NEW RESULTS ABOUT THE GEOMETRIC INVARIANTS ... - math.uaic

˘ ¸ II ”AL.I.CUZA” IAS ANALELE S ¸ TIINT ¸ IFICE ALE UNIVERSITAT ¸I Tomul XLVII, s.I a, Matematic˘ a, 2001, f.2.

NEW RESULTS ABOUT THE GEOMETRIC INVARIANTS IN KCC-THEORY BY † ˘ P.L. ANTONELLI∗ and I. BUCATARU

Abstract. The KCC-theory (Kosambi, [12], CARTAN, [7], and CHERN, [8]) of a system of second order ordinary differential equations (SODE) uses five geometric invariants that determine, up to a change of coordinates, the solutions of the system. Geometrically speaking to a SODE corresponds a vector field, called a semispray (or alternatively, a second order vector field) that lives on the tangent bundle of a manifold. For a semispray S it is well known that it determines a nonlinear connection N and a Finsler connection D, called the Berwald connection, both of them living on the tangent space of the given manifold. We prove that all five invariants of the system can be expressed in this geometric framework. Using the dynamical covariant derivative and the covariant derivative induced by the Berwald connection we determine two invariant equations for the variational equations of a SODE and for the symmetries of the associated semispray. The KCC-theory has significant applications in biology, [2], [14]. 2000 Mathematics Subject Classification: 53C60, 58A20, 58A30, 37N25. Key words and phrases: SODE, Jacobi field, Berwald connection.

Introduction. Second order systems of ODES are now recognized as being important in Volterra-Hamilton theory, [4] and in Analytical trophodynamics, [2], [14] where intrinsic properties like curvature determine the stability of production processes. From a geometric point of view, a system of second order ODE is equivalent to a vector field S, called a semispray that lies on the tangent bundle ∗ †

Partially supported by NSERC-7667 PIMS Postdoctoral Fellow

406

˘ P.L. ANTONELLI and I. BUCATARU

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T M of a manifold M . As it is well known, a semispray yields two other important geometrical objects, living also on the tangent bundle: a nonlinear connection, which is a distribution N supplementary to the vertical distribution V , and a linear connection D adapted to V and N . This linear connection was introduced in 1947 by Berwald in ([6]) and therefore is called the Berwald connection. A remarkable direction for the geometry of a SODE appeared in the thirties. The core of this new method defines the so-called ”KCC-theory”(Kosambi, [12], Cartan, [7], and Chern, [8]) which establishes at five the number of geometric invariants that determine, up to a change of coordinates, the solutions of the given system of second order. Our main result is an interpretation of these five KCC-invariants in terms of N and D, the curvature of N and the torsion and curvature of D. For a semispray S on the tangent bundle T M of a manifold M we consider the induced nonlinear connection N . The vertical component of S, with respect to this nonlinear connection N , gives the first invariant of S, called the deviation tensor of the semispray. The curvature of the nonlinear i is the third invariant. In section two we give a global exconnection, Rjk pression of the induced Berwald connection D. It is proved that this linear connection on the tangent bundle is a Finsler connection. With respect to an adapted basis to the nonlinear connection, the Berwald connection i and two nonzero comD has only one nonzero component of torsion Rjk i . These are the third, the fourth, and i and Djkl ponents of curvature Rjkl the fifth invariants of the semispray S, respectively. The second invariant is expressed using the third invariant and the h-covariant derivative of the first invariant. The Berwald connection appears also in [5], and [16] in local coordinates as a Finsler connection on the tangent bundle, and in [9] in the pullback bundle of the tangent bundle by its natural projection. Besides the dynamical covariant derivative induced by a nonlinear connection, or a semispray, ([16], [9]), we introduce a covariant derivative determined by the Berwald connection. Only in the particular case when the semispray is homogeneous, these two covariant derivatives coincide. The path equations and the variational equations (Jacobi equations) are written using both covariant derivatives. The relationship between Jacobi vector fields for a system of SODE and the dynamical symmetries of the associated semispray is given.

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GEOMETRIC INVARIANTS IN KCC- THEORY

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1. Dynamical covariant derivative induced by a semispray. Let M be a real, smooth, n-dimensional manifold, and (T M, π, M ) be its tangent bundle. For a local chart (U, φ = (xi )) on M , we denote by (π −1 (U ), Φ = (xi , y i )) the induced local chart on T M . The kernel of the linear map induced by the natural submersion π : T M → M , determines the vertical distribution V : u ∈ T M 7→ Vu = Kerπ∗,u ⊂ Tu T M . This is ∂ ∂ an n-dimensional integrable distribution. If { i |u , i |u } is the natural ∂x ∂y ∂ basis of the tangent space Tu T M , then { i |u } is a basis for Vu , ∀u ∈ T M . ∂y ∂ i Consider J = ⊗ dx , the almost tangent structure (J is also called the ∂y i ∂ vertical endomorphism of T M ), and Γ = y i i the Liouville vector field. A ∂y vector field S on T M is called a semispray (or a second order vector field) if ∂ ∂ JS = Γ. The local expression of a semispray is S = y i i − 2Gi (x, y) i . ∂x ∂y The functions Gi (x, y) are called the local coefficients of the semispray and these are defined on domains of local chart. An n-dimensional distribution N : u ∈ T M 7→ Nu ⊂ Tu T M that is supplementary to the vertical distribution V is called a nonlinear connection. For every u ∈ T M we have the direct sum Tu T M = Nu ⊕ Vu .

(1.1)

An adapted basis to the previous direct sum is {

δ ∂ ∂ = − Nij (x, y) j , δxi ∂xi ∂y

∂ }. We call this basis the Berwald basis of the nonlinear connection N. ∂y i The functions Nji (x, y) are defined on domains of local chart, and these are called the local coefficients of the nonlinear connection N . It is well known that every semispray S with local coefficients Gi , induces a nonlinear ∂Gi connection N with local coefficients Nji = , [11]. Next, we shall work ∂y j with this nonlinear connection. Denote by h and v the horizontal and the vertical projectors, that correspond to the decomposition (1.1). In the Berwald basis we have: (1.2)

h=

δ ∂ ⊗ dxi , v = i ⊗ δy i , i δx ∂y


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where (dxi , δy i = dy i + Nji (x, y)dxj ) is the dual basis of the Berwald basis. The vertical projector v is given by (1.3)

1 v(X) = (X + [S, JX] + J[X, S]), ∀X ∈ χ(T M ). 2

A tensor field of (r, s)-type on T M is called a Finsler tensor field, [15], (or a d-tensor field in [16]) if under a change of induced coordinates on T M , its components transform like the components of a (r, s)-type tensor field on the base manifold M . The local expression of a semispray in the Berwald basis is S = yi

(1.4)

δ ∂ − E i (x, y) i , δxi ∂y

where E i (x, y) = 2Gi (x, y) − Nji (x, y)y j

(1.5)

is a (1,0)-type tensor Finsler field. This is called the first invariant of the semispray in ([8], [7]), or the deviation tensor in [11]. ∂ For a vector field X = X i (x) i on the base manifold M , let us consider ∂x (1.6) X c = X i (x)

∂ ∂X i ∂ δ ∂ + (x)y j i , X h = X i i , and X v = X i (x) i , i j ∂x ∂x ∂y δx ∂y

the complete, the horizontal, and the vertical lift, respectively. It is very easy to check that these lifts of a vector field X ∈ χ(M ) are related by (1.6)0

X c = 2X h + [S, X v ] = 2X h + LS X v ,

where LS is the Lie derivative with respect to S. The dynamical covariant derivative of a Finsler vector field X i (x, y) is defined by, [9]: (1.7)

∇X i = S(X i ) + Nji X j =

∂X i j ∂Gi j ∂X i j y − 2 G + X . ∂xj ∂y j ∂y j

Remark 1.1. 1o For a Finsler vector field X i (x, y), its dynamical covariant derivative ∇X i satisfies (1.7)0

[S, X i

∂ δ ∂ ] = −X i i + ∇X i i . i ∂y δx ∂y

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As a consequence we have that v[S, X i

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∂ ∂ ] = ∇X i i , and then ∇X i is i ∂y ∂y

also a Finsler vector field. 2o We have the properties: ∇(X i + Y i ) = ∇X i + ∇Y i , and ∇(f X i ) = S(f )X i + f ∇X i . ∂ 3o If X = X i (x) i is a vector field on the base manifold M , then X c = ∂x δ ∂ i ∂ i X (x) i + ∇X , so v(X c ) = ∇X i i . i δx ∂y ∂y 4o The dynamical covariant derivative of the Liouville vector field (y i ) and the first invariant are related by ∇y i = −E i . A curve c : t ∈ I 7→ c(t) = (xi (t)) ∈ M is called a path of the semispray dxi (t)) ∈ T M is an integral S if its lift to T M : c˜ : t ∈ I 7→ c˜(t) = (xi (t), dt curve of S. In local coordinates the curve c(t) = (xi (t)) is a path of S if and only if: (1.8)

d2 xi dx + 2Gi (x, ) = 0. 2 dt dt

An equivalent invariant form of the system (1.8) is given by: (1.8)0

∇(

dxi dx ) = −E i (x, ). dt dt

Remark 1.2. The semispray S is called a spray if S is homogeneous of degree two with respect to y, and this is equivalent to E i = 0. In this case the paths of the semispray S are horizontal curves of the induced nonlinear connection, because S is a horizontal vector field. 2. The Berwald connection induced by a semispray. Consider S a semispray, and N the induced nonlinear connection with the Berwald δ ∂ δ basis { i , i }. Consider the tensor field θ = i ⊗ δy i . We can see that δx ∂y δx the restriction of θ to the vertical distribution is an isomorphism between this vertical distribution and the horizontal distribution.

Definition 2.1. A linear connection D (Koszul connection) on T M is called a Finsler connection if D preserves by parallelism the horizontal


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distribution N and the almost tangent structure J is absolutely parallel with respect to D. So, a linear connection D on T M is a Finsler connection if and only if Dh = 0 and DJ = 0. For a Finsler connection D it is very easy to check that D preserves also by parallelism the vertical distribution V , that is Dv = 0. Moreover a linear connection D on T M is a Finsler connection if and only if Dv = 0, and Dθ = 0. With respect to the Berwald basis, a Finsler connection D has the local expression, [16]:

(2.1)

 δ ∂ δ ∂  D δ = Fjik k , D δ = Fjik k ,   j j  δxi δx δx ∂y δxi ∂y     D

∂ ∂y i

δ ∂ k δ k ∂ = Cji = Cji ,D ∂ . j j k i δx δx ∂y k ∂y ∂y

k ). Next, a Finsler connection will be indicated also by the set D=(Nji , Fijk , Cij Under a change of induced coordinates on T M , the coefficients Fijk transform like the coefficients of a linear connection on the base manifold M . The k are the components of a (1,2)-type Finsler tensor field. coefficients Cij For a Finsler vector field X i (x, y), we define the covariant derivative k ) as: induced by a Finsler connection D = (Nji , Fijk , Cij

(2.2)

i i Xj E k, X j y k − Cjk DX i = S(X i ) + Fjk

or in the equivalent form (DX i )

(2.2)0

∂ ∂ = DS (X i i ). i ∂y ∂y

i , and X i | the horizontal, and the vertical covariant derivaDenote by X|k k i tives of X , respectively. These are given by:

(2.3)

i X|k =

∂X i δX i j i i i + F X , X | = + Cjk Xj. k jk δxk ∂y k

Then, the Finsler covariant derivative takes the form: (2.2)00

i k DX i = X|k y − X i |k E k .

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···ir are the components of a (r, s)-type Finsler tensor More generally, if Tji11···j s ···ir is also a Finsler tensor field, the Finsler covariant derivative of this DTji11···j s field of (r, s)-type, and it is defined by: ···ir ···ir ···ir | E k, y k − Tji11···j = Tji11···j DTji11···j s k s s |k

(2.2)000

···ir ···ir | are the h and v-covariant derivatives, respecand Tji11···j where Tji11···j s k s |k

tively ([16], p.40). We can remark here that this covariant derivative satisfies the Leibniz rule, that is for example D(Tji Skj ) = D(Tji )Skj + Tji D(Skj ). Theorem 2.1. The map D : χ(T M ) × χ(T M ) → χ(T M ), given by (2.4)

DX Y = v[hX, vY ] + h[vX, hY ] + J[vX, θY ] + θ[hX, JY ]

is a Finsler connection on T M . We call this the Berwald connection of the semispray S. Proof. As all the operators involved in (2.4) are additive we have that D is additive too, with respect to both arguments. To prove that Df X Y = f DX Y , ∀f ∈ F(T M ) we have to use that vh = hv = Jv = θh = 0. The other properties like DX f Y = X(f )Y + f DX Y , Dh = 0, and DJ = 0 follow easy using also Jθ = v, θJ = h, v 2 = v, and h2 = h. With respect to the Berwald basis, the Berwald connection D has the local coefficients, ([5], [16]): i = Fjk

(2.5)

∂Nji ∂ 2 Gi i = 0. = , Cjk ∂y k ∂y j ∂y k

For the Berwald connection D one considers typically the torsion T (X, Y ) = DX Y − DY X − [X, Y ]. With respect to the Berwald basis, there is only one nonzero component of the torsion, ([5], [16]): (2.6)

vT (

δNjk δ δ δNik ∂ k ∂ , ) =: R = ( − ) , (v)h − torsion. ij δxi δxj δxi δxj ∂y k ∂y k

k is also called the curvature tensor of the nonThe Finsler tensor field Rij linear connection N , because:

(2.7)

[

δ δ k ∂ , j ] = −Rij . i δx δx ∂y k

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Let us consider now R(X, Y )Z = DX DY Z − DY DX Z − D[X,Y ] Z, the curvature of the Berwald connection. With respect to the Berwald basis, there are only two nonzero components of the curvature ([5], [16]):  i δFjli δFjk  m i i i   − + Fjk Fml − Fjlm Fmk ; R =  jkl l k  δx δx     (the Riemann − Christoffel curvature tensor) (2.8)  i  ∂Fjk  ∂ 3 Gi i   D = =  jkl l j  ∂y ∂y ∂y k ∂y l   (the Douglas tensor of the semispray). Remark 2.1. If S is a spray, a consequence of the homogeneity is that i and Ri the Finsler tensor fields Rjk ljk are related by, ([6], [16]): i i yl . = Rljk Rjk

(2.9)

In the general case, when S is not homogeneous, we have the following result: Proposition 2.1. Let D be the Berwald connection associated to a i the (v)h-component of the torsion, and Ri the Riemannsemispray S, Rjk ljk Christoffel curvature tensor. Then these are related by: (2.10)

i i i i = y|j|k − y|k|j , y l − Rkj Rljk

or in the equivalent form (2.10)0

i i = E i |k|j − E i |j|k . y l − Rkj Rljk

Proof. Using the Ricci identities for the Berwald connection D ([17], p.78) and the Liouville vector field, we get (2.10). For (2.10)0 it is enough to prove that: (2.11)

y|ji = −E i |j .

This (1,1)-type Finsler tensor field is called the deflection tensor of the δy i i k i k i connection. According to (2.3) we have y|ji = δx j + Fkj y = Fkj y − Nj . As

9 E i = 2Gi − −y|ji .

GEOMETRIC INVARIANTS IN KCC- THEORY ∂Gi ∂y j

y j , the E i |j =

∂E i ∂y j

∂Gi

= 2 ∂yj −

∂Gi ∂y j

−

∂ 2 Gi ∂y j ∂y k

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i yk = y k = Nji − Fkj

Proposition 2.2. The dynamical covariant derivative (1.7) induced by a semispray S, and the Berwald covariant derivative are related by: (2.12)

∇X i = DX i − y|ji X j = DX i + E i |j X j .

Proof. From (2.2), the Berwald covariant derivative has the form: i y j X k , so DX i = S(X i ) + N i X k + (F i y j − N i )X k = DX i = S(X i ) + Fjk k jk k i y k , and the proof is completed. ∇X i + y|k Remark 2.2. Using the Berwald covariant derivative, we can write the path equation (1.8) in an equivalent form: (2.13)

D(

dxi dxj ) = −(E i + E i |j ). dt dt

3. Symmetries and Jacobi equations for a semispray. Consider a semispray S. A path of the semispray is a solution of the system (1.8), or of the equivalent forms (1.8)0 , or (2.13). Consider c(t) = (xi (t)) a trajectory of (1.8), and let vary it into nearby ones according to: (3.1)

x ˜i (t) = xi (t) + εξ i (t),

where ε denotes a scalar parameter with small value |ε|, and ξ i (t) are components of a contravariant vector field along c(t). Substitution of (3.1) into (1.8), asking for c˜(t) = (˜ xi (t)) to be also a trajectory of (1.8), and letting ε → 0 yields to the so-called variational equations: (3.2)

d2 ξ i ∂Gi ∂Gi dξ j + 2 j ξj + 2 j = 0. 2 dt ∂x ∂y dt

Theorem 3.1. For the variational equations (3.2) we have the equivalent invariant form (Jacobi equations): (3.3)

i ∇2 ξ i + (Rjk

dxk + E|ji )ξ j = 0. dt

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A vector field (ξ i (t)) along a path c of the semispray S is called a Jacobi vector field if it satisfies (3.3). Proof. Denote by: (3.4)

Bji = 2

∂Gi ∂Gi ∂Gi ∂Gr − S( ) − . ∂xj ∂y j ∂y r ∂y j

It can be proved that Bji is a (1,1)-type Finsler tensor field. It has been introduced in [6], for the homogeneous case. This tensor field is called the second invariant of the given SODE in [12], [7], and [8], or the Jacobi endomorphism in [9]. It is easy to check that the equations (3.2) are equivalent to: ∇2 ξ i + Bji ξ j = 0.

(3.5)

All we have to prove now is the following expression of the second invariant: i k y + E|ji . Bji = Rjk

(3.6)

Let X i (x, y) be an arbitrary Finsler vector field, and consider the vector field ˜ = X i ∂ + S(X i ) ∂ X ∂xi ∂y i

(3.7) on T M . We have then: (3.8)

˜ = (∇2 X i + Bji X j ) ∂ . [S, X] ∂y j

˜ in the Berwald basis S = y i δ i − If we consider the expression of S and X δx ˜ = X i δ i + ∇X i ∂ i , respectively, then the bracket [S, X] ˜ can E i ∂y∂ i and X δx ∂y be expressed as follows: (3.9)

˜ = {∇2 X i + (Ri y k + E i )X j } [S, X] jk |j

∂ . ∂y i

If we compare (3.8) and (3.9) and we take into account that X i (x, y) is an arbitrary Finsler vector field, then the second invariant Bji can be expressed as in (3.6).

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Definition 3.1. 1o A Lie symmetry of the semispray S is a vector field X on the base manifold M such that [S, X c ] = 0, where X c is the complete lift of X. ˜ on T M 2o A dynamical symmetry of the semispray S is a vector field X ˜ such that [S, X] = 0. If X ∈ χ(M ) is a Lie symmetry of S then X c is a dynamical symmetry of S. As for X ∈ χ(M ) we have that X c = 2X h + [S, X v ], then X is a Lie symmetry of S if and only if (3.10)

2[S, X h ] + [S, [S, X v ]] = 2LS X h + LS LS X v = 0.

Theorem 3.2. ˜ = X i (x, y) δ i + Y i (x, y) ∂ i is a dynamical symmetry of A vector field X δx ∂y S if and only if

1o

(3.11)

Y i = ∇X i , and ∇2 X i + Bji X j = 0.

˜ = X i (x, y) δ i + Y i (x, y) ∂ i is a dynamical symmetry of the semis2o If X δx ∂y pray S and c(t) = (xi (t)) is a path of S, then the restriction of X i (x, y) i along c˜(t) = (xi (t), dx dt (t)) is a Jacobi vector field for S. ˜ using the Berwald basis, Proof. 1o If we express the Lie bracket [S, X] we have (3.11)0

˜ = (∇X i − Y i ) [S, X]

∂ δ + (∇Y i + Bji X j ) j . δxi ∂y

˜ is a dynamical symmetry of S if and only if (3.11) is true. So, X o ˜ 2 If X i (x, y) are the horizontal components of a dynamical symmetry X, 2 i i k i j then ∇ X + (Rjk y + E|j )X = 0. The restriction of this along the curve c˜ give us the equations (3.3), and then X i is a Jacobi vector field along c. The Jacobi equations (3.3) are the invariant form of the variational equations (3.2) using the dynamical covariant derivative. Also, in (3.11) we found the invariant equations of dynamical symmetries (or Lie symmetries) in terms of dynamical covariant derivative. Next we shall rewrite these equations, using the Berwald covariant derivative. As ∇X i = DX i +E i |j X j , we have that ∇2 X i = ∇(DX i )+∇(E i |j X j ) = 2 D X i + 2E i |j DX j + [D(E i |j ) + E i |k E k |j ]X j .

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Here we have used that D(E i |j X j ) = D(E i |j )X j + E i |j DX j . Denote by (3.12)

i k Rji = D(E i |j ) + E i |k E k |j + Rjk y + E|ji .

We have that Rji is a (1, 1)-type Finsler tensor field. Thus we have the following theorem: Theorem 3.3. The Jacobi equations of the system (1.8) have the invariant form, using the Berwald covariant derivative:

1o

(3.13)

D2 ξ i + 2E i |j Dξ j + Rji ξ j = 0.

2o A Finsler vector field X i (x, y) is a dynamical symmetry of the semispray S if and only if: (3.14)

D2 X i + 2E i |j DX j + Rji X j = 0.

For the homogeneous case, that is E i = 0, we have that the dynamical covariant derivative and the Berwald covariant derivative coincide, and the (1, 1)-type Finsler tensor fields Bji and Rji are equal. 4. The geometric invariants of a semispray S. In KCC-theory of a semispray ([12], [7], [8]), there are five geometric invariants. The first KCC-invariant is E i , defined by (1.5). The second KCC-invariant is Bji , defined by (3.4). The third, fourth, and fifth invariants are:  ∂Bji  ∂Bki  i := 1 (  ), − B  jk  3 ∂y k ∂y j        i ∂Bjk i := (4.1) Blkj ,  ∂y l       i   ∂Fjk ∂ 3 Gi  i :=  = .  Djkl ∂y l ∂y j ∂y k ∂y l i The Tensor Djkl is called the Douglas tensor, and we already saw in (2.8) that it is one of the nonzero components of the curvature of the Berwald connection.

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Theorem 4.1. i of the nonlinear connection N (or the (v)h-torsion of 1o The curvature Rjk the Berwald connection D) is the third invariant of the semispray S. i of the Berwald connection 2o The Riemann-Christoffel curvature tensor Rjkl D is the fourth invariant of the semispray S. i = B i and Ri i Proof. We have to prove that Rjk jk jkl = Bjkl . First we i ∂Rjk i and Ri i = prove that Rjk satisfy (4.1) , that is R . From (2.6) we 2 jkl lkj ∂y l k ∂Rij δNjk δNik ∂ δNik δ ∂ ∂ δNjk k = − , so )− = ( ( j ). As [ j , i ] = have Rij i j i l l l δx δx δx ∂y ∂y ∂y δx ∂y δx k k k k k ∂R ∂N ∂N ∂ δ δ ∂N ij j j p p ∂Ni i Fjik k , we have that = ( )−F − ( )+F li ∂y p lj ∂y p = δxi ∂y l δxj ∂y l ∂y ∂y l δFjlk δFilk k k k − + Fjlp Fpi − Filp Fjp = Rlji . According to (3.6) we have for the δxi δxj ∂Bji i i y k + E i . So, second invariant Bji , the expression Bji = Rjk = Rjk + |j ∂y k ∂Bji ∂Bki i i i i i y l +E|ji |k −E|k |j . Using y l +Rjkl +Rklj − = 2Rjk y l +E|ji |k . Then, Rklj ∂y k ∂y j the Ricci identities for the Berwald connection D, ([17], p.78) we have that l i i E l , and E i | − E i | E|ji |k − E i |k|j = Dljk j|k = Dlkj E . As the Douglas tensor |k j i | − E i | . Consequently, we is symmetric we have that: E|ji |k − E i |k|j = E|k j j|k i | = E i| i| i y l − Ri . Finally, we have that: have, E|ji |k − E|k − E = R j j|k k|j kj lkj i ∂Bji ∂Bk i i i i − = 3Rjk + (Rklj + Rjkl + Rljk )y l . Using the Bianchi identity ∂y j ∂y k i + Ri + Ri = 0, so that for the Berwald connection D, we have that Rklj ljk jkl i i ∂B ∂B j k i = 1( Rjk − ), and the theorem is proved. 3 ∂y k ∂y j

5. A particular case. In this section we use the theory developed in the previous sections in order to study the system: (5.1)

dxj dxk dxj d2 xi i i + γ (x) + γ (x) + γ i (x) = 0, j jk dt2 dt dt dt

i (x) are the local coefficients of a linear symmetric connection on where γjk the base manifold M , γji (x) are the components of a (1,1)-type tensor field

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and γ i (x) are the components of a vector field. The system (5.1) occurs in modeling some processes in biology ([3]). i (x)y j y k + On a local chart on T M we define the functions 2Gi (x, y) = γjk γji (x)y j + γ i (x). It is very easy to check that Gi are the local components ∂ ∂ of a semispray S = y i i − 2Gi i on T M . Let N be the nonlinear ∂x ∂y connection induced by S. Then the local coefficients of N are Nji (x, y) = i (x)y k + 1 γ i (x). The first invariant of the system (5.1) is given by: γjk 2 j (5.2)

1 E i (x, y) = γji (x)y j + γ i (x). 2

The Berwald connection D, that corresponds to the semispray S has the i = γ i , C i = 0. So, an immediate consequence is that local coefficients Fjk jk jk i vanishes and the fifth invariant of the system (5.1), the Douglas tensor Djkl i the fourth invariant Rjkl is the curvature tensor of the linear connection i (x). Now let us express the second and the third invariants of the system. γjk First we have to remark that the h and v-covariant derivatives of the first invariant E i with respect to the Berwald D connection are given by:  1 i i    E |j = 2 γj , (5.3)    E i = 1 γ i yk + γ i − 1 γ i γ k . |j |j 2 k|j 4 k j i , the curvature As we saw in the previous section, the third invariant is Rjk k δNj δNik k = of the nonlinear connection N . As Rij − , by a straightforward δxi δxj calculation we get that

(5.4)

1 i i i i (x)y l + (γk|j (x, y) = Rlkj Rjk (x) − γj|k (x)). 2

The second invariant Bji can be expressed, using (3.6), as follows: (5.5)

1 i k 1 i k i i Bji = Rlkj y l y k + γk|j y k + γ|ji − γj|k y − γk γj . 2 4

All these invariants can be obtained also, using Schouten’s film space idea, ([18], [4]), and Kron’s work ([13]).

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Acknowledgements. This paper has been written during the second author’s PDF at the Department of Mathematics, University of Alberta, Edmonton, Canada.

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Received: 20.IV.2001

Department of Mathematical Sciences University of Alberta Edmonton, Alberta CANADA, T6G 2G1 [email protected] Faculty of Mathematics ”Al.I.Cuza” University Ia¸si, 6600 ROMANIA [email protected]