C-CONFORMAL METRIC TRANSFORMATIONS ON FINSLERIAN HYPERSURFACE | Narasimhamurthy | Journal of the Indonesian Mathematical Society 2 399 1 PB

J. Indones. Math. Soc.
Vol. 17, No. 2 (2011), pp. 59–66.

C-CONFORMAL METRIC TRANSFORMATIONS ON
FINSLERIAN HYPERSURFACE
S.K. Narasimhamurthy1 , Pradeep Kumar2 and C.S. Bagewadi3
1
Department of Mathematics, Kuvempu University,
Shankaraghatta-577451, Shimoga, Karnataka, India, nmurthysk@gmail.com
2
Department of Mathematics, Kuvempu University,
Shankaraghatta-577451, Shimoga, Karnataka, India
3
Department of Mathematics, Kuvempu University,

Shankaraghatta-577451, Shimoga, Karnataka, India

Abstract. The purpose of the paper is to give some relation between the original
Finslerian hypersurface and other C-conformal Finslerian hypersufaces. In this paper we define three types of hypersufaces, which were called a hyperplane of the 1st
kind, hyperplane of the 2nd kind and hyperplane of the 3rd kind under consideration
of C-conformal metric transformation.

Key words: Finsler spaces, Finsler hypersurface, Conformal, C-conformal, Hyperplane of 1st kind, 2nd kind and 3rd kind.

Abstrak. Tujuan dari paper ini adalah untuk memberikan beberapa kaitan antara hypersurface Finsler asal dengan hypersufaces C-konformal Finsler yang lain.
Dalam tulisan ini kami mendefinisikan tiga jenis hypersufaces, yang disebut hyperplane jenis pertama, hyperplane jenis kedua dan hyperplane jenis ketiga berdasarkan
transformasi metrik C-konformal.
Kata kunci: Ruang Finsler, hypersurface Finsler, konformal, C-konformal, hyperplane jenis pertama, jenis kedua dan jenis ketiga.

1. Introduction
The conformal theory and its related concepts of Finsler spaces was initiated
by Knebelman in 1929. M. Hashiguchi [1] introduced a special change called Cconformal change which satisfies C-condition. The theory of Special Finsler spaces
and their properties were studied by M. Matsumoto [8], C. Shibata [13] et al and authors like H. Izumi [2], S. Kikuchi [4] et al have given the condition for Finsler space
to be conformally flat. C. Shibata and H. Azuma [13] have studied C-conformal
2000 Mathematics Subject Classification: 53C60.
Received: 05-08-2011, revised: 08-08-2011, accepted: 08-08-2011.
59

S.K. Narasimhamurthy et al.

60

invariant tensor of Finsler metric. The author M. Kitayama ([5], [6], [7]) have
studied Finsler spaces admitting a parallel vector field and also studied Finslerian
hypersurface and metric transformations. The authors H.G. Nagaraja, C.S. Bagewadi and H. Izumi [9] have published a paper on infinitesimal h-conformal motions
of Finsler metric.
The authors S.K. Narasimhamurthy and C.S. Bagewadi ([10], [11]) have published a paper on C-conformal Special Finsler spaces admitting a parallel vector
field and the same authors have also studied on Infinitesimal C-conformal motions
of special Finsler spaces.
Throughout the paper, terminology and notations are referred to [1], [8] and
[12].

2. Preliminaries
A Finsler space, we mean a triple F n = (M, D, L), where M denotes ndimensional differentiable manifold, D is an open subset of a tangent vector bundle
T M endowed with the differentiable structure induced by the differentiable manifold T M and L : D → R is a differentiable mapping having the properties
i)

L(x, y) > 0, f or(x, y) ∈ D,

ii)

L(x, λy) = |λ|L(x, y), f or any (x, y) ∈ D and λ ∈ R, such that (x, λy) ∈ D,

∂
1
.
gij (x, y) = ∂˙i ∂˙j L2 , (x, y) ∈ D, is positive def inite, where ∂˙i =
2
∂y i

iii)

The metric tensor gij (x, y) and Cartan’s C-tensor Cijk are given by [12]:
1 ˙ ˙ 2
∂i ∂j L , g ij = (gij )−1 ,
2
1
1
i
= ∂˙k gij , Cjk
= g im (∂˙k gmj ),
2
2

gij (x, y) =
Cijk
where ∂˙j =

∂
∂y i

a)
b)
c)
d)
e)

and ∂˙i =

∂
∂xi .

We use the following [12]:

li = ∂˙i L,
li = y i /L,
hij = gij − li lj ,
1
i
γjk
= g ir (∂j grk + ∂k grj + ∂r gjk ),
2
1 i j k
i
G = γjk y y , Gij = ∂˙j Gi , Gijk = ∂˙k Gij , Gijkl = ∂˙l Gijk ,
2
1
i
Fjk = g ir (δj grk + δk grj − δr gjk ),
2
Nji = Nji − yj σ i + σ0 δji + σj y i ,

(1)

where δj = ∂j − Grj ∂r .
The Berwald connection and the Cartan connection of F n are given by BΓ =
i
i
) respectively.
, Nji , Cjk
(Gijk , Nji , 0) and CΓ = (Fjk

C-Conformal Metric Transformations

61

A hypersurface M n−1 of the underlying smooth manifold M n may be parametrically represented by the equation
xi = xi (uα ),
where uα are Gaussian coordinates on M n−1 and Greek indices take values 1 to
n-1. Here we shall assume that the matrix consisting of the projection factors
Bαi = ∂xi /∂uα is of rank (n-1). The following notations are also employed [6]:
i
Bαβ

= ∂xi /∂uα ∂uβ ,

i
i
B0β
= v α Bαβ
,

ij....
Bαβ....
= Bαi Bβj .....

If the supporting element y i at a point (uα ) of M n−1 is assumed to be tangential
to M n−1 , we may then write
y i = Bαi (u)v α ,
i.e., v α is thought of as the supporting element of M n−1 at a point (uα ). Since the
function L(u, v) = L(x(u), y(u, v)) gives rise to a Finsler matrix of M n−1 , we get a
(n-1)-dimensional Finsler space F n−1 = (M n−1 , L(u, v)).
At each point (uα ) of F n−1 , the unit normal vector N i (u, v) is defined by
gij Bαi N j = 0,

If

(Bαi , Ni )

is the inverse matrix of

Bαi Biβ

= δαβ ,

gij N i N j = 1.
(Biα , N i ),

Bαi Ni = 0,

(2)

we have

N i Biα = 0,

N i Ni = 1,

and further
Bαi Bjα + N i Nj = δji .
Making use of the inverse (g αβ ) of (gαβ ), we get
Biα = g αβ gij Bβj ,

Ni = gij N j .

α
α
For the induced Cartan connections ICΓ = (Fβγ
, Nβα , Cβγ
) on F n−1 , the
second fundamental h-tensor Hαβ and the normal curvature tensor Hα are given
by

i)
ii)

jk
i
i
Bαβ
) + Mα H β ,
Hαβ = Ni (Bαβ
+ Fjk

Hα =

i
Ni (B0α

+

(3)

Nji Bαj ),

i
i
respectively, where Mα = Cijk Bαi N j N k and B0α
= Bβα
v β . Transvecting Hαβ by
β
β
v , we get H0α = Hβα v = Hα .
Further more we have to put
ij
Mαβ = Cijk Bαβ
N k.

(4)

S.K. Narasimhamurthy et al.

62

3. C-Conformal Finsler Space
We shall consider conformal change of a Finsler metric formed by L → L =
eσ(x) L, where σ is conformal factor depends on the point x only and under this
n
change we have another Finsler space F = (M n , L) on the same underlying mann
ifold M .
M. Hashiguchi [1] introduced the special change named C-conformal change
which is by definition, a non-homothetic conformal change satisfying
i
Cjk
σ j = 0,

(5)

i
where Cjk
= g im (∂˙j gkm )/2, σ i = g im σm ,
σm = ∂σ/∂xm ,
(1) and by symmetry of lower indices of Cijk , we have

σ j = g ij σj . From

Cijk σ i = Cjik σ i = Cjki σ i = 0,
also we have
k i
k j
i
Cij
σ = Cij
σ = Cjk
σ k = 0.

In the following the quantity with bar will be defined in C-conformal Finsler
n
space F , and the quantity without bar will be defined in Finsler space F n .
Under the C-conformal change, we have the following [2], [13]:
a)

g ij = (L/L)2 gij ,

b)

yi = (L/L)2 yi ,

c)

C ijk = Cijk ,

d)

f)

i
γ ijk = γjk
+ (σj δki + σk δji − gjk σ i ),
1
i
G = Gi − L 2 σ i + σ0 y i ,
2
i
Gjk = Gijk − gjk σ i + σk δji + σj δki ,

g)

N j = Nji − yj σ i + σ0 δji + σj y i ,

h)

i
i
F jk = Fjk
− gjk σ i + σk δji + σj δki + σ0 Cjk
.

e)

g ij = (L/L)2 g ij ,
i

i
C jk = e2σ Cjk
,

C i = e−2σ Ci ,

(6)

i

i

4. Hypersurface Given by a C-Conformal Change
We now consider a Finsler hypersurface F n−1 = (M n−1 , L(u, v)) of the
n−1
= (M n−1 , L(u, v)) of
Finsler space F n and another Finsler hypersurface F
the Finsler space F n given by the C-conformal change.
Let N i (u, v) be a unit normal vector at each point of the F n−1 , and as component of n-1 linearly independent tangent vectors of F n−1 and they are invariant
under the C-conformal change. Thus we shall show that a unit normal vector
i
n−1
N (u, v) of F
is uniquely determined by
j

g ij Bαi N = 0,

i

j

g ij N N = 1.

(7)

C-Conformal Metric Transformations

63

By means of (2) and (6), we get
g ij (±e−σ N i )(±e−σ N j ) = 1.
Therefore we can put
i

N = e−σ N i ,
where we have chosen the sign ’+’ in order to fix an orientation. It is obvious that
N i (u, v) satisfies (2), hence we obtain:
i
Lemma 4.1. For a field of linear frame (B1i , B2i , ....Bn−1
, N i ) of F n , there exists
i

i
a field of linear frame (B1i , B2i , .....Bn−1
, N = e−σ N i ) of the F

conformal change such that (7) satisfied along F

n−1

given by the C-

.

α

The quantities B i are uniquely defined along F
α
Bi

n

n−1

by

= g αβ g ij Bβj ,
α

i

where (g αβ ) is the inverse metric of (g αβ ). If (B i , N ) is the inverse vector of
i

(B α , N i ), then we have
β

Bαi B i = δαβ ,

i

Bαi N i = 0,

and also

α

α

N B i = 0,

i

N N i = 1,

i

Bαi B j + N N j = δji .
j

Also we get N i = g ij N , that is
N i = e σ Ni .

(8)

We have from (6(e)),
L2 i
σ , where σ0 = σr y r .
2
Differentiating (9) by y j and from (6(f)), we obtain
i

D i = G − Gi = σ 0 y i −

Dji

=

i
D(j)
,

=

Gj − Gij ,

=

N j − Nji ,

=

−yj σ i + σ0 δji + σj y i ,

(9)

i

i

i
where D(j)
= ∂˙j Di . From (9), we have

L2
Ni σ i .
2
We assume that Ni σ i = 0. i.e., σ i (x) is tangential to F n−1 and using the condition
Ni y i = 0, then we have
Ni Di = 0.
(10)
Ni D i = σ0 Ni y i −

S.K. Narasimhamurthy et al.

64

Differentiating (10) by y j , we have
i
Ni D(j)
+ Di (Ni )(j) = 0,

Ni Dji + Di (∂˙j Ni ) = 0.
Transvecting above equation by Bαj , we get
Ni Dji Bαj + Di (∂˙j Ni )Bαj = 0,
Ni Dji Bαj = 0,
where we used

(11)

Bαj (∂˙j Ni ) = Mα Ni = Cijk Bαj N i N k Ni = 0.

Definition 4.1. If each path of the hypersurface F n−1 with respect to the induced
connection is also a path of the ambient space F n , then F n−1 is called a ‘hyperplane
of the 1st kind’.
A hyperplane of the 1st kind is characterized by Hα = 0.
From (3(ii)) and using (8), we have
i

i
H α = N i (B0α
+ N j Bαj ).

Thus
H α − e σ Hα

i

=

i
i
N i (B0α
+ N j Bαj ) − eσ Ni (B0α
+ Nji Bαj ),

=

i
i
eσ (Ni B0α
+ Ni N j Bαj ) − eσ (Ni B0α
+ Ni Nji Bαj ),

=

eσ Ni (N j − Nji )Bαj ,

=

eσ Ni Dji Bαj .

i

i

Thus we have
H α = eσ (Hα + Ni Dji Bαj ).
Thus from (11), we obtained
H α = e σ Hα .
Hence we state the following:
Theorem 4.1. A Finsler hypersurface F n−1 is a hyperplane of 1st kind if and
n−1
is a hyperplane of 1st kind, provided
only if C-conformal Finsler hypersurface F
i
i
n−1
Ni σ = 0, i.e., σ (x) is tangential to F
.
i
Now from (6(h)), the so called difference tensor Djk
has the following form
i
Djk

i

=

i
F jk − Fjk
,

=

i
−gij σ i + σk δji + σj δki + σ0 Cjk
.

Contracting above equation by Ni and Bαj , we get
i
Ni Djk
Bαj

=

i
−Ni gjk σ i Bαj + σk Ni δji Bαj + σj Ni δki Bαj + σ0 Cjk
Ni Bαi ,

=

0.

Where we use σ0 = σi y i and equation (5). Thus we state the following:

C-Conformal Metric Transformations

65

i
Lemma 4.2. Assuming that σi (x) is tangential to F n−1 , then the tensor Ni Djk
Bαj
is vanishes if and only if it satisfies (5).

Definition 4.2. If each h-path of a hypersurface F n−1 with respect to the induced
connection is also h-path of the ambient space F n , then F n−1 is called a ‘hyperplane
of the 2nd kind’.
A hyperplane of the 2nd kind is characterized by Hαβ = 0.
From (3(i)), we have
jk
i
i
Hαβ = Ni (Bαβ
+ Fjk
Bαβ
) + Mα H β .

(12)

Under the C-conformal change, (12) can be written as
i

jk
i
H αβ = N i (Bαβ
+ F jk Bαβ
) + M αH β .

(13)

Using equations (12) and (13), we get
H αβ − eσ Hαβ

=

i

jk
i
[N i (Bαβ
+ F jk Bαβ
) + M αH β ]

(14)

jk
i
i
−eσ Ni (Bαβ
+ Fjk
Bαβ
) − e σ Mα Hβ ,

using M α = Mα and H α = eσ Hα , we have
i

jk
i
H αβ − eσ Hαβ = eσ Ni (F jk − Fjk
)Bαβ
,

that implies
jk
i
H αβ − eσ Hαβ = eσ (Ni Djk
Bαβ
).

(15)

Thus by virtue of lemma (4.1), therefore we state the following:
Theorem 4.2. A Finsler hypersurface F n−1 is a hyperplane of the 2nd kind if and
n−1
is a hyperplane of the 2nd kind,
only if the C-conformal Finsler hypersurface F
n−1
provided σi (x) is tangential to F
.
Definition 4.3. If the unit normal vector of F n−1 is parallel along each curve of
F n−1 , then F n−1 is called a ‘hyperplane of the 3rd kind’.
A hyperplane of the 3rd kind is characterized by Hαβ = Mαβ = 0.
From (4), under C-conformal change the tensor Mαβ can be written as
M αβ

ij

k

=

C ijk B αβ N ,

=

ij
e−σ Cijk Bαβ
N k,
e−σ Mαβ .

=

(16)

By characterization of hyperplane of the 3rd kind and (15), we have H αβ = M αβ =
0.
Thus by virtue of lemma (4.1), we state the following:
Theorem 4.3. A Finsler hypersurface F n−1 is a hyperplane of the 3rd kind if
n−1
is a hyperplane of the 3rd kind,
and only if C-conformal Finsler hypersurface F
n−1
provided σi (x) is tangential to F
.

66

S.K. Narasimhamurthy et al.

Acknowledgement. The authors are thankful to the referees for their valuable
suggestions.

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