whereT is a function on M F . However, similarly to the case of Veronese web on the plane, it follows from X(T ) = 0 thatT is well defined on R 3 . Let us fixed a point x ∈ R 3 and takex = (x, 0) ∈ M F (x). Let us also choose a point ν in F N (x). We introduce the following coordinate system on R 3 : where pr : F N → R 3 is the projection. Note that at the beginning we go alongṼ 0 then alongṼ 2 and finally alongṼ 1 .
We are able to compute any derivative of the form ∂ a+b+cT /∂x a 0 ∂x b 2 ∂x c 1 at the origin of our coordinate system. Indeed, this is equivalent to (Ṽ 0 ) a (Ṽ 2 ) b (Ṽ 1 ) c (T ). Moreover, it follows from the structural equations and (81) that any derivative of the form (Ṽ 0 ) a (Ṽ 2 ) b (Ṽ 1 ) c (T ) with arbitrary c ∈ N can be written as a sum of derivatives (Ṽ 0 ) a (Ṽ 2 ) b (Ṽ 1 ) c (T ) where c = 1 or c = 0. Therefore if we knowT and S =Ṽ 1 (T ) on the plane {x 1 = 0} then we are able to recover all possible derivatives ∂ a+b+cT /∂x a 0 ∂x b 2 ∂x c 1 at the origin of our coordinate system. Hence, if all data are analytic then we can recoverT on R 3 .
The coordinate system is unique up to the choice of the point ν ∈ F N (x). If we change ν then every coordinate function is multiplied by G ∈ Gl(1) R * . We get Corollary 4.10 In the analytic category there is one-to-one correspondence between germs of Veronese webs at 0 ∈ R 3 and germs at 0 of two functionsT andS in two variables: x 0 and x 2 . The functions are given up to the following transformations: T (x 0 , x 2 ) → GT (Gx 0 , Gx 2 ),S(x 0 , x 2 ) → GS(Gx 0 , Gx 2 ).
Proof. The proof is similar to the proof of Corollary 4.7 and we skip it.