Acta Mathematica Academiae Paedagogicae Nyı́regyháziensis
32 (2016), 135–139
www.emis.de/journals
ISSN 1786-0091
ON A FINSLER SPACE WITH A SPECIAL METRICAL
CONNECTION
M. K. GUPTA AND ANIL K. GUPTA
Abstract. In this paper, we consider a Finsler space with a special metrical connection and find necessary and sufficient condition when the (v)hvi
torsion tensor ∗ Pjk
with respect to the special metrical connection coincides
i
with the (v)hv-torsion Pjk
with respect to general Finsler connection. The
relation in hv-curvature tensor, h-curvature tensor and v(h)-torsion tensor
with respect to these two connection are also obtained.
1. Introduction
Let F n = (M n , L) be an n-dimensional Finsler space equipped with the
fundamental function L(x, y). The metric tensor, the angular metric tensor
and Cartan tensor are defined by
1
gij = ∂˙i ∂˙j L2 ,
2
1
hij = L∂˙i ∂˙j L and Cijk = ∂˙k gij
2
respectively, where ∂˙i = ∂/∂y i .
i
A Finsler connection is a triad F Γ = (Γijk , Nji , Cjk
), where Γijk are connection coefficients of h-connection, Nji are connection coefficients of non-linear
i
connection and Cjk
are connection coefficients of v-connection. For a given
connection, the h- and v-covariant derivatives of any vector X i are given by
X|ki = δk X i + X r Γirk ,
and
i
X i |k = ∂˙k X i + X r Crk
,
where δk = ∂k − Nkr ∂˙r and ∂k =
∂
.
∂xk
2010 Mathematics Subject Classification. 53B40.
Key words and phrases. Finsler space, Finsler connection, metrical connection.
The first author is supported by UGC, Government of India.
135
136
M. K. GUPTA AND A. K. GUPTA
i
In 2006, H. S. Park et.al. [4] defined a new non-linear connection N j with
the help of given non-linear connection Nji for an (α, β)-metric as
yi
i
N j = Nji + ∇j L ,
L
where ‘∇j ’ denotes the covariant derivative with respect to the associated
Riemannian connection.
In 2008, H. G. Nagaraja [3] defined a new non-linear connection ∗ Nji with
the help of given non-linear connection for Randers space as
(1.1)
L|j y i
(1.2)
=
+
,
L
where ‘|j ’ denote the covariant derivative with respect to Finsler connection
i
F Γ, and find a new Finsler connection ∗ F Γ = (Γijk ,∗ Nji , Cjk
).
n
In this paper, we consider a Finsler space F admitting the Finsler connection ∗ F Γ and we find a relation between v(hv)-torsion tensors with respect to
these two Finsler connection connections F Γ and ∗ F Γ. We obtain necessary
and sufficient condition that two (v)hv-torsions coincides. We also find relation in hv-curvature tensor, h-curvature tensor and v(h) torsion tensor with
respect to these two Finsler connections.
The Terminology and notion are referred to [2, 5].
∗
Nji
Nji
2. A special metrical connection
i
Let F n = (M n , L) be an n-dimensional Finsler space and F Γ = (Γijk , Nji , Cjk
)
n
n
be a Finsler connection. Let the Finsler space F = (M , L) admits a new
i
Finsler connection ∗ F Γ = (∗ Γijk ,∗ Nji , Cjk
), which is h(h)-torsion free and non∗ i
linear coefficients Nj are given by (1.4). Then we have
L|k y r ˙
L|k y r ˙
∂r = δk −
∂r .
L
L
The h-covariant derivative of L with respect to ∗ F Γ is given by
(2.1)
∗
δk = ∂k −∗ Nkr ∂˙r = ∂k − Nkr ∂˙r −
L|k y r ˙
∂r L = δk L − L|k = 0.
L
Therefore the Finsler connection ∗ F Γ is h-metrical. Since ∗ F Γ is h-metrical
i
and h(h)- torsion ∗ Tjk
is zero, the linear connection coefficients ∗ Γijk of ∗ F Γ
are given in [1] by
(2.2)
L∗ |k =∗ δk L = δk L −
1
Γijk = g ir Γjrk = g ir [δj grk + δk grj − δr gjk ].
2
j
Using (2.1) and (∂˙j gik )y = 0 in (2.3), we have
(2.3)
(2.4)
∗ i
Γjk
= Γijk .
i
i
Thus, the new connection ∗ F Γ = (∗Γijk ,∗ Nji , Cjk
).
) reduces to ∗ F Γ = (Γijk ,∗ Nji , Cjk
ON A FINSLER SPACE WITH A SPECIAL METRICAL CONNECTION
137
i
The (v)hv-torsion tensor. The (v)hv-torsion tensor Pjk
of a Finsler space
with respect to connection F Γ is defined by
i
Pjk
= ∂˙k Nji − Γijk .
i
Therefore the (v)hv-torsion tensor ∗ Pjk
of a Finsler space with respect to the
∗
connection F Γ is given by
∗
i
Pjk
= ∂˙k∗Nji −∗ Γijk .
Using (1.2) and (2.4), we get
∗
(2.5)
i
i
Pjk
= Pjk
+ lk|j li + L|j
hik
.
L
hi
i
i
. Then equation (2.5) implies that lk|j li + L|j Lk = 0.
= Pjk
Let us suppose ∗ Pjk
Transvecting by yi and using hik yi = 0, we get Llk|j = 0, which gives lk|j = 0.
Conversely, let lk|j = 0. Also let us assume that deflection tensor Dji for the
Finsler connection F Γ is zero, i.e. Dji = 0. Then we get y|ji = 0. Again lk|j = 0
i
i
and y|ji = 0, implies that L|j = 0, and hence equation (2.5) yields ∗ Pjk
= Pjk
.
Thus, we have:
Theorem 2.1. Let the Finsler space F n admits a Finsler connection F Γ with
i
i
zero deflection tensor and a Finsler connection ∗ F Γ. Then ∗ Pjk
= Pjk
if and
only if lk|j = 0.
Rephrasing, the theorem concludes that (v)hv-torsion tensors with respect
to Finsler connections F Γ and ∗ F Γ are equal if and only if covariant differentiation of directional derivative of Fundamental metric function L vanishes.
i
The hv-curvature tensor. The hv-curvature tensor Phjk
of a Finsler space
with respect to connection F Γ is defined by
i
i
i
m
Phjk
= ∂˙k Γihj − Chk|j
+ Chm
Pjk
.
i
Therefore the hv-torsion tensor ∗ Pjk
of a Finsler space with respect to connec∗
tion F Γ is given by
∗
i
i
i ∗ m
Phjk
= ∂˙k∗ Γihj − Chk|j
+ Chm
P jk .
Using (2.4) and (2.5) we get
(2.6)
∗
i
i
i
Phjk
= Phjk
+ Chm
hm
k
L|j
.
L
Thus, we have:
Theorem 2.2. Let the Finsler space F n admits the Finsler connections F Γ
i
and ∗ F Γ. Then expression for ∗ Phjk
is given by (2.6).
138
M. K. GUPTA AND A. K. GUPTA
i
i
Let us suppose that ∗ Phjk
= Phjk
. Then equation (2.6) implies that
L|j
= 0,
L
= 0, i.e. the space is Riemannian. Thus, we have:
i
Chm
hm
k
i
which gives Chm
Theorem 2.3. If hv-curvature tensor with respect to Finsler connections F Γ
and ∗ F Γ coincides, then the space will be Riemannian.
i
of a Finsler space with
The v(h)-torsion tensor. The v(h)-torsion tensor Rjk
respect to connection F Γ is defined by
i
Rjk
= δk Nji − δj Nki .
i
Therefore the v(h)-torsion tensor ∗ Rjk
of a Finsler space with respect to con∗
nection F Γ is given by
∗
i
Rjk
= ∗ δ k ∗ N ij − ∗ δ j ∗ N ik .
Using (1.2) and (2.1) and simplification gives us
∗
(2.7)
i
i
i
Rjk
= Rjk
+ [L|j|k li + L|j l|k
− L|k lr lr|j li − j/k],
where −j/k denotes interchange of j and k and subtract the terms within the
bracket.
Theorem 2.4. Let the Finsler space F n admits the Finsler connections F Γ
i
and ∗ F Γ. Then expression for ∗ Rjk
is given by (2.7).
Equation (2.7) can be re-written as
∗
=
i
Rjk
+ [(L|j|k − L|k|j )l + L|j
i
y|k
y|ji
y|jr
r
y|k
− L|k + L|k l l − L|j lr li ].
L
L
L
L
i
Let us assume that deflection tensor Dj for the Finsler connection F Γ is zero
i.e. Dji = 0. Then we get y|ji = 0 and then equation (2.8) becomes
(2.8)
i
Rjk
i
∗
(2.9)
r i
i
i
Rjk
= Rjk
+ (L|j|k − L|k|j )li .
Thus, we have:
Corollary 1. Let the Finsler space F n admits the Finsler connection F Γ with
zero deflection tensor and a Finsler connection ∗ F Γ. Then v(h)-torsion tensor
∗ i
Rjk for the connection ∗ F Γ is given by (2.9).
i
The h-curvature tensor. The h-curvature tensor Rhjk
of a Finsler space
with respect to connection F Γ is defined by
m i
i
i
m
i
i
+ δk Γihj + Γm
Rjk
= Chm
Rhjk
hj Γmk − δj Γhk − Γhk Γmj .
i
Therefore the h-curvature tensor ∗ Rjk
of a Finsler space with respect to con∗
nection F Γ is given by
∗
m i
∗
i
i
i ∗ m
i
= Chm
Rjk + ∗ δ k Γihj + Γm
Rhjk
hj Γmk − δ j Γhk − Γhk Γmj .
ON A FINSLER SPACE WITH A SPECIAL METRICAL CONNECTION
139
Using (2.1) and (2.7) and simplifying, we obtain
∗ i
i
i
m
(2.10)
Rhjk = Rhjk
− (L|k lr ∂˙r Γihj − L|j lr ∂˙r Γihk ) + Chm
[L|j l|k
− L|k l|jm ].
Thus, we have:
Theorem 2.5. Let the Finsler space F n admits Finsler connections F Γ and
i
is given by (2.10).
F Γ. Then expression for ∗ Rhjk
∗
Equation (2.10) can be re-written as
(2.11)
∗
i
i
i
Rhjk
= Rhjk
− L|k lr ∂˙r Γihj + L|j lr ∂˙r Γihk + Chm
[L|j
m
y|k
− L|k
y|jm
].
L
L
Also let us assume that deflection tensor Dji for the Finsler connection F Γ is
zero, i.e. Dji = 0. Then we get y|ji = 0 and the equation (2.11) yields
(2.12)
∗
i
i
= Rhjk
− (L|k lr ∂˙r Γihj ) + (L|j lr ∂˙r Γihk ).
Rhjk
Thus, we have:
Corollary 2. Let the Finsler space F n admits the Finsler connections F Γ with
i
deflection zero and ∗ F Γ. Then expression for ∗ Rhjk
is given by (2.12).
References
[1] P. L. Antonelli, R. S. Ingarden, and M. Matsumoto. The theory of sprays and Finsler
spaces with applications in physics and biology, volume 58 of Fundamental Theories of
Physics. Kluwer Academic Publishers Group, Dordrecht, 1993.
[2] M. Matsumoto. Foundations of Finsler geometry and special Finsler spaces. Kaiseisha
Press, Shigaken, 1986.
[3] H. G. Nagaraja. Randers space with special nonlinear connection. Lobachevskii J. Math.,
29(1):27–31, 2008.
[4] H. S. Park, H. Y. Park, and B. D. Kim. On a Finsler space with (α, β)-metric and certain
metrical non-linear connection. Commun. Korean Math. Soc., 21(1):177–183, 2006.
[5] H. Rund. The differential geometry of Finsler spaces. Die Grundlehren der Mathematischen Wissenschaften, Bd. 101. Springer-Verlag, Berlin-Göttingen-Heidelberg, 1959.
Received September 9, 2015.
M. K. Gupta (corresponding author),
Department of Pure and Applied Mathematics,
Guru Ghasidas Vishwavidyalaya,
Bilaspur (C.G.), India
E-mail address: mkgiaps@gmail.com
Anil K. Gupta,
Department of Pure and Applied Mathematics,
Guru Ghasidas Vishwavidyalaya,
Bilaspur (C.G.), India
E-mail address: gupta.anil409@gmail.com