ON THE HYPERSURFACE OF A FINSLER SPACE

Journal of the Indian Math. Soc. Vol. 80, Nos. 3-4, (2013), 329-339.

ON THE HYPERSURFACE OF A FINSLER SPACE WITH THE SPECIAL (α, β)- METRIC α + β +

β N+1 αN

GAUREE SHANKER AND RAVINDRA Abstract. The purpose of the present paper is to investigate the various kinds of hypersurfaces of Finsler space with special (α, β) metric α+β+

β n+1 . αn

1. Introduction We consider an n-dimensional Finsler space F n = (M n , L), i.e., a pair consisting of an n-dimensional differential manifold M n equipped with a fundamental function L(x, y). The concept of the (α, β)-metric L(α, β) was introduced by M. Matsumoto [7] and has been studied by many authors [1], [2], [10]. The study of some well known (α, β)-metrics, the Randers metric α + β, the Kropina metric α2 /β and the generalized Kropina metric αm+1 /β m have greatly contributed to the growth of Finsler geometry and its applications to theory of relativity. Definition 1.1. A Finsler space F n of dimension n is a differentiable manifold R such that the length s of the curve xi (t) of F n is defined by S = L(x, y)dt. The fundamental function L(x, y) = L(x1 , ...xn , y 1 , ...y n ) is supposed to be differentiable for y 6= (0) and satisfies the following condition:(i) Positively homogeneous: L(x, py) = pL(x, y), p > 0. (ii) Positive: L(x, y) > 0, y 6= 0. 1 ∂ 2 L2 (x, y) (iii) gij = is positive definite. 2 ∂y i ∂y j Definition 1.2. A Finsler metric L(x, y) is called an (α, β)-metric L(α, β) if L is a positively homogeneous function of α and β of degree one, where α2 = aij (x)y i y j is a Riemannian metric and β = bi (x)y i is a 1-form on M n . 2010 Mathematics Subject Classification. 53B40, 53C60. Key words and phrases: Special Finsler hypersurface, (α, β)- metric, Hyperplane of 1st kind, Hyperplane of 2nd kind, Hyperplane of 3rd kind. ISSN 0019–5839

c Indian Mathematical Society, 2013 .

329

330

GAUREE SHANKER AND RAVINDRA

Definition 1.3. A Hypersurface is a generalization of the concept of hyperplane. Suppose an enveloping manifold M has n dimension, then any submanifold of M of n − 1 dimension is a hypersurface. The co-dimension of hypersurface is one. If m > n, then Vn is called a subspace of Vm . Vm is also enveloping space of Vn if m > n. In particular if m = n + 1, then Vn is called hypersurface of the enveloping space Vn+1 . In 1985, Matsumoto [5] studied the theory of Finslerian hypersurfaces and various types of Finslerian hypersurface called hyperplanes of the first, second and third kind. A hypersurface M n−1 of the M n may be represented parametrically by the equation xi = xi (uα ), α = 1, ..., n − 1, where uα are Gaussian coordinates on M n−1 . 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 , so that v α 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 metric of M n−1 , we get an (n − 1)-dimensional Finsler space F n−1 = (M n−1 , L(u, v)). In the present paper, we consider an n-dimensional Finsler space F n = n+1 n (M , L) with (α, β)- metric L(α, β) = α + β + βαn and the hypersurface of F n with bi (x) = ∂i b being the gradient of a scalar function b(x). We prove the conditions for this hypersurface to be a hyperplane of 1st kind, 2nd kind and we also prove that this hypersurface is not a hyperplane of 3rd kind. Throughout the present paper we use the terminology and notations of Matsumoto’s monograph [6]. 2. Preliminaries We are devoted to a special Finsler space F n = (M n , L) with the metric L(α, β) = α + β +

β n+1 . αn

(2.1)

The derivatives of the (2.1) with respect to α and β are given by αn + (n + 1)β n αn+1 − nβ n+1 , L = , β αn+1 αn n(n + 1)β n+1 n(n + 1)β n−1 = , L = , ββ αn+2 αn n −n(n + 1)β , = αn+1

Lα = Lαα Lαβ where Lα =

∂L ∂α , Lβ

=

∂L ∂β , Lαα

=

∂Lα ∂α , Lββ

=

∂Lβ ∂β

and Lαβ =

∂Lα ∂β .

ON THE HYPERSURFACE OF A FINSLER SPACE

331

In the special Finsler space F n = (M n , L) the normalized element of support li = ∂i L and the angular metric tensor hij are given by [9]: li = α−1 Lα Yi + Lβ bi ,

(2.2)

hij = paij + q0 bi bj + q1 (bi Yj + bj Yi ) + q2 Yi Yj ,

(2.3)

where Yi = aij y j , (αn+1 + αn β + β n+1 )(αn+1 − nβ n+1 ) , α2(n+1) n(n + 1)β n−1 (αn+1 + αn β + β n+1 ) = LLββ = , α2n −n(n + 1)β n (αn+1 + αn β + β n+1 ) , = LLαβ α−1 = α2(n+1) = Lα−2 (Lαα − Lα α−1 ), n+1 n(n + 2)β n+1 − αn+1 α + αn β + β n+1 . = αn+2 αn+2

p = LLα α−1 = q0 q1 q2 q2

(2.4)

The fundamental tensor gij = 12 ∂˙i ∂˙j L2 and its reciprocal tensor g ij is given by [9] gij = paij + p0 bi bj + p1 (bi Yj + bj Yi ) + p2 Yi Yj ,

(2.5)

where n(n + 1)β n−1 (αn+1 + αn β + β n+1 ) + (αn + (n + 1)β n )2 , α2n p1 = q1 + L−1 pLβ ,

p0 = q0 + L2β =

p1 =

−n(n + 1)β n (αn+1 + αn β + β n+1 ) + (αn+1 − nβ n+1 )(αn + (n + 1)β n ) , α2(n+1) (2.6)

p2 = q2 + p2 L−2 , p2 =

(αn+1 + αn β + β n+1 )(n(n + 2)β n+1 − αn+1 ) + (αn+1 − nβ n+1 )2 . α2(n+2) g ij = p−1 aij − S0 bi bj − S1 (bi y j + bj y i ) − S2 y i y j ,

(2.7)

where bi = aij bj ,

S0 = (pp0 + (p0 p2 − p21 )α2 )/ζp,

S1 = (pp1 + (p0 p2 − p21 )β)/ζp, S2 = (pp2 + (p0 p2 − 2

p21 )b2 )/ζp,

(2.8) 2

i j

b = aij b b ,

ζ = p(p + p0 b + p1 β) + (p0 p2 − p21 )(α2 b2 − β 2 ).

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The hv-torsion tensor Cijk = 12 ∂˙k gij is given by [10] 2pCijk = p1 (hij mk + hjk mi + hki mj ) + γ1 mi mj mk ,

(2.9)

where γ1 = p

∂p0 − 3p1 q0 , ∂β

mi = bi − α−2 βYi .

(2.10)

Here mi is a non-vanishing covariant vector orthogonal to the element of support ( yi. ) i Let be the components of Christoffel symbols of the associated jk Riemannian space Rn and ∇k be covariant differentiation with respect to xk relative to this Christoffel symbols. We put 2Eij = bij + bji ,

2Fij = bij − bji ,

(2.11)

where bij = ∇j bi . i Let CΓ = (Γ∗i Γ∗i ) be the Cartan connection of F n . The difference jk , ( ok , Γjk ) i i tensor Djk = Γ∗i of the special Finsler space F n is given by [6] jk − jk i Djk

= B i Ejk + Fki Bj + Fji Bk + Bji bok + Bki boj m is i m i −bom g im Bjk − Cjm Am k − Ckm Aj + Cjkm As g

(2.12)

m i m i m i Cms ), + Ckm Csj − Cjk Csk +λs (Cjm

where Bk = p0 bk + p1 Yk , B i = g ij Bj , Fik = g kj Fji ∂p0 −2 Bij = p1 (aij − α Yi Yj ) + mi mj /2 ∂β Bik = g kj Bji ,

Am k

=

Bkm E00

(2.13) m

+ B Ek0 +

λm = B m E00 + 2B0 F0m ,

Bk F0m

+

B0 Fkm ,

B0 = Bi y i .

where ′ 0′ denote contraction with y i except for the quantities p0 , q0 and S0 . 3. Finsler Hypersurface A hypersurface M n−1 of the underlying manifold M n may be represented parametrically by the equations xi = xi (uα ), where uα are the Gaussian coordinates on M n−1 . We assume that the matrix of projection factors Bαi = ∂xi /∂uα


333

is of rank n − 1. The element of support y i at a point u = uα of M n is to be taken tangential to M n−1 , that is y i = Bαi (u)v α .

(3.1)

so that v = v α is thought of as the supporting element of M n−1 at the point uα . The metric tensor gαβ and hv -torsion tensor Cαβγ of F n−1 are given by gαβ = gij Bαi Bβj ,

Cαβγ = Cijk Bαi Bβj Bγk .

(3.2)

At each point uα of F n−1 , a unit normal vector N i (u, v) is defined by gij (x(u, v), y(u, v))Bαi N j = 0,

gij (x(u, v), y(u, v))N i N j = 1.

(3.3)

As for the angular metric tensor hij , we have hαβ = hij Bαi Bβj ,

hij Bαi N j = 0,

hij N i N j = 1.

(3.4)

If (Biα , Ni ) denote the inverse of (Bαi , N i ), then we have Biα = g αβ gij Bβj , Bαi Biβ = δαβ , Biα N i = 0, Bαi Ni = 0, Ni = gij N j ,

(3.5)

Bik = g kj Bji ,

(3.6)

Bαi Bjα

i

+ N Nj =

δji .

n−1 α α induced from the The induced connection ICΓ = (Γ∗α βγ , Gβ , Cβγ ) of F ∗i ∗i i Cartan’s connection CΓ = (Γjk , Γ0k , Cjk ) is given in [5] α i ∗i j k α Γ∗α βγ = Bi (Bβγ + Γjk Bβ Bγ ) + Mβ Hγ , α i ∗i j Gα β = Bi (B0β + Γ0j Bβ ), α Cβγ =

i Biα Cjk Bβj Bγk ,

(3.7) (3.8) (3.9)

where i Mβγ = Ni Cjk Bβj Bγk , i Hβ = Ni (B0β +

Mβα = g αγ Mβγ ,

j Γ∗i 0j Bβ ),

(3.10) (3.11)

i i i and Bβγ = ∂Bβi /∂uγ , B0β = Bαβ v α . The quantities Mβγ and Hβ are called the second fundamental v-tensor and normal curvature vector respectively [5]. The second fundamental h-tensor Hβγ is defined in [5] j k i Hβγ = Ni (Bβγ + Γ∗i jk Bβ Bγ ) + Mβ Hγ ,

(3.12)

i Mβ = Ni Cjk Bβj N k .

(3.13)

where

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The relative h and v-covariant derivatives of projection factor Bαi with respect to ICΓ are given by i = Hαβ N i , Bα|β

Bαi |β = Mαβ N i .

(3.14)

The equation (3.12) shows that Hβγ is generally not symmetric and Hβγ − Hγβ = Mβ Hγ − Mγ Hβ .

(3.15)

The above equation yield H0γ = Hγ ,

Hγ0 = Hγ + Mγ H0 .

(3.16)

We shall use the following definitions and lemmas which are due to Matsumoto [5]: Definition 3.1. If each path of a hypersurface F n−1 with respect to the induced connection is also a path of the enveloping space F n , then F n−1 is called a hyperplane of the first kind. Definition 3.2. If each h- path of a hypersurface F n−1 with respect to the induced connection is also a h- path of the enveloping space F n , then F n−1 is called a hyperplane of the second kind. Definition 3.3. 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 third kind. Lemma 3.1. The normal curvature H0 = Hβ v β vanishes if and only if the normal curvature vector Hβ vanishes. Lemma 3.2. A hypersurface F n−1 is a hyperplane of the 1st kind if and only if Hα = 0. Lemma 3.3. A hypersurface F n−1 is a hyperplane of the 2nd kind with respect to the connection CΓ if and only if Hα = 0 and Hαβ = 0. Lemma 3.4. A hypersurface F n−1 is a hyperplane of the 3rd kind with respect to the connection CΓ if and only if Hα = 0 and Hαβ = Mαβ = 0. 4. Hypersurface F n−1 (c) of The Special Finsler Space n+1

Let us consider special Finsler metric L = α + β + βαn with a gradient bi (x) = ∂i b for a scalar function b(x) and a hypersurface F n−1 (c) given by the equation b(x) = c (constant) [11]. From parametric equation xi = xi (uα ) of F n−1 (c), we get


335

∂α b(x(u)) = 0 = bi Bαi , so that bi (x) are regarded as covariant components of a normal vector field of F n−1 (c). Therefore, along the F n−1 (c) we have bi Bαi = 0 and bi y i = 0.

(4.1)

The induced metric L(u, v) of F n−1 (c) is given by L(u, v) = aαβ v α v β ,

aαβ = aij Bαi Bβj

(4.2)

which is the Riemannian metric. At a point of F n−1 (c), from (2.4), (2.6) and (2.8), we have p = 1, q0 = 0, q1 = 0, q2 = −α−2 , p0 = 1, p1 = α−1 ,

(4.3)

p2 = 0, ζ = 1, S0 = 0, S1 = α−1 , S2 = −b2 /α2 . Therefore, from (2.7) we get g ij = aij −

b2 1 i j (b y + bj y i ) + 2 y i y j . α α

(4.4)

Thus along F n−1 (c), (4.4) and (4.3) lead to g ij bi bj = b2 . Therefore, we get bi (x(u)) =

√ b2 Ni , b2 = aij bi bj .

√ i.e.,bi (x(u)) = b2 Ni , where b is the length of the vector bi . Again from (4.4) and (4.5) we get √ bi = aij bj = b2 N i + b2 α−1 y i .

(4.5)

(4.6)

Thus we have Theorem 4.1. In the special Finsler hypersurface F n−1 (c), the induced Riemannian metric is given by (4.2) and the scalar function b(x) is given by (4.5) and (4.6). The angular metric tensor and metric tensor of F n are given by hij = aij −

Yi Yj , α2

(4.7)

1 (bi Yj + bj Yi ). (4.8) α From (4.1), (4.7) and (3.4) it follows that if haαβ denote the angular metric tensor of the Riemannian aij (x), then along F n−1 (c), hαβ = haαβ . From (2.6), we get gij = aij + bi bj +

n(n + 1) ∂p0 (n − 1)β n−2 (αn+1 + αn β + β n+1 ) + 3β n−1 (αn + (n + 1)β n ) . = 2n ∂β α

336


0 Thus along F n−1 (c), ∂p ∂β = 0 and therefore (2.10) gives γ1 = 0, mi = bi . Then the hv-torsion tensor becomes 1 Cijk = (hij bk + hjk bi + hki bj ) (4.9) 2α in a special Finsler hypersurface F n−1 (c).

Therefore, (3.4), (3.10), (3.13), (4.1) and (4.9) give 1 √ 2 ( b hαβ ) and Mα = 0. Mαβ = 2α From (3.15) it follows that Hαβ is symmetric. Thus we have

(4.10)

Theorem 4.2. The second fundamental v-tensor of special Finsler hypersurface F n−1 (c) is given by (4.10) and the second fundamental h-tensor Hαβ is symmetric. i Next from (4.1), we get bi|β Bαi + bi Bα|β = 0. Therefore, from (3.14) and

using bi|β = bi|j Bβj + bi |j N j Hβ , we get bi|j Bαi Bβj + bi |j Bαi N j Hβ + bi Hαβ N i = 0.

(4.11)

h Since bi |j = −bh Cij , we get

bi |j Bαi N j = 0. Thus (4.11) gives bHαβ + bi|j Bαi Bβj = 0.

(4.12)

It is noted that bi|j is symmetric. Furthermore, contracting (4.12) with v β and then with v α and using (3.1), (3.16) and (4.10), we get bHα + bi|j Bαi y j = 0,

(4.13)

bH0 + bi|j y i y j = 0.

(4.14)

In view of Lemmas (3.1) and (3.2), the hypersurface F n−1 (c) is hyperplane of the first kind if and only if H0 = 0. Thus from (4.14) it follows that F n−1 (c) is a hyperplane of the first kind if and only if bi|j y i y j = 0. Here bi|j being the covariant derivative with respect to CΓ of F n depends on y i . Since bi is a gradient vector, from (2.11) we have Eij = bij , Fij = 0 and Fji = 0. Thus (2.12) reduces to i Djk

=

B i bjk + Bji bok + Bki boj − bom g im Bjk

i i m m is −Cjm Am k − Ckm Aj + Cjkm As g s

+λ

i m (Cjm Csk

+

i m Ckm Csj

−

m i Cjk Cms ),

(4.15)


337

In view of (4.3) and (4.4), the relations in (2.13) become to Bi = bi + α−1 Yi , B i =

yi , α

(4.16)

1 (aij − α−2 Yi Yj ), 2α 1 1 1 δji − bk y i − 2 Yj y i , Bji = 2α α α

Bij =

m m m Am k = B bk0 , λ = B b00 .

m By virtue of (4.16) we have B0i = 0, Bi0 = 0 which leads Am 0 = B b00 . Therefore we have i i Dj0 = B i bj0 − B m Cjm b00 . i D00 = B i b00 =

(4.17)

i

y b00 . α

(4.18)

Thus from the relation (4.1), we get i bi Dj0 = 0,

(4.19)

i bi D00 = 0.

(4.20)

From (4.9) it follows that i bm bi Cjm Bαj = b2 Mα = 0. r Therefore, the relation bi|j = bij − br Dij and equations (4.19), (4.20) give r bi|j y i y j = b00 − br D00 = b00 .

Consequently, (4.13) and (4.14) may be written as bHα + bi0 Bαi = 0,

(4.21)

bH0 + b00 = 0.

(4.22)

Thus the condition H0 = 0 is equivalent to b00 = 0, where bij does not depend on y i . Since y i is to satisfy (4.1), the condition is written as bij y i y j = (bi y i )(cj y j ) for some cj (x), so that we have 2bij = bi cj + bj ci

(4.23)

From (4.1) and (4.23) it follows that b00 = 0, bij Bαi Bβj = 0, bij Bαi y j = 0. Hence 2

(4.21) gives Hα = 0. Again from (4.23) and (4.16) we get bi0 bi = c02b , λm = 1 hαβ. Thus (3.10), (4.4), (4.5), (4.6), (4.10) 0, Aij Bβj = 0 and Bij Bαi Bβj = 2α and (4.15) give r br Dij Bαi Bβj = −

C0 b2 hαβ . 4α

(4.24)

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Therefore, equation (4.12) reduces to bHαβ +

C0 b2 hαβ = 0. 4α

(4.25)

Hence the hypersurface F n−1 (c) is umbilic. Theorem 4.3. The necessary and sufficient condition for F n−1 (c) to be a hyperplane of 1s t kind is (4.23) and in this case the second fundamental tensor of F n−1 (c) is proportional to its angular metric tensor. In view of Lemma (3.3), F n−1 (c) is a hyperplane of second kind if and only if Hα = 0 and Hαβ = 0. Thus from (4.24), we get c0 = ci (x)y i = 0. Therefore, there exist a function e(x) such that ci (x) = e(x)bi (x). Thus (4.23) gives

bij = ebi bj .

(4.26)

Theorem 4.4. The necessary and sufficient condition for F n−1 (c) to be a hyperplane of 2nd kind is (4.25). Finally from (4.10) and Lemma (3.4) show that F n−1 (c) does not become a hyperplane of third kind. Theorem 4.5. The hypersurface F n−1 (c) is not a hyperplane of the 3rd kind. From the above obtained results we have the following:

5. Conclusion If F n = (M n , L be a special Finsler space with metric L(α, β) = α + β + β αn along with the gradient bi (x) = ∂i b for a scalar function b(x), then its hypersurface F n−1 (c) given by b(x) = c has the following properties: (i) It is a hyperplane of 1st kind iff there exist a function cj (x) such that n+1

2bij = bi cj + bj ci and the second fundamental tensor Hαβ of F n−1 (c) is proportional to its angular metric tensor hαβ . (ii) It is a hyperplane of 2nd kind iff there exist a function e(x) such that ci (x) = e(x)bi (x) and bij = ebi bj . (iii) It is not a hyperplane of third kind due to (4.10) and lemma (3.4).


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GAUREE SHANKER AND RAVINDRA, Department of Mathematics and Statistics, Banasthali University, Banasthali 304 022, INDIA. Email: [email protected], [email protected] Received: November 9, 2011. Accepted: June 20, 2012.