New results about the geometric invariants in KCC-theory

Abstract

The KCC-theory (Kosambi, [12], CARTAN, [7], and CHERN, [8]) of a system of second order ordinary differential equations (SODE) uses five geometric in-variants 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].

Full text

ANALELE S¸TIINT¸ IFICE ALE UNIVERSIT
˘
AT¸II ”AL.I.CUZA” IAS¸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. BUC
˘
ATARU
†
Abstract. The KCC-theory (Kosambi, [12], CARTAN, [7], and CHERN, [8]) of
a system of second order ordinary differential equations (SODE) uses five geometric in-
variants 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 al-
ternatively, 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 expresse d
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 f or 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 tropho-
dynamics, [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 equiv-
alent 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. BUC
˘
ATARU 2
T M of a manifold M . As it is well known, a s em ispray yields two other
important geometrical objects, living also on the tangent bundle: a non-
linear 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 con-
sider 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
connection, R
i
jk
is the third invariant. In section two we give a global ex-
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
D has only one nonzero component of torsion R
i
jk
and two nonzero com-
ponents of curvature R
i
jkl
and D
i
jkl
. These are the third, the fourth, and
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 con-
nection, or a semispray, ([16], [9]), we introduce a covariant derivative de-
termined 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.

3 GEOMETRIC INVARIANTS IN KCC- THEORY 407
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 = (x
i
)) on M, we denote by
(π
−1
(U), Φ = (x
i
, 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→ V
u
= Kerπ
∗,u
⊂ T
u
T M. This is
an n-dimensional integrable distribution. If {
∂
∂x
i
|
u
,
∂
∂y
i
|
u
} is the natural
basis of the tangent space T
u
T M, then {
∂
∂y
i
|
u
} is a basis for V
u
, ∀u ∈ T M .
Consider J =
∂
∂y
i
⊗ dx
i
, the almost tangent structure (J is also called the
vertical endomorphism of T M), and Γ = y
i
∂
∂y
i
the Liouville vector field. A
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
∂
∂x
i
− 2G
i
(x, y)
∂
∂y
i
.
The functions G
i
(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→ N
u
⊂ T
u
T M that is sup-
plementary to the vertical distribution V is called a nonlinear connection.
For every u ∈ T M we have the direct sum
(1.1) T
u
T M = N
u
⊕ V
u
.
An adapted basis to the previous direct sum is {
δ
δx
i
=
∂
∂x
i
− N
j
i
(x, y)
∂
∂y
j
,
∂
∂y
i
}. We call this basis the Berwald basis of the nonlinear connection N.
The functions N
i
j
(x, y) are defined on domains of loc al chart, and these
are called the local coefficients of the nonlinear connection N. It is well
known that every semispray S with local coefficients G
i
, induces a nonlinear
connection N with local coefficients N
i
j
=
∂G
i
∂y
j
, [11]. Next, we shall work
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
δx
i
⊗ dx
i
, v =
∂
∂y
i
⊗ δy
i
,

408 P.L. ANTONELLI and I. BUC
˘
ATARU 4
where (dx
i
, δy
i
= dy
i
+ N
i
j
(x, y)dx
j
) is the dual basis of the Berwald basis.
The vertical projector v is given by
(1.3) v(X) =
1
2
(X + [S, JX] + J[X, S]), ∀X ∈ χ(T M).
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
(1.4) S = y
i
δ
δx
i
− E
i
(x, y)
∂
∂y
i
,
where
(1.5) E
i
(x, y) = 2G
i
(x, y) − N
i
j
(x, y)y
j
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)
∂
∂x
i
on the base manifold M, let us consider
(1.6) X
c
= X
i
(x)
∂
∂x
i
+
∂X
i
∂x
j
(x)y
j
∂
∂y
i
, X
h
= X
i
δ
δx
i
, and X
v
= X
i
(x)
∂
∂y
i
,
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
+ L
S
X
v
,
where L
S
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
) + N
i
j
X
j
=
∂X
i
∂x
j
y
j
− 2
∂X
i
∂y
j
G
j
+
∂G
i
∂y
j
X
j
.
Remark 1.1.
1
o
For a Finsler vector field X
i
(x, y), its dynamical covariant derivative
∇X
i
satisfies
(1.7)
0
[S, X
i
∂
∂y
i
] = −X
i
δ
δx
i
+ ∇X
i
∂
∂y
i
.

5 GEOMETRIC INVARIANTS IN KCC- THEORY 409
As a consequence we have that v[S, X
i
∂
∂y
i
] = ∇X
i
∂
∂y
i
, and then ∇X
i
is
also a Finsler vector field.
2
o
We have the properties: ∇(X
i
+ Y
i
) = ∇X
i
+ ∇Y
i
, and ∇(fX
i
) =
S(f )X
i
+ f∇X
i
.
3
o
If X = X
i
(x)
∂
∂x
i
is a vector field on the base manifold M , then X
c
=
X
i
(x)
δ
δx
i
+ ∇X
i
∂
∂y
i
, so v(X
c
) = ∇X
i
∂
∂y
i
.
4
o
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) = (x
i
(t)) ∈ M is called a path of the semispray
S if its lift to T M : ˜c : t ∈ I 7→ ˜c(t) = (x
i
(t),
dx
i
dt
(t)) ∈ T M is an integral
curve of S. In local coordinates the curve c(t) = (x
i
(t)) is a path of S if
and only if:
(1.8)
d
2
x
i
dt
2
+ 2G
i
(x,
dx
dt
) = 0.
An equivalent invariant form of the system (1.8) is given by:
(1.8)
0
∇(
dx
i
dt
) = −E
i
(x,
dx
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
basis {
δ
δx
i
,
∂
∂y
i
}. Consider the tensor field θ =
δ
δx
i
⊗ δy
i
. We can see that
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 prese rves by parallelism the horizontal

410 P.L. ANTONELLI and I. BUC
˘
ATARU 6
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)









D
δ
δx
i
δ
δx
j
= F
k
ji
δ
δx
k
, D
δ
δx
i
∂
∂y
j
= F
k
ji
∂
∂y
k
,
D
∂
∂y
i
δ
δx
j
= C
k
ji
δ
δx
k
, D
∂
∂y
i
∂
∂y
j
= C
k
ji
∂
∂y
k
.
Next, a Finsler connection will be indicated also by the set D=(N
i
j
, F
k
ij
, C
k
ij
).
Under a change of induced c oordinates on T M, the coefficients F
k
ij
transform
like the coefficients of a linear connection on the base manifold M. The
coefficients C
k
ij
are the components of a (1,2)-type Finsler tensor field.
For a Finsler vector field X
i
(x, y), we define the covariant derivative
induced by a Finsler connection D = (N
i
j
, F
k
ij
, C
k
ij
) as:
(2.2) DX
i
= S(X
i
) + F
i
jk
X
j
y
k
− C
i
jk
X
j
E
k
,
or in the equivalent form
(2.2)
0
(DX
i
)
∂
∂y
i
= D
S
(X
i
∂
∂y
i
).
Denote by X
i
|k
, and X
i
|
k
the horizontal, and the vertical covariant deriva-
tives of X
i
, respectively. These are given by:
(2.3) X
i
|k
=
δX
i
δx
k
+ F
i
jk
X
j
, X
i
|
k
=
∂X
i
∂y
k
+ C
i
jk
X
j
.
Then, the Finsler covariant derivative takes the form:
(2.2)
00
DX
i
= X
i
|k
y
k
− X
i
|
k
E
k
.

7 GEOMETRIC INVARIANTS IN KCC- THEORY 411
More generally, if T
i
1
···i
r
j
1
···j
s
are the components of a (r, s)-type Finsler tensor
field, the Finsler covariant derivative of this DT
i
1
···i
r
j
1
···j
s
is also a Finsler tensor
field of (r, s)-type, and it is defined by:
(2.2)
000
DT
i
1
···i
r
j
1
···j
s
= T
i
1
···i
r
j
1
···j
s
|k
y
k
− T
i
1
···i
r
j
1
···j
s
|
k
E
k
,
where T
i
1
···i
r
j
1
···j
s
|k
and T
i
1
···i
r
j
1
···j
s
|
k
are the h and v-covariant derivatives, respec-
tively ([16], p.40). We can remark here that this covariant derivative satisfies
the Leibniz rule, that is for example D(T
i
j
S
j
k
) = D(T
i
j
)S
j
k
+ T
i
j
D(S
j
k
).
Theorem 2.1. The map D : χ(T M ) × χ(T M ) → χ(T M), given by
(2.4) D
X
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 b oth arguments. To prove that
D
fX
Y = fD
X
Y , ∀f ∈ F(T M ) we have to use that vh = hv = Jv = θh = 0.
The other properties like D
X
fY = X(f)Y + f D
X
Y , Dh = 0, and D J = 0
follow easy using also Jθ = v, θJ = h, v
2
= v, and h
2
= h.
With resp ec t to the Berwald basis, the Berwald connection D has the
local coefficients, ([5], [16]):
(2.5) F
i
jk
=
∂N
i
j
∂y
k
=
∂
2
G
i
∂y
j
∂y
k
, C
i
jk
= 0.
For the Berwald connection D one considers typically the torsion T (X, Y ) =
D
X
Y − D
Y
X − [X, Y ]. With respect to the Berwald basis, there is only one
nonzero component of the torsion, ([5], [16]):
(2.6) vT (
δ
δx
i
,
δ
δx
j
) =: R
k
ij
∂
∂y
k
= (
δN
k
j
δx
i
−
δN
k
i
δx
j
)
∂
∂y
k
, (v)h − torsion.
The Finsler tensor field R
k
ij
is also called the curvature tensor of the non-
linear connection N , because:
(2.7) [
δ
δx
i
,
δ
δx
j
] = −R
k
ij
∂
∂y
k
.

412 P.L. ANTONELLI and I. BUC
˘
ATARU 8
Let us consider now R(X, Y )Z = D
X
D
Y
Z − D
Y
D
X
Z − D
[X,Y ]
Z, the cur-
vature of the Berwald connection. With respect to the Berwald basis, there
are only two nonzero components of the curvature ([5], [16]):
(2.8)



















R
i
jkl
=
δF
i
jk
δx
l
−
δF
i
jl
δx
k
+ F
m
jk
F
i
ml
− F
m
jl
F
i
mk
;
(the Riemann − Christoffel curvature tensor)
D
i
jkl
=
∂F
i
jk
∂y
l
=
∂
3
G
i
∂y
j
∂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
the Finsler tensor fields R
i
jk
and R
i
ljk
are related by, ([6], [16]):
(2.9) R
i
jk
= R
i
ljk
y
l
.
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
semispray S, R
i
jk
the (v)h-component of the torsion, and R
i
ljk
the Riemann-
Christoffel curvature tensor. Then these are related by:
(2.10) R
i
ljk
y
l
− R
i
kj
= y
i
|j|k
− y
i
|k|j
,
or in the equivalent form
(2.10)
0
R
i
ljk
y
l
− R
i
kj
= E
i
|
k|j
− E
i
|
j|k
.
Proof. Using the Ricc i identities for the Berwald connection D ([17],
p.78) and the Liouville ve ctor field, we get (2.10). For (2.10)
0
it is enough
to prove that:
(2.11) y
i
|j
= −E
i
|
j
.
This (1,1)-type Finsler tensor field is called the deflection tensor of the
connection. According to (2.3) we have y
i
|j
=
δy
i
δx
j
+ F
i
kj
y
k
= F
i
kj
y
k
− N
i
j
. As

9 GEOMETRIC INVARIANTS IN KCC- THEORY 413
E
i
= 2G
i
−
∂G
i
∂y
j
y
j
, the E
i
|
j
=
∂E
i
∂y
j
= 2
∂G
i
∂y
j
−
∂G
i
∂y
j
−
∂
2
G
i
∂y
j
∂y
k
y
k
= N
i
j
− F
i
kj
y
k
=
−y
i
|j
.
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
i
|j
X
j
= DX
i
+ E
i
|
j
X
j
.
Proof. From (2.2), the Berwald covariant derivative has the form:
DX
i
= S(X
i
) + F
i
jk
y
j
X
k
, so DX
i
= S(X
i
) + N
i
k
X
k
+ (F
i
jk
y
j
− N
i
k
)X
k
=
∇X
i
+ y
i
|k
y
k
, and the proof is completed.
Remark 2.2. Using the Berwald covariant derivative, we can write the
path equation (1.8) in an equivalent form:
(2.13) D(
dx
i
dt
) = −(E
i
+ E
i
|
j
dx
j
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) = (x
i
(t)) a trajectory
of (1.8), and let vary it into nearby ones according to:
(3.1) ˜x
i
(t) = x
i
(t) + εξ
i
(t),
where ε denotes a scalar parameter with small value |ε|, and ξ
i
(t) are com-
ponents of a contravariant vector field along c(t). Substitution of (3.1) into
(1.8), asking for ˜c(t) = (˜x
i
(t)) to be also a trajectory of (1.8), and letting
ε → 0 yields to the so-called variational equations:
(3.2)
d
2
ξ
i
dt
2
+ 2
∂G
i
∂x
j
ξ
j
+ 2
∂G
i
∂y
j
dξ
j
dt
= 0.
Theorem 3.1. For the variational equations (3.2) we have the equiva-
lent invariant form (Jacobi equations):
(3.3) ∇
2
ξ
i
+ (R
i
jk
dx
k
dt
+ E
i
|j
)ξ
j
= 0.

414 P.L. ANTONELLI and I. BUC
˘
ATARU 10
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) B
i
j
= 2
∂G
i
∂x
j
− S(
∂G
i
∂y
j
) −
∂G
i
∂y
r
∂G
r
∂y
j
.
It can be proved that B
i
j
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 endo-
morphism in [9]. It is easy to check that the equations (3.2) are equivalent
to:
(3.5) ∇
2
ξ
i
+ B
i
j
ξ
j
= 0.
All we have to prove now is the following expression of the second invariant:
(3.6) B
i
j
= R
i
jk
y
k
+ E
i
|j
.
Let X
i
(x, y) be an arbitrary Finsler vector field, and consider the vector
field
(3.7)
˜
X = X
i
∂
∂x
i
+ S(X
i
)
∂
∂y
i
on T M . We have then:
(3.8) [S,
˜
X] = (∇
2
X
i
+ B
i
j
X
j
)
∂
∂y
j
.
If we consider the expression of S and
˜
X in the Berwald basis S = y
i
δ
δx
i
−
E
i
∂
∂y
i
and
˜
X = X
i
δ
δx
i
+ ∇ X
i
∂
∂y
i
, respectively, then the bracket [S ,
˜
X] can
be expressed as follows:
(3.9) [S,
˜
X] = {∇
2
X
i
+ (R
i
jk
y
k
+ E
i
|j
)X
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 B
i
j
can be expressed
as in (3.6).

11 GEOMETRIC INVARIANTS IN KCC- THEORY 415
Definition 3.1. 1
o
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.
2
o
A dynamical symmetry of the semispray S is a vector field
˜
X on T M
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
]] = 2L
S
X
h
+ L
S
L
S
X
v
= 0.
Theorem 3.2.
1
o
A vector field
˜
X = X
i
(x, y)
δ
δx
i
+ Y
i
(x, y)
∂
∂y
i
is a dynamical symmetry of
S if and only if
(3.11) Y
i
= ∇X
i
, and ∇
2
X
i
+ B
i
j
X
j
= 0.
2
o
If
˜
X = X
i
(x, y)
δ
δx
i
+ Y
i
(x, y)
∂
∂y
i
is a dynamical symmetry of the semis-
pray S and c(t) = (x
i
(t)) is a path of S, then the restriction of X
i
(x, y)
along ˜c(t) = (x
i
(t),
dx
i
dt
(t)) is a Jacobi vector field for S.
Proof. 1
o
If we express the Lie bracket [S,
˜
X] using the Berwald basis,
we have
(3.11)
0
[S,
˜
X] = (∇X
i
− Y
i
)
δ
δx
i
+ (∇Y
i
+ B
i
j
X
j
)
∂
∂y
j
.
So,
˜
X is a dynamical symmetry of S if and only if (3.11) is true.
2
o
If X
i
(x, y) are the horizontal components of a dynamical symmetry
˜
X,
then ∇
2
X
i
+ (R
i
jk
y
k
+ E
i
|j
)X
j
= 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
) =
D
2
X
i
+ 2E
i
|
j
DX
j
+ [D(E
i
|
j
) + E
i
|
k
E
k
|
j
]X
j
.

416 P.L. ANTONELLI and I. BUC
˘
ATARU 12
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) R
i
j
= D(E
i
|
j
) + E
i
|
k
E
k
|
j
+ R
i
jk
y
k
+ E
i
|j
.
We have that R
i
j
is a (1, 1)-type Finsler tensor field. Thus we have the
following theorem:
Theorem 3.3.
1
o
The Jacobi equations of the system (1.8) have the invariant form, using
the Berwald covariant derivative:
(3.13) D
2
ξ
i
+ 2E
i
|
j
Dξ
j
+ R
i
j
ξ
j
= 0.
2
o
A Finsler vector field X
i
(x, y) is a dynamical symmetry of the semispray
S if and only if:
(3.14) D
2
X
i
+ 2E
i
|
j
DX
j
+ R
i
j
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 B
i
j
and R
i
j
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 B
i
j
,
defined by (3.4). The third, fourth, and fifth invariants are:
(4.1)

























B
i
jk
:=
1
3
(
∂B
i
j
∂y
k
−
∂B
i
k
∂y
j
),
B
i
lkj
:=
∂B
i
jk
∂y
l
,
D
i
jkl
:=
∂F
i
jk
∂y
l
=
∂
3
G
i
∂y
j
∂y
k
∂y
l
.
The Tensor D
i
jkl
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.

13 GEOMETRIC INVARIANTS IN KCC- THEORY 417
Theorem 4.1.
1
o
The curvature R
i
jk
of t he nonlinear connection N (or the (v)h-torsion of
the Berwald connection D) is the third invariant of the semispray S.
2
o
The Riemann-Christoffel curvature tensor R
i
jkl
of the Berwald connection
D is the fourth invariant of the semispray S.
Proof. We have to prove that R
i
jk
= B
i
jk
and R
i
jkl
= B
i
jkl
. First we
prove that R
i
jk
and R
i
jkl
satisfy (4.1)
2
, that is R
i
lkj
=
∂R
i
jk
∂y
l
. From (2.6) we
have R
k
ij
=
δN
k
j
δx
i
−
δN
k
i
δx
j
, so
∂R
k
ij
∂y
l
=
∂
∂y
l
(
δN
k
j
δx
i
)−
∂
∂y
l
(
δN
k
i
δx
j
). As [
δ
δx
j
,
∂
∂y
i
] =
F
k
ji
∂
∂y
k
, we have that
∂R
k
ij
∂y
l
=
δ
δx
i
(
∂N
k
j
∂y
l
)−F
p
li
∂N
k
j
∂y
p
−
δ
δx
j
(
∂N
k
i
∂y
l
)+F
p
lj
∂N
k
i
∂y
p
=
δF
k
jl
δx
i
−
δF
k
il
δx
j
+ F
p
jl
F
k
pi
− F
p
il
F
k
jp
= R
k
lji
. According to (3.6) we have for the
second invariant B
i
j
, the expression B
i
j
= R
i
jk
y
k
+ E
i
|j
. So,
∂B
i
j
∂y
k
= R
i
jk
+
R
i
klj
y
l
+E
i
|j
|
k
. Then,
∂B
i
j
∂y
k
−
∂B
i
k
∂y
j
= 2R
i
jk
+R
i
klj
y
l
+R
i
jkl
y
l
+E
i
|j
|
k
−E
i
|k
|
j
. Using
the Ricci identities for the B erwald connection D, ([17], p.78) we have that
E
i
|j
|
k
− E
i
|
k|j
= D
i
ljk
E
l
, and E
i
|k
|
j
− E
i
|
j|k
= D
i
lkj
E
l
. As the Douglas tensor
is symmetric we have that: E
i
|j
|
k
− E
i
|
k|j
= E
i
|k
|
j
− E
i
|
j|k
. Consequently, we
have, E
i
|j
|
k
− E
i
|k
|
j
= E
i
|
j|k
− E
i
|
k|j
= R
i
lkj
y
l
− R
i
kj
. Finally, we have that:
∂B
i
j
∂y
k
−
∂B
i
k
∂y
j
= 3R
i
jk
+ (R
i
klj
+ R
i
jkl
+ R
i
ljk
)y
l
. Using the Bianchi identity
for the Berwald connection D, we have that R
i
klj
+ R
i
jkl
+ R
i
ljk
= 0, so that
R
i
jk
=
1
3
(
∂B
i
j
∂y
k
−
∂B
i
k
∂y
j
), and the theorem is proved.
5. A particular case. In this section we use the theory developed in
the previous sections in order to study the system:
(5.1)
d
2
x
i
dt
2
+ γ
i
jk
(x)
dx
j
dt
dx
k
dt
+ γ
i
j
(x)
dx
j
dt
+ γ
i
(x) = 0,
where γ
i
jk
(x) are the lo c al coeffic ients of a linear symmetric connection on
the base manifold M , γ
i
j
(x) are the components of a (1,1)-type tensor field

418 P.L. ANTONELLI and I. BUC
˘
ATARU 14
and γ
i
(x) are the components of a vector field. The system (5.1) occurs in
modeling some processes in biology ([3]).
On a local chart on T M we define the functions 2G
i
(x, y) = γ
i
jk
(x)y
j
y
k
+
γ
i
j
(x)y
j
+ γ
i
(x). I t is very easy to check that G
i
are the local components
of a semispray S = y
i
∂
∂x
i
− 2G
i
∂
∂y
i
on T M. Let N be the nonlinear
connection induced by S. Then the local coefficients of N are N
i
j
(x, y) =
γ
i
jk
(x)y
k
+
1
2
γ
i
j
(x). The first invariant of the system (5.1) is given by:
(5.2) E
i
(x, y) =
1
2
γ
i
j
(x)y
j
+ γ
i
(x).
The Berwald connection D, that corresponds to the semispray S has the
local coefficients F
i
jk
= γ
i
jk
, C
i
jk
= 0. So, an immediate consequence is that
the fifth invariant of the system (5.1), the Douglas tensor D
i
jkl
vanishes and
the fourth invariant R
i
jkl
is the curvature tensor of the linear connection
γ
i
jk
(x). Now let us express the second and the third invariants of the system.
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:
(5.3)







E
i
|
j
=
1
2
γ
i
j
,
E
i
|j
=
1
2
γ
i
k|j
y
k
+ γ
i
|j
−
1
4
γ
i
k
γ
k
j
.
As we saw in the previous section, the third invariant is R
i
jk
, the curvature
of the nonlinear connection N . As R
k
ij
=
δN
k
j
δx
i
−
δN
k
i
δx
j
, by a straightforward
calculation we get that
(5.4) R
i
jk
(x, y) = R
i
lkj
(x)y
l
+
1
2
(γ
i
k|j
(x) − γ
i
j|k
(x)).
The second invariant B
i
j
can be expressed, using (3.6), as follows:
(5.5) B
i
j
= R
i
lkj
y
l
y
k
+ γ
i
k|j
y
k
+ γ
i
|j
−
1
2
γ
i
j|k
y
k
−
1
4
γ
i
k
γ
k
j
.
All these invariants can be obtained also, using Schouten’s film space idea,
([18], [4]), and Kron’s work ([13]).

15 GEOMETRIC INVARIANTS IN KCC- THEORY 419
Acknowledgements. This paper has be en 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
pa2@gpu.srv.ualberta.ca
Faculty of Mathematics
”Al.I.Cuza” University
Ia¸si, 6600
ROMANIA
bucataru@mail.uaic.ro