1 Geometrical objects on 1-jet spaces

Vladimir Balan and Mircea Neagu

Dedicated to the 70-th anniversary of Professor Constantin Udriste

Abstract. In this paper we construct the jet geometrical extensions of the KCC-invariants, which characterize a given second-order system of differential equations on the 1-jet spaceJ¹(R, M). A generalized theorem of characterization of our jet geometrical KCC-invariants is also presented.

M.S.C. 2000: 58B20, 37C10, 53C43.

Key words: 1-jet spaces; temporal and spatial semisprays; nonlinear connections;

SODE; jeth−KCC-invariants.

1 Geometrical objects on 1-jet spaces

We remind first several differential geometrical properties of the 1-jet spaces. The 1-jet bundle

ξ= (J¹(R, M), π1,R×M)

is a vector bundle over the product manifold R×M, having the fibre of type Rⁿ, wheren is the dimension of the spatialmanifold M. If the spatial manifold M has the local coordinates (xⁱ)_i=1,n, then we shall denote the local coordinates of the 1-jet total spaceJ¹(R, M) by (t, xⁱ, xⁱ₁); these transform by the rules [13]

(1.1)













et=et(t) e

xⁱ=xeⁱ(x^j) e

xⁱ₁= ∂exⁱ

∂x^j dt det·x^j₁.

In the geometrical study of the 1-jet bundle, a central role is played by the distin- guished tensors(d−tensors).

Definition 1.1. A geometrical objectD=³

D1i(j)(1)...

1k(1)(l)...

´

on the 1-jet vector bundle, whose local components transform by the rules

(1.2) D1i(j)(1)...

1k(1)(u)...=De1p(m)(1)...

1r(1)(s)...

dt det

∂xⁱ

∂ex^p Ã∂x^j

∂xe^m det dt

!det

dt

∂ex^r

∂x^k µ∂ex^s

∂x^u dt det

¶ ..., is called ad−tensor field.

Balkan Journal of Geometry and Its Applications, Vol.15, No.1, 2010, pp. 8-16.∗

°c Balkan Society of Geometers, Geometry Balkan Press 2010.

Remark 1.2. The use of parentheses for certain indices of the local components D1i(j)(1)...

1k(1)(l)...

of the distinguished tensor fieldD on the 1-jet space is motivated by the fact that the pair of indices ”^(j)₍₁₎ ” or ”⁽¹⁾_(l) ” behaves like a single index.

Example 1.3. The geometrical object

C=C⁽ⁱ⁾₍₁₎ ∂

∂xⁱ₁,

where C⁽ⁱ⁾₍₁₎ = xⁱ₁, represents a d−tensor field on the 1-jet space; this is called the canonical Liouville d−tensor field of the 1-jet bundle and is a global geometrical object.

Example 1.4. Leth= (h11(t)) be a Riemannian metric on the relativistic time axis R. The geometrical object

Jh=J_(1)1j⁽ⁱ⁾ ∂

∂xⁱ₁ ⊗dt⊗dx^j,

whereJ_(1)1j⁽ⁱ⁾ =h11δⁱ_jis ad−tensor field onJ¹(R, M), which is called theh-normalization d−tensor fieldof the 1-jet space and is a global geometrical object.

In the Riemann-Lagrange differential geometry of the 1-jet spaces developed in [12], [13] important rˆoles are also played by geometrical objects as thetemporal or spatial semisprays, together with thejet nonlinear connections.

Definition 1.5. A set of local functionsH =

³ H₍₁₎₁^(j)

´

onJ¹(R, M),which transform by the rules

(1.3) 2He₍₁₎₁^(k) = 2H₍₁₎₁^(j) µdt

det

¶2

∂ex^k

∂x^j −dt det

∂xe^k₁

∂t , is called atemporal semisprayonJ¹(R, M).

Example 1.6. Let us consider a Riemannian metric h= (h11(t)) on the temporal manifoldRand let

H₁₁¹ =h¹¹ 2

dh11

dt ,

whereh¹¹= 1/h11, be its Christoffel symbol. Taking into account that we have the transformation rule

(1.4) He₁₁¹ =H₁₁¹ dt

det+det dt

d²t det², we deduce that the local components

H˚₍₁₎₁^(j) =−1 2H₁₁¹x^j₁ define a temporal semispray ˚H =

³H˚₍₁₎₁^(j)

´

onJ¹(R, M). This is called thecanonical temporal semispray associated to the temporal metrich(t).

Definition 1.7. A set of local functionsG=

³ G^(j)₍₁₎₁

´

,which transform by the rules

(1.5) 2Ge^(k)₍₁₎₁= 2G^(j)₍₁₎₁ µdt

det

¶₂

∂ex^k

∂x^j −∂x^m

∂ex^j

∂ex^k₁

∂x^mxe^j₁,

is called aspatial semisprayonJ¹(R, M).

Example 1.8. Letϕ= (ϕij(x)) be a Riemannian metric on the spatial manifoldM and let us consider

γ_jkⁱ = ϕ^im 2

µ∂ϕjm

∂x^k +∂ϕkm

∂x^j −∂ϕjk

∂x^m

¶

its Christoffel symbols. Taking into account that we have the transformation rules (1.6) eγ_qr^p =γ_jkⁱ ∂ex^p

∂xⁱ

∂x^j

∂xe^q

∂x^k

∂ex^r +∂xe^p

∂x^l

∂²x^l

∂ex^q∂ex^r, we deduce that the local components

G˚^(j)₍₁₎₁= 1

2γ_kl^jx^k₁x^l₁ define a spatial semispray ˚G = ³

G˚^(j)₍₁₎₁´

on J¹(R, M). This is called the canonical spatial semispray associated to the spatial metricϕ(x).

Definition 1.9. A set of local functions Γ = ³

M₍₁₎₁^(j), N_(1)i^(j)´

on J¹(R, M), which transform by the rules

(1.7) Mf₍₁₎₁^(k) =M₍₁₎₁^(j) µdt

det

¶₂

∂ex^k

∂x^j −dt det

∂ex^k₁

∂t

and

(1.8) Ne_(1)l^(k) =N_(1)i^(j)dt det

∂xⁱ

∂ex^l

∂xe^k

∂x^j −∂x^m

∂xe^l

∂xe^k₁

∂x^m, is called anonlinear connectionon the 1-jet spaceJ¹(R, M).

Example 1.10. Let us consider that (R, h₁₁(t)) and (M, ϕ_ij(x)) are Riemannian manifolds having the Christoffel symbolsH₁₁¹ (t) and γ_jkⁱ (x). Then, using the trans- formation rules (1.1), (1.4) and (1.6), we deduce that the set of local functions

˚Γ =

³M˚₍₁₎₁^(j),N˚_(1)i^(j)

´ , where

M˚₍₁₎₁^(j) =−H₁₁¹x^j₁ and N˚_(1)i^(j) =γ_im^j x^m₁ ,

represents a nonlinear connection on the 1-jet space J¹(R, M). This jet nonlinear connection is called thecanonical nonlinear connection attached to the pair of Rie- mannian metrics(h(t), ϕ(x)).

In the sequel, let us study the geometrical relations between temporalor spatial semispraysandnonlinear connectionson the 1-jet spaceJ¹(R, M). In this direction, using the local transformation laws (1.3), (1.7) and (1.1), respectively the transfor- mation laws (1.5), (1.8) and (1.1), by direct local computation, we find the following geometrical results:

Theorem 1.11. a) Thetemporal semisprays H = (H₍₁₎₁^(j)) and the sets of temporal components of nonlinear connections Γtemporal = (M₍₁₎₁^(j)) are in one-to-one corre- spondence on the 1-jet spaceJ¹(R, M), via:

M₍₁₎₁^(j) = 2H₍₁₎₁^(j), H₍₁₎₁^(j) = 1 2M₍₁₎₁^(j) .

b) The spatial semisprays G = (G^(j)₍₁₎₁) and the sets of spatial components of nonlinear connections Γspatial= (N_(1)k^(j) )are connected on the 1-jet space J¹(R, M), via the relations:

N_(1)k^(j) = ∂G^(j)₍₁₎₁

∂x^k₁ , G^(j)₍₁₎₁= 1

2N_(1)m^(j) x^m₁.

2 Jet geometrical KCC-theory

In this Section we generalize on the 1-jet spaceJ¹(R, M) the basics of the KCC- theory ([1], [4], [7], [14]). In this respect, let us consider onJ¹(R, M) a second-order system of differential equations of local form

(2.1) d²xⁱ

dt² +F₍₁₎₁⁽ⁱ⁾ (t, x^k, x^k₁) = 0, i= 1, n,

wherex^k₁=dx^k/dtand the local componentsF₍₁₎₁⁽ⁱ⁾ (t, x^k, x^k₁) transform under a change of coordinates (1.1) by the rules

(2.2) Fe₍₁₎₁^(r) =F₍₁₎₁^(j) µdt

det

¶₂

∂xe^r

∂x^j −dt det

∂ex^r₁

∂t −∂x^m

∂xe^j

∂xe^r₁

∂x^mxe^j₁.

Remark 2.1. The second-order system of differential equations (2.1) is invariant under a change of coordinates (1.1).

Using a temporal Riemannian metric h11(t) on R and taking into account the transformation rules (1.3) and (1.5), we can rewrite the SODEs (2.1) in the following form:

d²xⁱ

dt² −H₁₁¹xⁱ₁+ 2G⁽ⁱ⁾₍₁₎₁(t, x^k, x^k₁) = 0, i= 1, n, where

G⁽ⁱ⁾₍₁₎₁=1

2F₍₁₎₁⁽ⁱ⁾ +1 2H₁₁¹ xⁱ₁

are the components of a spatial semispray onJ¹(R, M). Moreover, the coefficients of the spatial semispray G⁽ⁱ⁾₍₁₎₁ produce the spatial components N_(1)j⁽ⁱ⁾ of a nonlinear

connection Γ on the 1-jet spaceJ¹(R, M), by putting N_(1)j⁽ⁱ⁾ =∂G⁽ⁱ⁾₍₁₎₁

∂x^j₁ =1 2

∂F₍₁₎₁⁽ⁱ⁾

∂x^j₁ +1 2H₁₁¹ δⁱ_j.

In order to find the basic jet differential geometrical invariants of the system (2.1) (see Kosambi [11], Cartan [9] and Chern [10]) under the jet coordinate transformations (1.1), we define theh−KCC-covariant derivative of a d−tensor of kindT₍₁₎⁽ⁱ⁾(t, x^k, x^k₁) on the 1-jet spaceJ¹(R, M) via

DTh ₍₁₎⁽ⁱ⁾

dt = dT₍₁₎⁽ⁱ⁾

dt +N_(1)r⁽ⁱ⁾ T₍₁₎^(r)−H₁₁¹T₍₁₎⁽ⁱ⁾=

= dT₍₁₎⁽ⁱ⁾ dt +1

2

∂F₍₁₎₁⁽ⁱ⁾

∂x^r₁ T₍₁₎^(r)−1

2H₁₁¹T₍₁₎⁽ⁱ⁾, where the Einstein summation convention is used throughout.

Remark 2.2. Theh−KCC-covariant derivativecomponents DTh ₍₁₎⁽ⁱ⁾

dt transform under a change of coordinates (1.1) as ad−tensor of typeT₍₁₎₁⁽ⁱ⁾.

In such a geometrical context, if we use the notationxⁱ₁=dxⁱ/dt, then the system (2.1) can be rewritten in the following distinguished tensorial form:

Dxh ⁱ₁

dt = −F₍₁₎₁⁽ⁱ⁾ (t, x^k, x^k₁) +N_(1)r⁽ⁱ⁾ x^r₁−H₁₁¹xⁱ₁=

= −F₍₁₎₁⁽ⁱ⁾ +1 2

∂F₍₁₎₁⁽ⁱ⁾

∂x^r₁ x^r₁−1 2H₁₁¹xⁱ₁, Definition 2.3. The distinguished tensor

hε⁽ⁱ⁾₍₁₎₁=−F₍₁₎₁⁽ⁱ⁾ +1 2

∂F₍₁₎₁⁽ⁱ⁾

∂x^r₁ x^r₁−1 2H₁₁¹ xⁱ₁

is called thefirsth−KCC-invarianton the 1-jet spaceJ¹(R, M) of the SODEs (2.1), which is interpreted as anexternal force[1], [7].

Example 2.4. It can be easily seen that for the particular first order jet rheonomic dynamical system

(2.3) dxⁱ

dt =X₍₁₎⁽ⁱ⁾(t, x^k)⇒ d²xⁱ

dt² = ∂X₍₁₎⁽ⁱ⁾

∂t +∂X₍₁₎⁽ⁱ⁾

∂x^m x^m₁,

whereX₍₁₎⁽ⁱ⁾(t, x) is a givend−tensor onJ¹(R, M), the firsth−KCC-invariant has the form

hε⁽ⁱ⁾₍₁₎₁ =∂X₍₁₎⁽ⁱ⁾

∂t +1 2

∂X₍₁₎⁽ⁱ⁾

∂x^r x^r₁−1 2H₁₁¹ xⁱ₁.

In the sequel, let us vary the trajectoriesxⁱ(t) of the system (2.1) by the nearby trajectories (xⁱ(t, s))s∈(−ε,ε), where xⁱ(t,0) = xⁱ(t). Then, considering the variation d−tensor field

ξⁱ(t) = ∂xⁱ

∂s

¯¯

¯¯

s=0

,

we get thevariational equations

(2.4) d²ξⁱ

dt² +∂F₍₁₎₁⁽ⁱ⁾

∂x^j ξ^j+∂F₍₁₎₁⁽ⁱ⁾

∂x^r₁ dξ^r

dt = 0.

In order to find other jet geometrical invariants for the system (2.1), we also introduce the h−KCC-covariant derivative of a d−tensor of kind ξⁱ(t) on the 1-jet spaceJ¹(R, M) via

Dξh ⁱ

dt =dξⁱ

dt +N_(1)m⁽ⁱ⁾ ξ^m=dξⁱ dt +1

2

∂F₍₁₎₁⁽ⁱ⁾

∂x^m₁ ξ^m+1 2H₁₁¹ξⁱ. Remark 2.5. Theh−KCC-covariant derivativecomponents

Dξh ⁱ

dt transform under a change of coordinates (1.1) as ad−tensorT₍₁₎⁽ⁱ⁾.

In this geometrical context, the variational equations (2.4) can be rewritten in the following distinguished tensorial form:

Dh

dt



 Dξh ⁱ

dt



=P^hi_m11ξ^m, where

Phⁱ_j11 = −∂F₍₁₎₁⁽ⁱ⁾

∂x^j +1 2

∂²F₍₁₎₁⁽ⁱ⁾

∂t∂x^j₁ +1 2

∂²F₍₁₎₁⁽ⁱ⁾

∂x^r∂x^j₁x^r₁−1 2

∂²F₍₁₎₁⁽ⁱ⁾

∂x^r₁∂x^j₁F₍₁₎₁^(r) + +1

4

∂F₍₁₎₁⁽ⁱ⁾

∂x^r₁

∂F₍₁₎₁^(r)

∂x^j₁ +1 2

dH₁₁¹ dt δ_jⁱ−1

4H₁₁¹H₁₁¹δⁱ_j.

Definition 2.6. Thed−tensor P^{h i}_j11 is called the second h−KCC-invariant on the 1-jet spaceJ¹(R, M) of the system (2.1), or thejeth−deviation curvatured−tensor.

Example 2.7. If we consider the second-order system of differential equations of the harmonic curves associated to the pair of Riemannian metrics(h11(t), ϕij(x)),system which is given by (see the Examples 1.6 and 1.8)

d²xⁱ

dt² −H₁₁¹(t)dxⁱ

dt +γ_jkⁱ (x)dx^j dt

dx^k dt = 0,

whereH₁₁¹(t) andγ_jkⁱ (x) are the Christoffel symbols of the Riemannian metricsh11(t) andϕij(x),then the secondh−KCC-invariant has the form

Phⁱ_j11=−Rⁱ_pqjx^p₁x^q₁,

where

Rⁱ_pqj= ∂γⁱ_pq

∂x^j −∂γⁱ_pj

∂x^q +γ_pq^r γ_rjⁱ −γ^r_pjγ_rqⁱ

are the components of the curvature of the spatial Riemannian metricϕ_ij(x).Conse- quently, the variational equations (2.4) become the followingjet Jacobi field equations:

Dh

dt



 Dξh ⁱ

dt



+Rⁱ_pqmx^p₁x^q₁ξ^m= 0,

where

Dξh ⁱ

dt = dξⁱ

dt +γⁱ_jmx^j₁ξ^m.

Example 2.8. For the particular first order jet rheonomic dynamical system (2.3) the jeth−deviation curvatured−tensor is given by

Phⁱ_j11= 1 2

∂²X₍₁₎⁽ⁱ⁾

∂t∂x^j +1 2

∂²X₍₁₎⁽ⁱ⁾

∂x^j∂x^rx^r₁+1 4

∂X₍₁₎⁽ⁱ⁾

∂x^r

∂X₍₁₎^(r)

∂x^j +1 2

dH₁₁¹ dt δⁱ_j−1

4H₁₁¹H₁₁¹δ_jⁱ. Definition 2.9. The distinguished tensors

Rhⁱ_jk1= 1 3



∂P^hi_j11

∂x^k₁ −∂P^hi_k11

∂x^j₁



, B^hi_jkm=∂R^hi_jk1

∂x^m₁

and

Dⁱ¹_jkm= ∂³F₍₁₎₁⁽ⁱ⁾

∂x^j₁∂x^k₁∂x^m₁

are called the third, fourth and fifth h−KCC-invariant on the 1-jet vector bundle J¹(R, M) of the system (2.1).

Remark 2.10. Taking into account the transformation rules (2.2) of the components F₍₁₎₁⁽ⁱ⁾ , we immediately deduce that the components Dⁱ¹_jkmbehave like ad−tensor.

Example 2.11. For the first order jet rheonomic dynamical system (2.3) the third, fourth and fifthh−KCC-invariants are zero.

Theorem 2.12 (of characterization of the jeth−KCC-invariants). All the five h−KCC-invariants of the system (2.1) cancel onJ¹(R, M)if and only if there exists a flat symmetric linear connectionΓⁱ_jk(x)on M such that

(2.5) F₍₁₎₁⁽ⁱ⁾ = Γⁱ_pq(x)x^p₁x^q₁−H₁₁¹ (t)xⁱ₁. Proof. ”⇐” By a direct calculation, we obtain

hε⁽ⁱ⁾₍₁₎₁ = 0, P^hi_j11=−Rⁱ_pqjx^p₁x^q₁= 0 and Dⁱ¹_jkl= 0,

where Rⁱ_pqj = 0 are the components of the curvature of the flat symmetric linear connection Γⁱ_jk(x) onM.

”⇒” By integration, the relation

Dⁱ¹_jkl= ∂³F₍₁₎₁⁽ⁱ⁾

∂x^j₁∂x^k₁∂x^l₁ = 0 subsequently leads to

∂²F₍₁₎₁⁽ⁱ⁾

∂x^j₁∂x^k₁ = 2Γⁱ_jk(t, x)⇒ ∂F₍₁₎₁⁽ⁱ⁾

∂x^j₁ = 2Γⁱ_jpx^p₁+U_(1)j⁽ⁱ⁾ (t, x)⇒

⇒ F₍₁₎₁⁽ⁱ⁾ = Γⁱ_pqx^p₁x^q₁+U_(1)p⁽ⁱ⁾ x^p₁+V₍₁₎₁⁽ⁱ⁾ (t, x),

where the local functions Γⁱ_jk(t, x) are symmetrical in the indicesj andk.

The equality^hε⁽ⁱ⁾₍₁₎₁ = 0 onJ¹(R, M) leads us to V₍₁₎₁⁽ⁱ⁾ = 0 and to U_(1)j⁽ⁱ⁾ =−H₁₁¹δⁱ_j. Consequently, we have

∂F₍₁₎₁⁽ⁱ⁾

∂x^j₁ = 2Γⁱ_jpx^p₁−H₁₁¹ δⁱ_j and F₍₁₎₁⁽ⁱ⁾ = Γⁱ_pqx^p₁x^q₁−H₁₁¹xⁱ₁.

The condition P^hi_j11 = 0 on J¹(R, M) implies the equalities Γⁱ_jk = Γⁱ_jk(x) and Rⁱ_pqj+Rⁱ_qpj= 0, where

Rⁱ_pqj= ∂Γⁱ_pq

∂x^j −∂Γⁱ_pj

∂x^q + Γ^r_pqΓⁱ_rj−Γ^r_pjΓⁱ_rq.

It is important to note that, taking into account the transformation laws (2.2), (1.3) and (1.1), we deduce that the local coefficients Γⁱ_jk(x) behave like a symmetric linear connection onM.Consequently,Rⁱ_pqj represent the curvature of this symmetric linear connection.

On the other hand, the equalityR^hi_jk1= 0 leads us toRⁱ_qjk= 0,which infers that the symmetric linear connection Γⁱ_jk(x) onM is flat. ¤ Acknowledgements. The present research was supported by the Romanian Academy Grant 4/2009.

References

[1] P.L. Antonelli,Equivalence Problem for Systems of Second Order Ordinary Dif- ferential Equations, Encyclopedia of Mathematics, Kluwer Academic, Dordrecht, 2000.

[2] P.L. Antonelli, L. Bevilacqua, S.F. Rutz,Theories and models in symbiogenesis, Nonlinear Analysis: Real World Applications4(2003), 743-753.

[3] P.L. Antonelli, I. Buc˘ataru,New results about the geometric invariants in KCC- theory, An. S¸t. Univ. ”Al.I.Cuza” Ia¸si. Mat. N.S. 47 (2001), 405-420.

[4] P.L. Antonelli, R.S. Ingarden, M. Matsumoto,The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Fundamental Theories of Physics, vol. 58, Kluwer Academic Publishers, Dordrecht, 1993.

[5] Gh. Atanasiu, V. Balan, N. Brˆınzei, M. Rahula, The Differential-Geometric Structures: Tangent Bundles, Connections in Bundles and The Exponential Law in Jet-Spaces (in Russian), LKI, Moscow, 2010.

[6] V. Balan, I.R. Nicola, Berwald-Moor metrics and structural stability of conformally-deformed geodesic SODE, Appl. Sci. 11 (2009), 19-34.

[7] V. Balan, I.R. Nicola, Linear and structural stability of a cell division process model, Hindawi Publishing Corporation, International Journal of Mathematics and Mathematical Sciences, vol. 2006, 1-15.

[8] I. Buc˘ataru, R. Miron, Finsler-Lagrange Geometry. Applications to Dynamical Systems, Romanian Academy Eds., 2007.

[9] E. Cartan,Observations sur le m´emoir pr´ec´edent, Mathematische Zeitschrift 37, 1 (1933), 619-622.

[10] S.S. Chern, Sur la g´eom´etrie d’un syst`eme d’equations differentialles du second ordre, Bulletin des Sciences Math´ematiques 63 (1939), 206-212.

[11] D.D. Kosambi, Parallelism and path-spaces, Mathematische Zeitschrift 37, 1 (1933), 608-618.

[12] M. Neagu, Riemann-Lagrange Geometry on 1-Jet Spaces, Matrix Rom, Bucharest, 2005.

[13] M. Neagu, C. Udri¸ste, A. Oan˘a, Multi-time dependent sprays and h-traceless maps on J¹(T, M), Balkan J. Geom. Appl. 10, 2 (2005), 76-92.

[14] V.S. Sab˘au,Systems biology and deviation curvature tensor, Nonlinear Analysis.

Real World Applications 6, 3 (2005), 563-587.

Authors’ addresses:

Vladimir Balan

University Politehnica of Bucharest, Faculty of Applied Sciences, Department of Mathematics-Informatics I,

313 Splaiul Independentei, 060042 Bucharest, Romania.

E-mail: [email protected]

Website: http://www.mathem.pub.ro/dept/vbalan.htm Mircea Neagu

University Transilvania of Bra¸sov, Faculty of Mathematics and Informatics, Department of Algebra, Geometry and Differential Equations,

B-dul Iuliu Maniu, Nr. 50, BV 500091, Bra¸sov, Romania.

E-mail: [email protected]

Website: http://www.2collab.com/user:mirceaneagu