Directory UMM :Journals:Journal_of_mathematics:BJGA:

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, x}i_{, x}i

1); these transform by the rules [13]

(1.1)

          

e

t=et(t)

e

xi₌

e

xi_(xj₎

e

xi

1= ∂xei

∂xj

dt det·x

j

1.

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=³D1_1ki(₍₁₎₍j)(1)_l)..._...´on the 1-jet vector bundle, whose local components transform by the rules

(1.2) D₁1_ki(₍₁₎₍j)(1)_u)..._...=De₁1_rp₍₁₎₍(m)(1)_s)......dt det

∂xi

∂exp

Ã

∂xj

∂xem

det dt

!

det dt

∂xer

∂xk

µ

∂exs

∂xu

dt det

¶

...,

is called ad−_{tensor field.} ∗

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

Remark 1.2. The use of parentheses for certain indices of the local components

D1₁i_k(₍₁₎₍j)(1)_l)..._...

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

Example 1.3. The geometrical object

C=C(₍₁₎i) ∂

∂xi

1 ,

where C(₍₁₎i) = xi

1, 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( i) (1)1j

∂ ∂xi

1

⊗_dt⊗_dxj_,

whereJ₍₁₎₁(i) _j=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 =³H₍₁₎₁(j) ´onJ1_{(R, M),which transform}

by the rules

(1.3) 2He₍₁₎₁(k) = 2H₍₁₎₁(j)

µ

dt det

¶2 ∂exk

∂xj −

dt det

∂xek

1 ∂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

d2_t det2,

we deduce that the local components

˚

H₍₁₎₁(j) =−1

2H

1 11x

j

1

Definition 1.7. A set of local functionsG=³G(₍₁₎₁j) ´,which transform by the rules

(1.5) 2Ge(₍₁₎₁k) = 2G(₍₁₎₁j)

µ

dt det

¶2 ∂exk

∂xj −

∂xm

∂exj

∂xek

1 ∂xmxe

j

1,

is called aspatial semisprayonJ1_{(R, M).}

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

and let us consider

γjki =

ϕim

2

µ

∂ϕjm

∂xk +

∂ϕkm

∂xj −

∂ϕjk

∂xm

¶

its Christoffel symbols. Taking into account that we have the transformation rules

(1.6) eγp

qr=γ i jk

∂xep

∂xi

∂xj

∂xeq

∂xk

∂exr +

∂xep

∂xl

∂2_xl

∂exq_∂exr,

we deduce that the local components

˚

G(₍₁₎₁j) = 1 2γ

j klx

k

1xl1

define a spatial semispray ˚G = ³G˚₍₁₎₁(j) ´ 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 = ³M₍₁₎₁(j), N₍₁₎(j)_i´ on J1_{(R, M), which}

transform by the rules

(1.7) Mf₍₁₎₁(k) =M₍₁₎₁(j)

µ_dt

det

¶2 ∂xek

∂xj −

dt det

∂exk

1 ∂t

and

(1.8) Ne₍₁₎(k)_l=N₍₁₎(j)_idt det

∂xi

∂exl

∂xek

∂xj −

∂xm

∂xel

∂xek

1 ∂xm,

is called anonlinear connectionon the 1-jet spaceJ1_{(R, M).}

Example 1.10. Let us consider that (R, h11(t)) and (M, ϕij(x)) are Riemannian

manifolds having the Christoffel symbolsH1

11(t) and γjki (x). Then, using the

trans-formation rules (1.1), (1.4) and (1.6), we deduce that the set of local functions

˚_{Γ =}³_M˚(j) (1)1,N˚

(j) (1)i

´

,

where

˚

M₍₁₎₁(j) =−H111x

j

1 and N˚ (j) (1)i=γ

j imx

m

1 ,

represents a nonlinear connection on the 1-jet space J1_{(R, M). This jet nonlinear}

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 Γ_temporal = (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

(j) (1)1.

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

N₍₁₎(j)_k= ∂G

(j) (1)1 ∂xk

1

, G(₍₁₎₁j) = 1 2N

(j) (1)mx

m

1.

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) d

2_xi

dt2 +F (i) (1)1(t, x

k_{, x}k

1) = 0, i= 1, n,

wherexk

1=dxk/dtand the local componentsF (i) (1)1(t, x

k_{, x}k

1) transform under a change

of coordinates (1.1) by the rules

(2.2) Fe₍₁₎₁(r) =F₍₁₎₁(j)

µ

dt det

¶2 ∂xer

∂xj −

dt det

∂exr

1 ∂t −

∂xm

∂xej

∂xer

1 ∂xmxe

j

1.

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:

d2_xi

dt2 −H 1

11xi1+ 2G (i) (1)1(t, x

k_{, x}k

1) = 0, i= 1, n,

where

G(₍₁₎₁i) =1 2F

(i) (1)1+

1 2H

1 11x

i

1

are the components of a spatial semispray onJ1_{(R, M). Moreover, the coefficients}

connection Γ on the 1-jet spaceJ1_{(R, M), by putting}

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, xk_{, x}k

where the Einstein summation convention is used throughout.

Remark 2.2. Theh−KCC-covariant derivativecomponents

h

DT₍₁₎(i)

dt transform under

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

In such a geometrical context, if we use the notationxi

1=dxi/dt, then the system

(2.1) can be rewritten in the following distinguished tensorial form:

h

Definition 2.3. The distinguished tensor

h

which is interpreted as anexternal force[1], [7].

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) =} ∂x

we get thevariational equations

(2.4) d

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 ξ}i_{(t) on the 1-jet}

Remark 2.5. Theh−KCC-covariant derivativecomponents

h

Dξi

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:

h

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)

where

are the components of the curvature of the spatial Riemannian metricϕij(x).

Conse-quently, the variational equations (2.4) become the followingjet Jacobi field equations:

h

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

h

Definition 2.9. The distinguished tensors

h

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 onJ}1_{(R, M)if and only if there exists} a flat symmetric linear connectionΓi

jk(x)on M such that

Proof. ”⇐_{” By a direct calculation, we obtain}

where Ri

pqj = 0 are the components of the curvature of the flat symmetric linear

connection Γi

jk(x) onM.

”⇒_{” By integration, the relation}

Dijkl1 =

∂3_F(i) (1)1 ∂xj1∂xk1∂xl1

= 0

subsequently leads to

∂2_F(i) (1)1 ∂xj₁∂xk

1

= 2Γi

jk(t, x)⇒

∂F₍₁₎₁(i) ∂xj₁ = 2Γ

i jpx

p

1+U (i)

(1)j(t, x)⇒

⇒ _F(i) (1)1= Γ

i pqx

p

1x

q

1+U (i) (1)px

p

1+V (i) (1)1(t, x),

where the local functions Γi

jk(t, x) are symmetrical in the indicesj andk.

The equalityhε(₍₁₎₁i) = 0 onJ1_{(R, M) leads us to V}(i)

(1)1= 0 and to U (i) (1)j =−H

1 11δij.

Consequently, we have

∂F₍₁₎₁(i) ∂xj₁ = 2Γ

i jpx

p

1−H111 δij and F

(i) (1)1 = Γ

i pqx

p

1x

q

1−H111xi1.

The condition Phi

j11 = 0 on J1(R, M) implies the equalities Γijk = Γ i

jk(x) and

Ri

pqj+Riqpj= 0, where

Ri_pqj= ∂Γ

i pq

∂xj −

∂Γi pj

∂xq + Γ r

pqΓirj−ΓrpjΓirq.

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 equalityRhi

jk1= 0 leads us toRiqjk= 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.

[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: vladimir.balan@upb.ro

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: mircea.neagu@unitbv.ro