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Jet geometrical extension of the KCC-invariants
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 space J 1 (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; jet h−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 = (J 1 (R, M ), π1 , R × M )
is a vector bundle over the product manifold R × M , having the fibre of type Rn ,
where n is the dimension of the spatial manifold 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 space J 1 (R, M ) by (t, xi , xi1 ); these transform by the rules [13]
e
t=e
t(t)
i
x
e =x
ei (xj )
(1.1)
∂e
xi dt j
x
· x1 .
ei1 =
∂xj de
t
In the geometrical study of the 1-jet bundle, a central role is played by the distinguished tensors (d−tensors).
³
´
1i(j)(1)...
Definition 1.1. A geometrical object D = D1k(1)(l)... on the 1-jet vector bundle,
whose local components transform by the rules
!
Ã
µ s ¶
j e
i
de
t ∂e
dt
∂x
d
t
x dt
xr ∂e
∂x
1i(j)(1)...
1p(m)(1)...
e
(1.2)
D1k(1)(u)... = D
...,
1r(1)(s)...
p
m
k
x
∂e
x dt dt ∂x
∂xu de
de
t ∂e
t
is called a d−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.
°
Jet geometrical extension of the KCC-invariants
9
Remark 1.2. The use of parentheses for certain indices of the local components
1i(j)(1)...
D1k(1)(l)...
of the distinguished tensor field D on the 1-jet space is motivated by the fact that
(j)
(1)
the pair of indices ” (1) ” or ” (l) ” behaves like a single index.
Example 1.3. The geometrical object
(i)
C = C(1)
∂
,
∂xi1
(i)
where C(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. Let h = (h11 (t)) be a Riemannian metric on the relativistic time axis
R. The geometrical object
(i)
Jh = J(1)1j
∂
⊗ dt ⊗ dxj ,
∂xi1
(i)
where J(1)1j = h11 δji is a d−tensor field on J 1 (R, M ), which is called the h-normalization
d−tensor field of 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 the temporal or
spatial semisprays, together with the jet nonlinear connections.
³
´
(j)
Definition 1.5. A set of local functions H = H(1)1 on J 1 (R, M ), which transform
by the rules
µ ¶2 k
dt ∂e
xk1
∂e
x
e (k) = 2H (j) dt
−
,
(1.3)
2H
(1)1
(1)1
j
∂x
de
t
de
t ∂t
is called a temporal semispray on J 1 (R, M ).
Example 1.6. Let us consider a Riemannian metric h = (h11 (t)) on the temporal
manifold R and let
h11 dh11
1
,
H11
=
2 dt
where h11 = 1/h11 , be its Christoffel symbol. Taking into account that we have the
transformation rule
e 2
e 1 = H 1 dt + dt d t ,
H
11
11
de
t dt de
t2
we deduce that the local components
(1.4)
˚(j) = − 1 H 1 xj
H
(1)1
2 11 1
³
´
˚(j) on J 1 (R, M ). This is called the canonical
˚= H
define a temporal semispray H
(1)1
temporal semispray associated to the temporal metric h(t).
10
Vladimir Balan and Mircea Neagu
³
´
(j)
Definition 1.7. A set of local functions G = G(1)1 , which transform by the rules
(1.5)
(k)
(j)
e
2G
(1)1 = 2G(1)1
µ
¶2 k
∂xm ∂e
xk1 j
dt
∂e
x
−
x
e ,
j
j
∂x
∂e
x ∂xm 1
de
t
is called a spatial semispray on J 1 (R, M ).
Example 1.8. Let ϕ = (ϕij (x)) be a Riemannian metric on the spatial manifold M
and let us consider
¶
µ
ϕim ∂ϕjm
∂ϕkm
∂ϕjk
i
γjk
=
+
−
2
∂xk
∂xj
∂xm
its Christoffel symbols. Taking into account that we have the transformation rules
(1.6)
p
i
γ
eqr
= γjk
∂e
xp ∂ 2 xl
∂e
xp ∂xj ∂xk
+
,
i
q
r
∂x ∂e
x ∂e
x
∂xl ∂e
xq ∂e
xr
we deduce that the local components
˚(j) = 1 γ j xk xl
G
(1)1
2 kl 1 1
³
´
˚(j) on J 1 (R, M ). This is called the canonical
˚= G
define a spatial semispray G
(1)1
spatial semispray associated to the spatial metric ϕ(x).
³
´
(j)
(j)
Definition 1.9. A set of local functions Γ = M(1)1 , N(1)i on J 1 (R, M ), which
transform by the rules
(1.7)
and
(1.8)
f(k) = M (j)
M
(1)1
(1)1
µ
¶2 k
xk1
∂e
x
dt
dt ∂e
−
∂xj
de
t
de
t ∂t
i
xk
xk1
∂xm ∂e
e (k) = N (j) dt ∂x ∂e
−
,
N
(1)l
(1)i e ∂e
l
j
l
∂e
x ∂xm
dt x ∂x
is called a nonlinear connection on the 1-jet space J 1 (R, M ).
Example 1.10. Let us consider that (R, h11 (t)) and (M, ϕij (x)) are Riemannian
1
i
manifolds having the Christoffel symbols H11
(t) and γjk
(x). Then, using the transformation rules (1.1), (1.4) and (1.6), we deduce that the set of local functions
³
´
˚ (j) , N
˚(j) ,
˚
Γ= M
(1)1
where
1 j
˚ (j) = −H11
M
x1
(1)1
(1)i
˚(j) = γ j xm
and N
im 1 ,
(1)i
represents a nonlinear connection on the 1-jet space J 1 (R, M ). This jet nonlinear
connection is called the canonical nonlinear connection attached to the pair of Riemannian metrics (h(t), ϕ(x)).
11
Jet geometrical extension of the KCC-invariants
In the sequel, let us study the geometrical relations between temporal or spatial
semisprays and nonlinear connections on the 1-jet space J 1 (R, M ). In this direction,
using the local transformation laws (1.3), (1.7) and (1.1), respectively the transformation laws (1.5), (1.8) and (1.1), by direct local computation, we find the following
geometrical results:
(j)
Theorem 1.11. a) The temporal semisprays H = (H(1)1 ) and the sets of temporal
(j)
components of nonlinear connections Γtemporal = (M(1)1 ) are in one-to-one correspondence on the 1-jet space J 1 (R, M ), via:
(j)
(j)
(j)
M(1)1 = 2H(1)1 ,
H(1)1 =
1 (j)
M .
2 (1)1
(j)
b) The spatial semisprays G = (G(1)1 ) and the sets of spatial components of
(j)
nonlinear connections Γspatial = (N(1)k ) are connected on the 1-jet space J 1 (R, M ),
via the relations:
(j)
∂G(1)1
1 (j)
(j)
(j)
,
G(1)1 = N(1)m xm
N(1)k =
1 .
2
∂xk1
2
Jet geometrical KCC-theory
In this Section we generalize on the 1-jet space J 1 (R, M ) the basics of the KCCtheory ([1], [4], [7], [14]). In this respect, let us consider on J 1 (R, M ) a second-order
system of differential equations of local form
(2.1)
d2 xi
(i)
+ F(1)1 (t, xk , xk1 ) = 0,
dt2
i = 1, n,
(i)
where xk1 = dxk /dt and the local components F(1)1 (t, xk , xk1 ) transform under a change
of coordinates (1.1) by the rules
(2.2)
(r)
(j)
Fe(1)1 = F(1)1
µ
¶2 r
xr1 j
dt ∂e
∂xm ∂e
xr1
∂e
x
dt
−
−
x
e .
∂xj
∂e
xj ∂xm 1
de
t
de
t ∂t
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
(i)
1 i
− H11
x1 + 2G(1)1 (t, xk , xk1 ) = 0, i = 1, n,
dt2
where
1 1 i
1 (i)
(i)
x1
G(1)1 = F(1)1 + H11
2
2
are the components of a spatial semispray on J 1 (R, M ). Moreover, the coefficients
(i)
(i)
of the spatial semispray G(1)1 produce the spatial components N(1)j of a nonlinear
12
Vladimir Balan and Mircea Neagu
connection Γ on the 1-jet space J 1 (R, M ), by putting
(i)
(i)
N(1)j
=
∂G(1)1
∂xj1
(i)
1 ∂F(1)1
1 1 i
=
+ H11
δj .
j
2 ∂x1
2
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
(i)
(1.1), we define the h−KCC-covariant derivative of a d−tensor of kind T(1) (t, xk , xk1 )
on the 1-jet space J 1 (R, M ) via
h
(i)
DT(1)
dt
(i)
=
=
dT(1)
dt
(i)
dT(1)
dt
(i)
(r)
(i)
1
+ N(1)r T(1) − H11
T(1) =
(i)
+
1 ∂F(1)1 (r) 1 1 (i)
T − H11 T(1) ,
2 ∂xr1 (1)
2
where the Einstein summation convention is used throughout.
h
Remark 2.2. The h−KCC-covariant derivative components
a change of coordinates (1.1) as a d−tensor of type
(i)
DT(1)
(i)
T(1)1 .
dt
transform under
In such a geometrical context, if we use the notation xi1 = dxi /dt, then the system
(2.1) can be rewritten in the following distinguished tensorial form:
h
Dxi1
dt
(i)
(i)
=
1 i
−F(1)1 (t, xk , xk1 ) + N(1)r xr1 − H11
x1 =
=
−F(1)1 +
(i)
(i)
1 ∂F(1)1 r 1 1 i
x − H x ,
2 ∂xr1 1 2 11 1
Definition 2.3. The distinguished tensor
(i)
h(i)
ε(1)1
(i)
= −F(1)1 +
1 ∂F(1)1 r 1 1 i
x − H x
2 ∂xr1 1 2 11 1
is called the first h−KCC-invariant on the 1-jet space J 1 (R, M ) of the SODEs (2.1),
which is interpreted as an external force [1], [7].
Example 2.4. It can be easily seen that for the particular first order jet rheonomic
dynamical system
(i)
(i)
∂X(1)
∂X(1) m
dxi
d2 xi
(i)
= X(1) (t, xk ) ⇒
=
+
x ,
2
dt
dt
∂t
∂xm 1
(2.3)
(i)
where X(1) (t, x) is a given d−tensor on J 1 (R, M ), the first h−KCC-invariant has the
form
(i)
(i)
∂X(1)
1 ∂X(1) r 1 1 i
h(i)
+
x − H x .
ε(1)1 =
∂t
2 ∂xr 1 2 11 1
13
Jet geometrical extension of the KCC-invariants
In the sequel, let us vary the trajectories xi (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
¯
∂xi ¯¯
i
,
ξ (t) =
∂s ¯
s=0
we get the variational equations
(i)
(i)
∂F(1)1 j ∂F(1)1 dξ r
d2 ξ i
+
ξ +
= 0.
2
dt
∂xj
∂xr1 dt
(2.4)
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
space J 1 (R, M ) via
h
(i)
dξ i
dξ i
1 ∂F(1)1 m 1 1 i
Dξ i
(i)
=
+ N(1)m ξ m =
+
ξ + H11 ξ .
dt
dt
dt
2 ∂xm
2
1
h
Dξ i
Remark 2.5. The h−KCC-covariant derivative components
transform under a
dt
(i)
change of coordinates (1.1) as a d−tensor T(1) .
In this geometrical context, the variational equations (2.4) can be rewritten in the
following distinguished tensorial form:
h
h
D Dξ i h i
= P m11 ξ m ,
dt dt
where
h
P ij11
(i)
= −
∂F(1)1
∂xj
(i)
+
(i)
(i)
(i)
2
2
2
1 ∂ F(1)1
1 ∂ F(1)1 r 1 ∂ F(1)1 (r)
+
+
x −
F
+
2 ∂t∂xj1
2 ∂xr ∂xj1 1 2 ∂xr1 ∂xj1 (1)1
(r)
1
1 dH11
1
1 ∂F(1)1 ∂F(1)1
+
δi − H 1 H 1 δi .
r
j
4 ∂x1 ∂x1
2 dt j 4 11 11 j
h
Definition 2.6. The d−tensor P ij11 is called the second h−KCC-invariant on the
1-jet space J 1 (R, M ) of the system (2.1), or the jet h−deviation curvature d−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)
dxi
dxj dxk
d2 xi
1
i
(t)
(x)
− H11
+ γjk
= 0,
2
dt
dt
dt dt
1
i
where H11
(t) and γjk
(x) are the Christoffel symbols of the Riemannian metrics h11 (t)
and ϕij (x), then the second h−KCC-invariant has the form
h
i
P ij11 = −Rpqj
xp1 xq1 ,
14
Vladimir Balan and Mircea Neagu
where
i
i
∂γpj
∂γpq
r i
r i
−
+ γpq
γrj − γpj
γrq
∂xj
∂xq
are the components of the curvature of the spatial Riemannian metric ϕij (x). Consequently, the variational equations (2.4) become the following jet Jacobi field equations:
i
Rpqj
=
h
h
D Dξ i
i
+ Rpqm
xp1 xq1 ξ m = 0,
dt dt
where
h
Dξ i
dξ i
i
=
+ γjm
xj1 ξ m .
dt
dt
Example 2.8. For the particular first order jet rheonomic dynamical system (2.3)
the jet h−deviation curvature d−tensor is given by
h
P ij11
(i)
(i)
(i)
(r)
2
2
1
1 ∂ X(1)
1 ∂ X(1) r 1 ∂X(1) ∂X(1)
1 dH11
1 1 1 i
=
+
x
+
+
δji − H11
H11 δj .
1
j
j
r
r
j
2 ∂t∂x
2 ∂x ∂x
4 ∂x
∂x
2 dt
4
Definition 2.9. The distinguished tensors
h
h
Rijk1
and
h
h
∂ P ik11
1 ∂ P ij11
−
=
,
k
3
∂x1
∂xj1
h
B ijkm
∂ Rijk1
=
∂xm
1
(i)
i1
Djkm
=
∂ 3 F(1)1
∂xj1 ∂xk1 ∂xm
1
are called the third, fourth and fifth h−KCC-invariant on the 1-jet vector bundle
J 1 (R, M ) of the system (2.1).
Remark 2.10. Taking into account the transformation rules (2.2) of the components
(i)
i1
F(1)1 , we immediately deduce that the components Djkm
behave like a d−tensor.
Example 2.11. For the first order jet rheonomic dynamical system (2.3) the third,
fourth and fifth h−KCC-invariants are zero.
Theorem 2.12 (of characterization of the jet h−KCC-invariants). All the five
h−KCC-invariants of the system (2.1) cancel on J 1 (R, M ) if and only if there exists
a flat symmetric linear connection Γijk (x) on M such that
(i)
1
F(1)1 = Γipq (x)xp1 xq1 − H11
(t)xi1 .
(2.5)
Proof. ”⇐” By a direct calculation, we obtain
h(i)
ε(1)1
= 0,
h
i1
P ij11 = −Ripqj xp1 xq1 = 0 and Djkl
= 0,
15
Jet geometrical extension of the KCC-invariants
where Ripqj = 0 are the components of the curvature of the flat symmetric linear
connection Γijk (x) on M.
”⇒” By integration, the relation
(i)
i1
Djkl
=
∂ 3 F(1)1
∂xj1 ∂xk1 ∂xl1
=0
subsequently leads to
(i)
(i)
∂ 2 F(1)1
∂xj1 ∂xk1
∂F(1)1
(i)
= 2Γijp xp1 + U(1)j (t, x) ⇒
=
2Γijk (t, x)
⇒
F(1)1 = Γipq xp1 xq1 + U(1)p xp1 + V(1)1 (t, x),
⇒
∂xj1
(i)
(i)
(i)
where the local functions Γijk (t, x) are symmetrical in the indices j and k.
h(i)
(i)
(i)
1 i
The equality ε(1)1 = 0 on J 1 (R, M ) leads us to V(1)1 = 0 and to U(1)j = −H11
δj .
Consequently, we have
(i)
∂F(1)1
∂xj1
1 i
δj
= 2Γijp xp1 − H11
and
(i)
1 i
x1 .
F(1)1 = Γipq xp1 xq1 − H11
h
The condition P ij11 = 0 on J 1 (R, M ) implies the equalities Γijk = Γijk (x) and
i
Rpqj + Riqpj = 0, where
Ripqj =
∂Γipj
∂Γipq
−
+ Γrpq Γirj − Γrpj Γirq .
∂xj
∂xq
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 Γijk (x) behave like a symmetric linear
connection on M. Consequently, Ripqj represent the curvature of this symmetric linear
connection.
h
On the other hand, the equality Rijk1 = 0 leads us to Riqjk = 0, which infers that
the symmetric linear connection Γijk (x) on M is flat.
¤
Acknowledgements. The present research was supported by the Romanian
Academy Grant 4/2009.
References
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Vladimir Balan and Mircea Neagu
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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
Website: http://www.2collab.com/user:mirceaneagu
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 space J 1 (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; jet h−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 = (J 1 (R, M ), π1 , R × M )
is a vector bundle over the product manifold R × M , having the fibre of type Rn ,
where n is the dimension of the spatial manifold 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 space J 1 (R, M ) by (t, xi , xi1 ); these transform by the rules [13]
e
t=e
t(t)
i
x
e =x
ei (xj )
(1.1)
∂e
xi dt j
x
· x1 .
ei1 =
∂xj de
t
In the geometrical study of the 1-jet bundle, a central role is played by the distinguished tensors (d−tensors).
³
´
1i(j)(1)...
Definition 1.1. A geometrical object D = D1k(1)(l)... on the 1-jet vector bundle,
whose local components transform by the rules
!
Ã
µ s ¶
j e
i
de
t ∂e
dt
∂x
d
t
x dt
xr ∂e
∂x
1i(j)(1)...
1p(m)(1)...
e
(1.2)
D1k(1)(u)... = D
...,
1r(1)(s)...
p
m
k
x
∂e
x dt dt ∂x
∂xu de
de
t ∂e
t
is called a d−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.
°
Jet geometrical extension of the KCC-invariants
9
Remark 1.2. The use of parentheses for certain indices of the local components
1i(j)(1)...
D1k(1)(l)...
of the distinguished tensor field D on the 1-jet space is motivated by the fact that
(j)
(1)
the pair of indices ” (1) ” or ” (l) ” behaves like a single index.
Example 1.3. The geometrical object
(i)
C = C(1)
∂
,
∂xi1
(i)
where C(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. Let h = (h11 (t)) be a Riemannian metric on the relativistic time axis
R. The geometrical object
(i)
Jh = J(1)1j
∂
⊗ dt ⊗ dxj ,
∂xi1
(i)
where J(1)1j = h11 δji is a d−tensor field on J 1 (R, M ), which is called the h-normalization
d−tensor field of 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 the temporal or
spatial semisprays, together with the jet nonlinear connections.
³
´
(j)
Definition 1.5. A set of local functions H = H(1)1 on J 1 (R, M ), which transform
by the rules
µ ¶2 k
dt ∂e
xk1
∂e
x
e (k) = 2H (j) dt
−
,
(1.3)
2H
(1)1
(1)1
j
∂x
de
t
de
t ∂t
is called a temporal semispray on J 1 (R, M ).
Example 1.6. Let us consider a Riemannian metric h = (h11 (t)) on the temporal
manifold R and let
h11 dh11
1
,
H11
=
2 dt
where h11 = 1/h11 , be its Christoffel symbol. Taking into account that we have the
transformation rule
e 2
e 1 = H 1 dt + dt d t ,
H
11
11
de
t dt de
t2
we deduce that the local components
(1.4)
˚(j) = − 1 H 1 xj
H
(1)1
2 11 1
³
´
˚(j) on J 1 (R, M ). This is called the canonical
˚= H
define a temporal semispray H
(1)1
temporal semispray associated to the temporal metric h(t).
10
Vladimir Balan and Mircea Neagu
³
´
(j)
Definition 1.7. A set of local functions G = G(1)1 , which transform by the rules
(1.5)
(k)
(j)
e
2G
(1)1 = 2G(1)1
µ
¶2 k
∂xm ∂e
xk1 j
dt
∂e
x
−
x
e ,
j
j
∂x
∂e
x ∂xm 1
de
t
is called a spatial semispray on J 1 (R, M ).
Example 1.8. Let ϕ = (ϕij (x)) be a Riemannian metric on the spatial manifold M
and let us consider
¶
µ
ϕim ∂ϕjm
∂ϕkm
∂ϕjk
i
γjk
=
+
−
2
∂xk
∂xj
∂xm
its Christoffel symbols. Taking into account that we have the transformation rules
(1.6)
p
i
γ
eqr
= γjk
∂e
xp ∂ 2 xl
∂e
xp ∂xj ∂xk
+
,
i
q
r
∂x ∂e
x ∂e
x
∂xl ∂e
xq ∂e
xr
we deduce that the local components
˚(j) = 1 γ j xk xl
G
(1)1
2 kl 1 1
³
´
˚(j) on J 1 (R, M ). This is called the canonical
˚= G
define a spatial semispray G
(1)1
spatial semispray associated to the spatial metric ϕ(x).
³
´
(j)
(j)
Definition 1.9. A set of local functions Γ = M(1)1 , N(1)i on J 1 (R, M ), which
transform by the rules
(1.7)
and
(1.8)
f(k) = M (j)
M
(1)1
(1)1
µ
¶2 k
xk1
∂e
x
dt
dt ∂e
−
∂xj
de
t
de
t ∂t
i
xk
xk1
∂xm ∂e
e (k) = N (j) dt ∂x ∂e
−
,
N
(1)l
(1)i e ∂e
l
j
l
∂e
x ∂xm
dt x ∂x
is called a nonlinear connection on the 1-jet space J 1 (R, M ).
Example 1.10. Let us consider that (R, h11 (t)) and (M, ϕij (x)) are Riemannian
1
i
manifolds having the Christoffel symbols H11
(t) and γjk
(x). Then, using the transformation rules (1.1), (1.4) and (1.6), we deduce that the set of local functions
³
´
˚ (j) , N
˚(j) ,
˚
Γ= M
(1)1
where
1 j
˚ (j) = −H11
M
x1
(1)1
(1)i
˚(j) = γ j xm
and N
im 1 ,
(1)i
represents a nonlinear connection on the 1-jet space J 1 (R, M ). This jet nonlinear
connection is called the canonical nonlinear connection attached to the pair of Riemannian metrics (h(t), ϕ(x)).
11
Jet geometrical extension of the KCC-invariants
In the sequel, let us study the geometrical relations between temporal or spatial
semisprays and nonlinear connections on the 1-jet space J 1 (R, M ). In this direction,
using the local transformation laws (1.3), (1.7) and (1.1), respectively the transformation laws (1.5), (1.8) and (1.1), by direct local computation, we find the following
geometrical results:
(j)
Theorem 1.11. a) The temporal semisprays H = (H(1)1 ) and the sets of temporal
(j)
components of nonlinear connections Γtemporal = (M(1)1 ) are in one-to-one correspondence on the 1-jet space J 1 (R, M ), via:
(j)
(j)
(j)
M(1)1 = 2H(1)1 ,
H(1)1 =
1 (j)
M .
2 (1)1
(j)
b) The spatial semisprays G = (G(1)1 ) and the sets of spatial components of
(j)
nonlinear connections Γspatial = (N(1)k ) are connected on the 1-jet space J 1 (R, M ),
via the relations:
(j)
∂G(1)1
1 (j)
(j)
(j)
,
G(1)1 = N(1)m xm
N(1)k =
1 .
2
∂xk1
2
Jet geometrical KCC-theory
In this Section we generalize on the 1-jet space J 1 (R, M ) the basics of the KCCtheory ([1], [4], [7], [14]). In this respect, let us consider on J 1 (R, M ) a second-order
system of differential equations of local form
(2.1)
d2 xi
(i)
+ F(1)1 (t, xk , xk1 ) = 0,
dt2
i = 1, n,
(i)
where xk1 = dxk /dt and the local components F(1)1 (t, xk , xk1 ) transform under a change
of coordinates (1.1) by the rules
(2.2)
(r)
(j)
Fe(1)1 = F(1)1
µ
¶2 r
xr1 j
dt ∂e
∂xm ∂e
xr1
∂e
x
dt
−
−
x
e .
∂xj
∂e
xj ∂xm 1
de
t
de
t ∂t
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
(i)
1 i
− H11
x1 + 2G(1)1 (t, xk , xk1 ) = 0, i = 1, n,
dt2
where
1 1 i
1 (i)
(i)
x1
G(1)1 = F(1)1 + H11
2
2
are the components of a spatial semispray on J 1 (R, M ). Moreover, the coefficients
(i)
(i)
of the spatial semispray G(1)1 produce the spatial components N(1)j of a nonlinear
12
Vladimir Balan and Mircea Neagu
connection Γ on the 1-jet space J 1 (R, M ), by putting
(i)
(i)
N(1)j
=
∂G(1)1
∂xj1
(i)
1 ∂F(1)1
1 1 i
=
+ H11
δj .
j
2 ∂x1
2
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
(i)
(1.1), we define the h−KCC-covariant derivative of a d−tensor of kind T(1) (t, xk , xk1 )
on the 1-jet space J 1 (R, M ) via
h
(i)
DT(1)
dt
(i)
=
=
dT(1)
dt
(i)
dT(1)
dt
(i)
(r)
(i)
1
+ N(1)r T(1) − H11
T(1) =
(i)
+
1 ∂F(1)1 (r) 1 1 (i)
T − H11 T(1) ,
2 ∂xr1 (1)
2
where the Einstein summation convention is used throughout.
h
Remark 2.2. The h−KCC-covariant derivative components
a change of coordinates (1.1) as a d−tensor of type
(i)
DT(1)
(i)
T(1)1 .
dt
transform under
In such a geometrical context, if we use the notation xi1 = dxi /dt, then the system
(2.1) can be rewritten in the following distinguished tensorial form:
h
Dxi1
dt
(i)
(i)
=
1 i
−F(1)1 (t, xk , xk1 ) + N(1)r xr1 − H11
x1 =
=
−F(1)1 +
(i)
(i)
1 ∂F(1)1 r 1 1 i
x − H x ,
2 ∂xr1 1 2 11 1
Definition 2.3. The distinguished tensor
(i)
h(i)
ε(1)1
(i)
= −F(1)1 +
1 ∂F(1)1 r 1 1 i
x − H x
2 ∂xr1 1 2 11 1
is called the first h−KCC-invariant on the 1-jet space J 1 (R, M ) of the SODEs (2.1),
which is interpreted as an external force [1], [7].
Example 2.4. It can be easily seen that for the particular first order jet rheonomic
dynamical system
(i)
(i)
∂X(1)
∂X(1) m
dxi
d2 xi
(i)
= X(1) (t, xk ) ⇒
=
+
x ,
2
dt
dt
∂t
∂xm 1
(2.3)
(i)
where X(1) (t, x) is a given d−tensor on J 1 (R, M ), the first h−KCC-invariant has the
form
(i)
(i)
∂X(1)
1 ∂X(1) r 1 1 i
h(i)
+
x − H x .
ε(1)1 =
∂t
2 ∂xr 1 2 11 1
13
Jet geometrical extension of the KCC-invariants
In the sequel, let us vary the trajectories xi (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
¯
∂xi ¯¯
i
,
ξ (t) =
∂s ¯
s=0
we get the variational equations
(i)
(i)
∂F(1)1 j ∂F(1)1 dξ r
d2 ξ i
+
ξ +
= 0.
2
dt
∂xj
∂xr1 dt
(2.4)
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
space J 1 (R, M ) via
h
(i)
dξ i
dξ i
1 ∂F(1)1 m 1 1 i
Dξ i
(i)
=
+ N(1)m ξ m =
+
ξ + H11 ξ .
dt
dt
dt
2 ∂xm
2
1
h
Dξ i
Remark 2.5. The h−KCC-covariant derivative components
transform under a
dt
(i)
change of coordinates (1.1) as a d−tensor T(1) .
In this geometrical context, the variational equations (2.4) can be rewritten in the
following distinguished tensorial form:
h
h
D Dξ i h i
= P m11 ξ m ,
dt dt
where
h
P ij11
(i)
= −
∂F(1)1
∂xj
(i)
+
(i)
(i)
(i)
2
2
2
1 ∂ F(1)1
1 ∂ F(1)1 r 1 ∂ F(1)1 (r)
+
+
x −
F
+
2 ∂t∂xj1
2 ∂xr ∂xj1 1 2 ∂xr1 ∂xj1 (1)1
(r)
1
1 dH11
1
1 ∂F(1)1 ∂F(1)1
+
δi − H 1 H 1 δi .
r
j
4 ∂x1 ∂x1
2 dt j 4 11 11 j
h
Definition 2.6. The d−tensor P ij11 is called the second h−KCC-invariant on the
1-jet space J 1 (R, M ) of the system (2.1), or the jet h−deviation curvature d−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)
dxi
dxj dxk
d2 xi
1
i
(t)
(x)
− H11
+ γjk
= 0,
2
dt
dt
dt dt
1
i
where H11
(t) and γjk
(x) are the Christoffel symbols of the Riemannian metrics h11 (t)
and ϕij (x), then the second h−KCC-invariant has the form
h
i
P ij11 = −Rpqj
xp1 xq1 ,
14
Vladimir Balan and Mircea Neagu
where
i
i
∂γpj
∂γpq
r i
r i
−
+ γpq
γrj − γpj
γrq
∂xj
∂xq
are the components of the curvature of the spatial Riemannian metric ϕij (x). Consequently, the variational equations (2.4) become the following jet Jacobi field equations:
i
Rpqj
=
h
h
D Dξ i
i
+ Rpqm
xp1 xq1 ξ m = 0,
dt dt
where
h
Dξ i
dξ i
i
=
+ γjm
xj1 ξ m .
dt
dt
Example 2.8. For the particular first order jet rheonomic dynamical system (2.3)
the jet h−deviation curvature d−tensor is given by
h
P ij11
(i)
(i)
(i)
(r)
2
2
1
1 ∂ X(1)
1 ∂ X(1) r 1 ∂X(1) ∂X(1)
1 dH11
1 1 1 i
=
+
x
+
+
δji − H11
H11 δj .
1
j
j
r
r
j
2 ∂t∂x
2 ∂x ∂x
4 ∂x
∂x
2 dt
4
Definition 2.9. The distinguished tensors
h
h
Rijk1
and
h
h
∂ P ik11
1 ∂ P ij11
−
=
,
k
3
∂x1
∂xj1
h
B ijkm
∂ Rijk1
=
∂xm
1
(i)
i1
Djkm
=
∂ 3 F(1)1
∂xj1 ∂xk1 ∂xm
1
are called the third, fourth and fifth h−KCC-invariant on the 1-jet vector bundle
J 1 (R, M ) of the system (2.1).
Remark 2.10. Taking into account the transformation rules (2.2) of the components
(i)
i1
F(1)1 , we immediately deduce that the components Djkm
behave like a d−tensor.
Example 2.11. For the first order jet rheonomic dynamical system (2.3) the third,
fourth and fifth h−KCC-invariants are zero.
Theorem 2.12 (of characterization of the jet h−KCC-invariants). All the five
h−KCC-invariants of the system (2.1) cancel on J 1 (R, M ) if and only if there exists
a flat symmetric linear connection Γijk (x) on M such that
(i)
1
F(1)1 = Γipq (x)xp1 xq1 − H11
(t)xi1 .
(2.5)
Proof. ”⇐” By a direct calculation, we obtain
h(i)
ε(1)1
= 0,
h
i1
P ij11 = −Ripqj xp1 xq1 = 0 and Djkl
= 0,
15
Jet geometrical extension of the KCC-invariants
where Ripqj = 0 are the components of the curvature of the flat symmetric linear
connection Γijk (x) on M.
”⇒” By integration, the relation
(i)
i1
Djkl
=
∂ 3 F(1)1
∂xj1 ∂xk1 ∂xl1
=0
subsequently leads to
(i)
(i)
∂ 2 F(1)1
∂xj1 ∂xk1
∂F(1)1
(i)
= 2Γijp xp1 + U(1)j (t, x) ⇒
=
2Γijk (t, x)
⇒
F(1)1 = Γipq xp1 xq1 + U(1)p xp1 + V(1)1 (t, x),
⇒
∂xj1
(i)
(i)
(i)
where the local functions Γijk (t, x) are symmetrical in the indices j and k.
h(i)
(i)
(i)
1 i
The equality ε(1)1 = 0 on J 1 (R, M ) leads us to V(1)1 = 0 and to U(1)j = −H11
δj .
Consequently, we have
(i)
∂F(1)1
∂xj1
1 i
δj
= 2Γijp xp1 − H11
and
(i)
1 i
x1 .
F(1)1 = Γipq xp1 xq1 − H11
h
The condition P ij11 = 0 on J 1 (R, M ) implies the equalities Γijk = Γijk (x) and
i
Rpqj + Riqpj = 0, where
Ripqj =
∂Γipj
∂Γipq
−
+ Γrpq Γirj − Γrpj Γirq .
∂xj
∂xq
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 Γijk (x) behave like a symmetric linear
connection on M. Consequently, Ripqj represent the curvature of this symmetric linear
connection.
h
On the other hand, the equality Rijk1 = 0 leads us to Riqjk = 0, which infers that
the symmetric linear connection Γijk (x) on M is flat.
¤
Acknowledgements. The present research was supported by the Romanian
Academy Grant 4/2009.
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Vladimir Balan and Mircea Neagu
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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
Website: http://www.2collab.com/user:mirceaneagu