Research Article On a Hypersurface of a Finsler Space with Randers Change of Matsumoto Metric

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1 Geometry Volume 013, Article ID 84573, 6 pages Research Article On a Hypersurface of a Finsler Space with Randers Change of Matsumoto Metric M. K. Gupta, 1 Abhay Singh, 1 andp.n.pandey 1 Department of Pure & Applied Mathematics, Guru Ghasidas Vishwavidyalaya, Bilaspur, India Department of Mathematics, University of Allahabad, Allahabad, India Correspondence should be addressed to M. K. Gupta; mkgiaps@gmail.com Received 9 January 013; Accepted 11 June 013 Academic Editor: Anna Fino Copyright 013 M. K. Gupta et al. This is an open access article distributed under the Creative Commons Attribution License, which permits unrestricted use, distribution, and reproduction in any medium, provided the original work is properly cited. The present paper contains certain geometrical properties of a hypersurface of a Finsler space with Randers change of Matsumoto metric. 1. Introduction The concept of Finsler spaces with (α, β)-metric was introduced by Matsumoto [1]. It was later discussed by several authors such as Shibata and others [ 4]. It has plenty of applications in various fields such as physics, mechanics, seismology, biology, and ecology [5 8]. Matsumoto introduced a special type of (α, β)-metric of the form L=α /(α β), α = a ij (x)y i y j,andβ = b i (x)y i, which is slope-of-amountain metric and is known as Matsumoto metric [9]. This metric has enriched Finsler geometry and it has provided researchers an important tool to work with significantly in this field [7, 10]. A change of Finsler metric L(x, y) L(x, y) = L(x, y)+ b i (x)y i is called Randers change of metric. The notion of a Randers change was proposed by Matsumoto, named by Hashiguchi and Ichijyo [11] and studied in detail by Shibata [1]. A Randers change of Matsumoto metric is given by L(x, y) = α /(α β) + β. Recently, Nagaraja and Kumar [13] studied the properties of a Finsler space with the Randers change of Matsumoto metric. Matsumoto [14] presented the theory of Finslerian hypersurface. The present authors (Gupta and Pandey [15, 16]) obtained certain geometrical properties of hypersurfaces of some special Finsler spaces. Singh and Kumari [17]discussed a hypersurface of a Finsler space with Matsumoto metric. In this paper, we consider an n-dimensional Finsler space F n =(M n,l)with the Randers change of Matsumoto metric L=α /(α β) + β and find certain geometrical properties of a hypersurface of the Finsler space with above metric. The paper is organized as follows. Section consists of Preliminaries relevant to the subsequent sections. The induced Cartan connection for hypersurface of a Finsler space is defined in Section 3. Necessary and sufficient conditions under which the hypersurface of the above Finsler space is a hyperplane of first, second and third kind are obtained in Section 4.. Preliminaries Let M n be an n-dimensional smooth manifold and let F n = (M n,l) be an n-dimensional Finsler space equipped with Randers change of Matsumoto metric function L= α +β. (1) α β The derivative of above Randers change of Matsumoto metric with respect to α and β is given by L α = α αβ (α β), L β = α +β αβ (α β),

2 Geometry β L αα = (α β) 3, L α ββ = (α β) 3, L αβ = αβ (α β) 3, () b i =a ij b j, s 0 = {pp 0 +(p 0 p p 1 )α }, ζp L α = L α, L ββ = L β β, L β = L β, L αβ = L α β. L αα = L α α, The normalized element of support l i = i L is given by (3) l i =α 1 L α y i +L β b i, (4) y i =a ij y j.theangularmetrictensorh ij =L i j L is given by h ij =pa ij +q 0 b i b j +q 1 (b i y j +b j y i )+q y i y j, (5) s 1 = {pp 1 (p 0 p p 1 )β}, ζp s = {pp +(p 0 p p 1 ) }, b =a ζp ij b i b j, ζ=p(p+p 0 b +p 1 β) + (p 0 p p 1 )(α b β ). The Cartan tensor C ijk = (1/) k g ij is given by pc ijk =p 1 (h ij m k +h jk m i +h ki m j ) +γ 1 m i m j m k, (10) (11) p=ll α α 1 = α3 α β+β 3 3αβ (α β) 3, q 0 =LL ββ = α (α +αβ β ) q 1 =LL αβ α 1 = β (α +αβ β ) q =Lα (L αα L α α 1 )= (α +αβ β )(3β α) α(α β) 4. (6) γ 1 =p p 0 β 3p 1q 0, m i =b i α βy i. (1) Let { i jk } be the components of Christoffel symbols of the associated Riemannian space R n and let k be the covariant differentiation with respect to x k relative to this Christoffel symbols. We will use the following tensors: E ij =b ij +b ji, F ij =b ij b ji, (13) b ij = j b i. If we denote the Cartan connection in F n as CΓ = (F i jk,gi j,ci jk ), then the difference tensor Di jk =Fi jk {i jk } of the Finsler space F n is given by The fundamental metric tensor g ij = (1/) i j L is given by g ij =pa ij +p 0 b i b j +p 1 (b i y j +b j y i )+p y i y j, (7) D i jk =Bi E jk +F i k B j +F i j B k +B i j b 0k +B i k b 0j b 0m g im B jk C i jm Am k Ci km Am j +C jkma m s gis (14) p 0 =q 0 +L β = 6α4 6α 3 β+6α β 4αβ 3 +β 4 p 1 =q 1 +L 1 pl β = α(α +3β 8αβ) p =q +p L = β (8αβ α 3β ) α(α β) 4. Moreover, the reciprocal tensor g ij of g ij is given by (8) g ij =p 1 a ij s 0 b i b j s 1 (b i y j +b j y i ) s y i y j, (9) +λ s (C i jm Cm sk +Ci km Cm sj Cm jk Ci ms ), B k =p 0 b k +p 1 y k, B i =g ij B j, F k i =g kj F ji, B ij = {p 1 (a ij α y i y j )+( p 0 / β) m i m j }, A m k =Bm k E 00 +B m E k0 +B k F m 0 +B 0F m k, B k i =g kj B ji, λ m =B m E 00 +B 0 F m 0. (15) The suffix 0 denotes the transvection by the supporting element y i except for the quantities p 0, q 0,ands 0.

3 Geometry 3 3. Induced Cartan Connection A hypersurface M n 1 of the underlying manifold M n may be represented parametrically by x i =x i (u α ),u α are the Gaussian coordinates on M n 1 (Latin indices run from 1 to n, whilegreekindicestakevaluesfrom1ton 1). We assume that the matrix of projection factors B i α = xi / u α is of rank n 1. If the supporting element y i at a point u = (u α ) of M n 1 is assumed to be tangent to M n 1,wemaythenwrite y i =B i α (u)vα so that V =(V α ) is thought of as the supporting element of M n 1 at the point u α. Since the function L (u, V) = L(x(u), y(u, V)) gives rise to a Finsler metric on M n 1,weget an (n 1)-dimensional Finsler space F n 1 =(M n 1,L(u, V)). The metric tensor g αβ and the Cartan tensor C αβγ are given by g αβ =g ij B i α Bj β, C αβγ =C ijk B i α Bj β Bk γ. (16) At each point u α of F n 1,aunitnormalvectorN i (u, V) is defined by g ij B i α Nj =0, g ij N i N j =1. (17) For the angular metric tensor h ij,wehave h αβ =h ij B i α Bj β, h ijb i α Nj =0, h ij N i N j =1. (18) The inverse projection factors B α i (u, V) of Bi α are defined as B α i =g αβ g ij B j β, (19) g αβ is the inverse of the metric tensor g αβ of F n 1. From (17)and(19), it follows that B i α Bβ i =δ β α, Bi α N i =0, N i B α i =0, N i N i =1, (0) and further B i α Bα j +Ni N j =δ i j. (1) FortheinducedCartanconnectionICΓ = (F α βγ,gα β,cα βγ ) on F n 1, the second fundamental h-tensor H αβ and the normal curvature vector H α are given by H αβ =N i (B i αβ +Fi jk Bj α Bk β )+M αh β, () H α =N i (B i 0α +Gi j Bj α ), M α =C ijk B i α Nj N k, B i αβ = x i / u α u β,andb i 0α = B i βα Vβ.ItisclearthatH αβ is not symmetric and The second fundamental V-tensor M αβ is defined as: M αβ =C ijk B i α Bj β Nk. (5) The relative h- andv-covariant derivatives of B i α and Ni are given by B i α β =H αβn i, B i α β =M αβ N i, N i β = H αβb α j gij, N i β = M αβ B α j gij. (6) Let X i (x, y) be a vector field of F n.therelativeh- andvcovariant derivatives of X i are given by X i β =X i j B j β +X i j N j H β, X i β =X i j B j β. (7) Matsumoto [14] defined different kinds of hyperplanes and obtained their characteristic conditions, which are given in the following lemmas. Lemma 1. AhypersurfaceF n 1 is a hyperplane of the first kind ifandonlyifh α =0or equivalently H 0 =0. Lemma. AhypersurfaceF n 1 is a hyperplane of the second kind if and only if H αβ =0. Lemma 3. AhypersurfaceF n 1 is a hyperplane of the third kind if and only if H αβ =0=M αβ. 4. Hypersurface F n 1 (c) of the Finsler Space with Randers Change of Matsumoto Metric Let us consider the Randers change of Matsumoto metric L = α /(α β) + β with the gradient b i (x) = i b for a scalar function b(x) and a hypersurface F n 1 (c) given by the equation b(x) = c (constant). From parametric equation x i =x i (u α ) of F n 1 (c), weget α b(x(u)) = 0 = b i B i α,sothat b i (x) are regarded as covariant components of a normal vector field of F n 1 (c). Therefore, along F n 1 (c),wehave b i B i α =0, b iy i =0. (8) Therefore the induced metric L(u, V) of F n 1 (c) is given by L (u, V) = a αβ V α V β, a αβ =a ij B i α Bj β, (9) which is a Riemannian metric. At a point of F n 1 (c),from(6), (8), and (10), we have p=1, q 0 =, q 1 =0, q = 1 α, H αβ H βα =M α H β M β H α. (3) Equation ()yields H 0α =H βα V β =H α, (4) H α0 =H αβ V β =H α +M α H 0. p 0 =6, p 1 = α, p =0, ζ=1+b, s 0 = (1 + b ), s 1 = α(1+b ), s 4b = α (1 + b ). (30)

4 4 Geometry Therefore (9)gives g ij =a ij using (8)we get which gives (1 + b ) bi b j α(1+b ) (b i y j +b j y i )+ 4b α (1 + b ) yi y j, b (31) g ij b i b j = 1+b, (3) b i (x (u)) = 1+b N (33) i, b is the length of the vector b i.using(31) and(33) we get b i =a ij b j = b (1 + b )N i +b α 1 y i. (34) Theorem 4. Let F n be a Finsler space with Randers change of Matsumoto metric L=α /(α β) + β with a gradient b i (x) = i b(x) and let F n 1 (c) be a hypersurface of F n,whichisgiven by b(x) = c (constant). Then the induced metric on F n 1 (c) is Riemannian metric given by (9), and the scalar function b(x) is given by (33) and (34). Along F n 1 (c), the angular metric tensor and the metric tensor of F n are given by h ij =a ij +b i b j y iy j α, (35) g ij =a ij +6b i b j + α (b iy j +b j y i ). (36) If h (a) αβ denote the angular metric tensor of the Riemannian metric a ij (x),then,using(8), (35), and (18), we get From (8), we get h αβ =h (a) αβ. (37) p 0 β = 18α4 6α 3 β (α β) 5. (38) Thus, along F n 1 (c), p 0 / β = 18/α, and therefore (1) gives γ 1 =6/α, m i =b i. Then the Cartan tensor becomes C ijk = 1 α (h ijb k +h jk b i +h ki b j ) + 3 α b ib j b k, (39) and therefore, using (18), (5), and (8), we get M αβ = 1 α 1+b h αβ, M α =0, (40) and hence from (3) itfollowsthath αβ is symmetric. Thus we have the following. Theorem 5. The second fundamental v-tensor of Finsler hypersurface F n 1 (c) of Finsler space with Randers change of Matsumoto metric, is given by (40), andthesecondfundamental h-tensor is symmetric. Taking h-covariant derivative of (8) with respect to the induced connection, we get Applying (7)forthevectorb i,weget b i β B i α +b ib i α β =0. (41) b i β =b i j B j β +b i j N j H β. (4) Using this and B i α β =H αβn i,(41)becomes b i j B i α Bj β +b i j B i α Nj H β +b i H αβ N i =0. (43) Since b i j = b h C h ij,using(33)and(40), we get b i j B i α Nj = 1+b M α =0. (44) Thus (43)gives 1+b H αβ +b i j B i α Bj β =0. (45) Since H αβ is symmetric, it is clear that b i j is symmetric. Further contracting (45)withV β andthenwithv α,weget 1+b H α +b i j B i α yj =0, 1+b H 0 +b i j y i y j =0. (46) In view of Lemma 1,thehypersurfaceF n 1 (c) is a hyperplane ofthefirstkindifandonlyifb i j y i y j =0.Hereb i j being the covariant derivative with respect to the Cartan connection of F n may depend on y i. Since b i is a gradient vector, from (13), we have E ij = b ij, F ij =0.Thus(14)reducesto D i jk =Bi b jk +B i j b 0k +B i k b 0j b 0m g im B jk C i jm Am k Ci km Am j +C jkma m s gis +λ s (C i jm Cm sk +Ci km Cm sj Cm jk Ci ms ). In view of (30)and(31), the relations in (15)becometo B i =6b i + α y i, B i = 1+b bi + α(1+b ) yi, B ij = 1 α a ij 1 α 3 y iy j + 9 α b ib j, B i j =gki B kj, A m k =Bm k b 00 +B m b k0, λ m =B m b 00. (47) (48)

5 Geometry 5 By virtue of (8)wehaveB i0 =0, B i 0 =0 which leads Am 0 = B m b 00. Therefore we have D i j0 =Bi b j0 +B i j b 00 B m C i jm b 00, D i 00 =Bi b 00 =[ bi 1+b + y i α(1+b ) ]b 00. Using the relation (8), we get b i D i j0 = b 1+b b j α(1+b ) b 00b j 1+b bm b i C i jm b 00, (49) (50) b i D i 00 = b 1+b b 00. (51) b i j is the covariant derivative of b i with respect to x j relative to the Cartan connection of F n,andb ij = j b i is the covariant derivative of b i with respect to x j relative to the Riemannian connection: b i j b ij = ( j b i F r ij b r) ( j b i { r ij }b r) Using (51), we get = (F r ij {r ij }) b r = D r ij b r, that is, b i j =b ij D r ij b r. (5) b i j y i y j =b 00 D r 00 b 1 r = 1+b b 00. (53) Consequently, (46)maybewrittenas 1+b H b b 00 =0. (54) Thus the condition H 0 =0is equivalent to b 00 =0, b ij does not depend upon y i.sincey i is to satisfy (8), the condition is written as b ij y i y j =(b i y i )(c j y j ) for some c j (x), so that we have Using (8), it follows that b ij =b i c j +b j c i. (55) b 00 =0, b ij B i α Bj β =0, b ijb i α yj =0. (56) Again (48)and(55)gives b i0 b i = c 0b, λm =0, A i j Bj β =0, B ij B i α Bj β = 1 α h αβ. Using the (5), (31), (34), (40), and (47), we get b r D r ij Bi α Bj β = c 0 b (57) α(1 + b ) h αβ. (58) Therefore in view of (5), (45)reducesto 1+b H c αβ + 0 b α(1 + b ) h αβ =0. (59) Theorem 6. The necessary and sufficient condition for the hypersurface F n 1 (c) of Finsler space with Randers change of Matsumotometrictobehyperplaneofthefirstkindis(55) and in this case the second fundamental h-tensor of hypersurface F n 1 (c) is proportional to its angular metric tensor. In view of Lemma,thehypersurfaceF n 1 (c) is a hyperplane of the second kind if and only if H αβ =0.Thusfrom (59)wegetc 0 =c i (x)y i =0. Therefore there exists a function e(x) such that c i (x) = e(x)b i (x).thus(55)gives b ij =eb i b j. (60) Theorem 7. The necessary and sufficient condition for the hypersurface F n 1 (c) of Finsler space with Randers change of Matsumoto metric, to be hyperplane of the second kind is (60). In view of (40)andLemma3,wehavethefollowing. Theorem 8. The hypersurface F n 1 (c) of Finsler space with Randers change of Matsumoto metric is not a hyperplane of the third kind. Acknowledgment The authors are thankful to the referee for his/her valuable suggestions. References [1] M. Matsumoto, On C-reducible finsler spaces, Tensor N.S., vol. 4, pp. 9 37, 197. [] M. Hashiguchi, S.-i. Hōjō,andM.Matsumoto, OnLandsberg spaces of two dimensions with (α, β)-metric, the Korean Mathematical Society,vol.10,pp.17 6,1973. [3] M. Matsumoto, Projectively flat Finsler spaces with (α, β)- metric, Reports on Mathematical Physics,vol.30,no.1,pp.15 0, [4] C. Shibata, On Finsler spaces with an (α, β)-metric, Hokkaido University of Education. Section II A,vol.35,no.1,pp. 1 16, [5] P. L. Antonelli, A. Bóna, and M. A. Slawiński, Seismic rays as Finsler geodesics, Nonlinear Analysis RWA, vol. 4, no. 5, pp , 003. [6] P.L.Antonelli,R.S.Ingarden,andM.Matsumoto,The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology, Kluwer Academic, Dordrecht, The Netherlands, [7] M. Rafie-Rad and B. Rezaei, On Einstein Matsumoto metrics, Nonlinear Analysis RWA, vol. 13, no., pp , 01. [8] H. Shimada and S. V. Sabau, Introduction to Matsumoto metric, Nonlinear Analysis,vol.63,pp ,005. [9] M. Matsumoto, A slope of a mountain is a Finsler surface with respect to a time measure, Mathematics of Kyoto University,vol.9,no.1,pp.17 5,1989.

6 6 Geometry [10] A. Tayebi, E. Peyghan, and H. Sadeghi, On Matsumoto-type Finsler metrics, Nonlinear Analysis, vol. 13, no. 6, pp , 01. [11] M. Hashiguchi and Y. Ichijyo, Randers spaces with rectilinear geodesics, Reports of the Faculty of Science of Kagoshima University, no. 13, pp , [1] C. Shibata, On invariant tensors of β-changes of Finsler metrics, Mathematics of Kyoto University, vol.4,no.1, pp , [13] H. G. Nagaraja and P. Kumar, On Randers change of Matsumoto metric, Bulletin of Mathematical Analysis and Applications, vol. 4, no. 1, pp , 01. [14] M. Matsumoto, The induced and intrinsic Finsler connections of a hypersurface and Finslerien projective geometry, Journal of Mathematics of Kyoto University, vol.5,no.1,pp , [15] M. K. Gupta and P. N. Pandey, On hypersurface of a Finsler space with a special metric, Acta Mathematica Hungarica,vol. 10,no.1-,pp ,008. [16] M. K. Gupta and P. N. Pandey, Hypersurfaces of conformally and h-conformally related Finsler spaces, Acta Mathematica Hungarica,vol.13,no.3,pp.57 64,009. [17] U. P. Singh and B. Kumari, On a hypersurface of a Matsumoto space, Indian Pure and Applied Mathematics,vol.3, no. 4, pp , 001.

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