On a deformation of the Kropina metric
Marcel Roman
α2
we study its
β
deformation given by F (α, β) = K(α, β) + ϕ(x)β, where ϕ(x) depends
on the position only. From the variational problem we obtain the Lorentz
equations. Consequently, we determine the canonical nonlinear connection
N of the considered Finsler space F n = (M, F (x, y)).
Abstract. Considering the Kropina metric K(α, β) =
M.S.C. 2000: 53B40,53C60.
Key words: Finsler spaces, Kropina metric.
1
Preliminaries
Let M be a n-dimensional, real, differentiable manifold and π : T M −→ M be the
tangent bundle of M.
Definition 1.1 Let F : Tg
M = T M \ {0} −→ IR be a Finsler metric. The Finsler
space F n = (M, F (x, y)) is called with (α, β)-metric if the fundamental function F is
written in the form
F (x, y) = Fe(α(x, y), β(x, y)),
(1.1)
where
(1.2)
α(x, y) =
q
aij (x)y i y j with aij (x) a Riemannian metric on M
and
(1.3)
β(x, y) = bi (x)y i
with bi (x)dxi a 1 − form field on T M.
The theory of Finsler spaces with (α, β)-metric was studied by M. Matsumoto [1],
R. Miron [3], M. Roman [6], [7], [8], V.S. Sab˘au and H. Shimada [5], [9], and many
others mathematicians.
Since α(x, y) and β(x, y) are 1-homogeneous with respect to y i :
α(x, ty) = tα(x, y),
β(x, ty) = tβ(x, y),
Differential Geometry - Dynamical Systems, Vol.8, 2006, pp.
c Balkan Society of Geometers, Geometry Balkan Press 2006.
°
∀t ∈ IR+ ,
236-243.
On a deformation of the Kropina metric
237
it follows
Fe(tα(x, y), tβ(x, y)) = tFe (α(x, y), β(x, y)),
(1.4)
∀t ∈ IR+ ,
that is the function Fe (α, β) is positively 1-homogeneous in both of the arguments.
Taking into account the 2-homogeneity of the Lagrangian L(α, β) = Fe2 (α, β) we
obtain:
αLα + βLβ = 2L,
αLαα + βLαβ = Lα ,
(1.5)
αLαβ + βLββ = Lβ , α2 Lαα + 2αβLαβ + β 2 Lββ = 2L,
where
(1.6)
Lα =
∂L
∂L
∂2L
∂2L
∂2L
, Lβ =
, Lαα =
,
L
=
,
L
=
.
αβ
ββ
∂α
∂β
∂α2
∂α∂β
∂β 2
In order to determine the fundamental tensor gij (x, y) of a Finsler space with
(α, β)-metric, we introduce the following invariants, [1],
ρ=
(1.7)
ρ0 =
1
Lββ ,
2
1
Lα ,
2α
ρ−1 =
ρ1 =
1
Lαβ ,
2α
1
Lβ ,
2
ρ−2 =
1 ¡
1 ¢
Lαα − Lα ,
2
2α
α
where the subscripts 1, 0, -1, -2 are the homogeneity degree of these invariants.
As we know, denoting by
(1.8)
pi = α
∂α
= aij y j ,
∂y i
and using the invariants from 1.7, the fundamental tensor is obtained in the following
theorem, [1] :
Theorem 1.1 The fundamental tensor field gij of the Finsler space F n with (α, β)metric is represented by
(1.9)
2
gij = ρaij + ρ0 bi bj + ρ−1 (bi pj + bj pi ) + ρ−2 pi pj .
The deformation of the Kropina metric K(α, β)
Let us consider the Kropina metric K(α, β) =
α2
where α and β are given by 1.2
β
and 1.3.
In the following we will study the perturbation of this metric given by
(2.1)
F (α, β) = K(α, β) + ϕ(x)β,
∗
where ϕ : M −→ IR is a differentiable function.
We remark that the particular case ϕ(x) = ±1 was studied by R.Miron, H.Shimada
and V.S. Sab˘au in [5].
In order for F to be positive on T M \ {0} we obtain the following proposition:
238
Marcel Roman
Proposition 2.1 The positivity of the metric F (α, β) given in 2.1 holds if and only
if
(2.2)
1
||b|| < p
|ϕ|
and
β > 0.
Proof.
The positivity of F on T M \ {0} means that
K(α, β) + ϕ(x)β > 0,
∀y 6= 0.
It follows
(2.3)
α2 + ϕβ 2
> 0,
β
∀y =
6 0.
→ Suppose positivity holds. Substitute y i = −bi = −aij bj in the previous in1
equality we obtain the relation ||b|| < p
and as necessary condition β > 0.
|ϕ|
← Conversely, using the relations 2.2 and starting with the Cauchy-BuniakowskiSchwarz inequality:
p
√
|aij bi y j | ≤ apq bp bq ars y r y s
we obtain the positivity of F.
Regarding to the fundamental tensor of the metric F we state:
¤
Proposition 2.2 The following formula holds good:
(2.4)
where
det||gij || =
b2 An−1 (B − C)
det||aij ||,
αn−1 β 2(n+1)
A = 2α(α2 + ϕβ 2 ),
B = 3α4 + ϕ2 β 4 ,
C = −2α2 (α2 − ϕβ 2 ).
Resuming the previous propositions,we have the following theorem:
Theorem 2.1 The pair (M, F ) with the fundamental function F defined in 2.1 which
satisfies the conditions 2.2 is a Finsler space.
Proof. As we know, a Finsler metric F : T M −→ IR satisfies the following conditions:
1◦ . F is positive.
2◦ . F is positive 1-homogeneous with respect to y i .
1 ∂2F 2
is positively defined.
3◦ . the fundamental tensor gij =
2 ∂y i ∂y j
1◦ . The positivity of F is given in Proposition 2.1. Indeed, from the positive
1-homogeneity of α and β it follows 2◦ .
On a deformation of the Kropina metric
239
For 3◦ . we shall use the Proposition 2.2. We obtain
det||gij || =
Since
2n−1 b2 F n−1 α 4
α
[5( ) − 2ϕ( )2 + ϕ2 ]det||aij ||.
n−1
β
β
β
α
α
α
α
5( )4 − 2ϕ( )2 + ϕ2 = 4( )4 + [( )2 − ϕ]2 > 0,
β
β
β
β
we obtain 3◦ .
¤
1
Using the homogeneous frame {bi , li }, [6], with li := pi we express the fundaα
mental tensor gij .
Proposition 2.3 The fundamental tensor gij of the Finsler space (M, F ) is given by
gij =
(2.5)
2F
α
α
α
aij + 4( )2 li lj − 4( )3 (bi lj + bj li ) + [ϕ2 + 3( )4 ]bi bj ,
β
β
β
β
1
1
aij y j = pi .
α
α
The contravariant tensor g ij is expressed in the following form:
where li =
(2.6)
g ij =
β ij
a − Θ[β 2 F 2 bi bj − 2αβ 2 F (bi lj + bj li ) − 2α2 (b2 (α2 − ϕβ 2 ) − 2β 2 )li lj ]
2F
where b2 = aij bi bj and
Θ=
αβ 2
.
b2 (AB − C)
Proof. Taking into account the invariants:
α
α
4α2
4
ρ = 2( )2 + 2ϕ, ρ0 = 3( )4 + ϕ2 , ρ−1 = − 3 , ρ−2 = 2 ,
β
β
β
β
we obtain 2.5. Next we have gij g jk = δik and 2.6 holds true.
By straightforward calculation, we get:
¤
Proposition 2.4 If kij is the fundamental tensor of the Kropina space (M, K), then
the
fundamental
tensor
field
gij
of
the
deformation
space
(M, F ) is expressed by:
(2.7)
3
gij = kij + 2ϕaij + ϕ2 bi bj .
The variational problem
We shall study the variational problem of the space F n = (M, F (x, y)) started from
the integral of action for the Lagrangian L(α, β) = F 2 (x, y).
Let c : t ∈ [0, 1] 7−→ c(t) ∈ M be a smooth curve on M having the imagine
in a local chart domain of manifold M. The curve c can be express analiticaly by
xi = xi (t), t ∈ [0, 1].
240
Marcel Roman
The integral of action for the nondegenerate Lagrangian F 2 (x, y) = L(α(x, y), β(x, y))
is given by the functional
Z
(3.1)
I(c) =
1
F 2 (x,
0
dx
)dt.
dt
The variational problem for the functional I(c), (the points x0 = (xi (0)) and
x1 = (xi (1)) being fixed) follows to the Euler-Lagrange equations, [2]:
Ei (F 2 ) :=
(3.2)
∂F 2
d ∂F 2
dxi
i
.
−
=
0,
y
=
∂xi
dt ∂y i
dt
The curves c, solutions of the differential equations above-mentioned is calling
extremals.
Proposition 3.1 The Euler-Lagrange equations 2.2 are equivalent with the following
differential equations:
(3.3)
Ei (α2 ) + 2
ρ1
dα ∂α
1 dLα ∂α
dLβ ∂β
dxi
i
Ei (β) + 2
=
{
+
},
y
=
.
ρ
dt ∂y i
ρ dt ∂y i
dt ∂y i
dt
Proof. We remark the relation between Ei (α) and Ei (α2 ) :
2αEi (α) = Ei (α2 ) + 2
dα ∂α
.
dt ∂y i
Consequently, we have
Ei (L) =
1
1
dα ∂α
dLα ∂α
dLβ ∂β
Lα Ei (α2 ) + Lα
+ Lβ Ei (β) − {
+
}.
i
i
2α
α
dt ∂y
dt ∂y
dt ∂y i
and Ei (L) = 0 is exactly 3.3 .
¤
Let fix now a parameterization on curve c and let consider the arc length of curve
dx
dx
given by ds2 = α2 (x, )dt2 . In this case α2 (x, ) = 1. Then, s will be calling
dt
ds
canonical parameter. Along the curve c we have
dα
= 0.
ds
(3.4)
Proposition 3.2 In
curves c, we have
(3.5)
the
canonical
parameterization,
along
the
extremal
dL
dLα
dLβ
dβ
= 0,
= 0,
= 0,
= 0.
ds
ds
ds
ds
dL
= 0 follows from the previous proposition. Taking into account
ds
dα
dβ
dβ
dL
= Lα
+ Lβ
= Lβ
= 0.
that along the extremals c, (2.11) hold, it follows
ds
ds
ds
ds
dβ
Remarking that Lβ 6= 0, we obtain
= 0.
ds
Proof. Indeed,
On a deformation of the Kropina metric
241
dLα
dLβ
dα
dβ
= Lαα
+ Lαβ
= 0. Analogously for
= 0.
¤
ds
ds
ds
ds
Let Fij (x) be the electromagnetic tensorial field which correspond to the electromagnetic function β = bi (x)y i :
Finally,
(3.6)
Fij (x) =
∂bj
∂bi
−
.
i
∂x
∂xj
Taking into account the expression of Ei (β), we can write:
(3.7)
Ei (β) = Fij (x)
dxj
.
ds
Finally, we obtain the Lorentz equations of the space (M, F )
Theorem 3.1 The Lorentz equations of the Finsler space F n = (M, F ) are given in
the form:
dxj dxk
1
dxj
d2 xi
i
i
+
γ
(x)
=
[ϕβ
−
F
]F
(x)
jk
j
ds2
ds ds
2
ds
(3.8)
where
(3.9)
Fji (x) = ais (x)Fsj (x)
i
and γjk
are the Christoffel symbols of the Riemannian metric tensor aij (x).
4
The Lorentz nonlinear connection of the Finsler
space (M, F ).
The variational problem allows to introduce the Lorentz nonlinear connection, [4], [7],
having the coefficients:
(4.1)
1
N ij (x, y) = γ ijk (x)y k + [ F (x, y) − ϕ(x)β(x, y)]Fji (x).
2
The system of functions N ij from 4.1 determines a canonical nonlinear connection
N, which depends only on the fundamental function F (x, y) of the Finsler spaces F n .
The Lorentz equations 3.8 can be written in the form
(4.2)
d2 xi
dxj
+ N ij (x, y)
= 0,
2
ds
ds
yi =
dxi
.
ds
It follows that 4.2 gives us the autoparallel curves of the nonlinear connection N.
The canonical nonlinear connection N determines a differentiable distribution
which is supplementary to the vertical distribution V on the manifold T M :
(4.3)
Tu (T M ) = Nu ⊕ Vu , ∀u ∈ T M.
242
Let (
Marcel Roman
δ
∂
, i ) be the local basis adapted to N and V, and (dxi , δy i ) its dual basis:
i
δx ∂y
δ
∂
∂
=
− Nij (x, y) j ,
δxi
∂xi
∂y
(4.4)
δy i = dy i + Nji (x, y)dxj .
While the vertical distribution V is integrable, the horizontal distribution N has
not this property.
The tensor of integrability of N , [2] is:
i
=
Rjk
(4.5)
δNji
δNki
−
.
δxk
δxj
i
be the covariant derivative of Fji with respect to the Levi-Civita connecLet Fj|k
i
tion γ jk .
We have, [7]:
i
Proposition 4.1 The tensor of integrability Rjk
of the nonlinear connection N is
given by
(4.6)
where σ =
i
i
i
Rjk
= −ρj i km y m + σ(Fj|k
− Fk|j
) + (Fji
δσ
δσ
− Fki j )
δxk
δx
1
F − ϕβ and ρj i km is the curvature tensor of Levi-Civita connection.
2
Acknowledgement The author would like to thanks Academician Radu Miron
for many useful suggestions and comments.
References
[1] M. Matsumoto, Theory of Finsler spaces with (α, β)-metrics, Rep. of Math. Phys.
31(1991), 43-83.
[2] R. Miron and M. Anastasiei, The Geometry of Lagrange spaces: Theory and
Applications, Kluwer Acad. Publ. FTPH, no.59,(1994).
[3] R. Miron, General Randers spaces, Lagrange and Finsler geometry, vol.76, Applications to Physics and Biology, (P.L. Antonelli and R. Miron eds.), Kluwer
Acad. Publ. FTPH,(1996), 126-140.
On a deformation of the Kropina metric
243
[4] R.Miron and B.T. Hassan, Variational problems in Finsler spaces with (α, β)metric, Algebras, Groups and Geometry, Hadronic Press, (2002).
[5] R. Miron, H. Shimada and V.S.Sab˘au, Two new classes of Finsler spaces with
(α, β)-metric, (to appear).
[6] M. Roman, Special Higher Order Lagrange Spaces. Applications, (Ph.D. Thesis),
Geometry Balkan Press, (2002).
[7] M. Roman, The variational problem for Finsler spaces with (α, β) - metric,
Finsler and Lagrange Geometries, (edited by M. Anastasiei and P. L. Antonelli),
Kluwer Academic Publishers, (2003), 171-179.
[8] M. Roman, General Randers spaces and its homogeneous nonlinear connection,
Algebras Groups and Geometries, Hadronic Press, vol.20(3), (2003), 313-322.
[9] V.S. Sab˘au and H. Shimada, Classes of Finsler spaces with (α, β)-metrics, Reports on Mathematical Physics, vol.47(1), (2001).
Author’s address:
Roman Marcel
Technical University ”Gh. Asachi” of Iasi,
Faculty of Electronics and Telecommunications,
Department of Mathematics, Iasi, Roamnia
email: [email protected]
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