Some intrinsic properties of h-Randers conformal change
aa r X i v : . [ m a t h . G M ] J un Some intrinsic properties ofh-Randers conformal change H. S. Shukla, V. K. Chaubey and Arunima Mishra Department of Mathematics and Statistics, D. D. U. GorakhpurUniversity, Gorakhpur (U.P.)-273009, India,E-mail:- profhsshuklagkp@rediffmail.com Department of Applied Sciences, Buddha Institute of Technology,Sector-7, Gida, Gorakhpur,(U.P.) India’E-Mail: [email protected] Department of Mathematics, St. Joseph’s College for Women,Civil Lines, Gorakhpur, (U.P.), India,E-Mail: [email protected]
Abstract
In the present paper we have considered h-Randers conformal changeof a Finsler metric L , which is defined as L ( x, y ) → ¯ L ( x, y ) = e σ ( x ) L ( x, y ) + β ( x, y ) , where σ ( x ) is a function of x, β ( x, y ) = b i ( x, y ) y i is a 1- form on M n and b i satisfies the condition of being an h-vector. We have ob-tained the expressions for geodesic spray coefficients under this change.Further we have studied some special Finsler spaces namely quasi-C-reducible, C-reducible, S3-like and S4-like Finsler spaces arising fromthis metric. We have also obtained the condition under which thischange of metric leads a Berwald (or a Landsberg) space into a spaceof the same kind. Mathematics subject Classification:
Keywords: h-vector; special Finsler spaces; geodesic; conformal change. Introduction
Let M n be an n-dimensional differentiable manifold and F n be a Finslerspace equipped with a fundamental function L ( x, y ) , ( y i = ˙ x i ) of M n . Ifa differential 1-form β ( x, y ) = b i ( x ) y i is given on M n , M. Matsumoto [7]introduced another Finsler space whose fundamental function is given by¯ L ( x, y ) = L ( x, y ) + β ( x, y )This change of Finsler metric has been called β -change [10, 11].The conformal theory of Finsler spaces was initiated by M.S. Knebelman [6]in 1929 and has been investigated in detail by many authors [1, 2, 3, 4]. Theconformal change is defined as¯ L ( x, y ) → e σ ( x ) L ( x, y ),where σ ( x ) is a function of position only and known as conformal factor.In 1980, Izumi [3] introduced the h-vector b i which is v-covariantly constantwith respect to Cartan’s connection C Γ (i.e. b i | j = 0) and satisfies therelation LC hij b h = ρh ij , where C hij are components of (h)hv-torsion tensorand h ij are components of angular metric tensor. Thus the h-vector is notonly a function of coordinates x i , but it is also a function of directionalarguments satisfying L ˙ ∂ j b i = ρh ij .In the paper [14] S. H. Abed generalized the above two changes and haveintroduced another Finsler metric named as Conformal β - change and furtherGupta and Pandey [15] renamed it Randers conformal change and obtainedvarious important result in the filed of Finsler spaces. Recently we [13] havegeneralized the metric given by S. H. Abed with the help of h-vector andhave introduced another Finsler metric which is defined as¯ L ( x, y ) = e σ ( x ) L ( x, y ) + β ( x, y ) , (1.1)where σ ( x ) is a function of x and β ( x, y ) = b i ( x, y ) y i is a 1- form on M n and b i satisfies the condition of being an h-vector, We call the change L ( x, y ) → ¯ L ( x, y ) as h-Randers conformal change. This change generalizesvarious types of changes. When β = 0, it reduces to a conformal change.When σ = 0, it reduces to a h-Randers change [9]. When β = 0 and σ is anon-zero constant then it reduces to a homothetic change. When b i is func-tion of position only and σ = 0, it reduces to Randers change[12]. When b i and σ are functions of position only, it reduces to Randers conformal change214, 15].In the present paper we have obtained the expressions for geodesic spraycoefficients under this change. Further we have studied some special Finslerspaces namely quasi C-reducible, C-reducible, S3-like and S4-like Finslerspaces arising from this metric. We have also obtained the conditions underwhich this change of metric leads a Berwald (or a Landsberg) space into aspace of the same kind. Let the Cartan’s connection of Finsler space F n be denoted by C Γ = ( F ijk , G ij , C ijk ).Since b i ( x, y ) are components of h-vector, we have( a ) b i | j = ˙ ∂ j b i − b h C hij = 0 ( b ) LC hij b h = ρh ij (2.1)Hence we obtain ˙ ∂ j b i = L − ρh ij (2.2)Since h ij are components of an indicatory tensor i.e. h ij y j = 0, we have˙ ∂ i β = b i . Definition 2.1.
Let M n be an n-dimensional differentiable manifold and F n be a Finsler space equipped with a fundamental function L ( x, y ) , ( y i = ˙ x i ) of M n . A change in the fundamental function L by the equation (1.1) on thesame manifold M n is called h-Randers conformal change. A space equippedwith fundamental metric ¯ L is called h-Randers conformally changed Finslerspace ¯ F n . Differentiating equation (1.1) with respect to y i , the normalized support-ing element ¯ l i = ˙ ∂ i ¯ L is given by¯ l i = e σ l i + b i , (2.3)where l i = ˙ ∂ i L is the normalized supporting element l i of F n .Differentiating (2.3) with respect to y j and using (2.2) and the fact that˙ ∂ j l i = L − h ij , we get ¯ h ij = φh ij , (2.4)where φ = L − ¯ L ( e σ + ρ ) and h ij = L ˙ ∂ i ˙ ∂ j L is the angular metric tensor inthe Finsler space F n . 3ince h ij = g ij − l i l j , from (2.3) and (2.4) the fundamental tensor ¯ g ij =˙ ∂ i ˙ ∂ j ¯ L = ¯ h ij + ¯ l i ¯ l j is given as¯ g ij = φg ij + b i b j + e σ ( b i l j + b j l i ) + ( e σ − φ ) l i l j (2.5)It is easy to see that the det(¯ g ij ) does not vanish, and the reciprocal tensorwith components ¯ g ij of ¯ F n , obtainable from ¯ g ij ¯ g jk = δ ik , is given by¯ g ij = φ − g ij − µl i l j − φ − ( e σ + ρ )( l i b j + l j b i ) , (2.6)where µ = ( e σ + ρ ) φ − ( e σ − b − φ ), b = b i b i , b i = g ij b j and g ij is thereciprocal tensor of g ij of F n .We have following lemma [13]: Lemma 2.1.
The scalar ρ used in the condition of h-vector is a function ofcoordinates x i only. From equations (1.1), (2.3) and lemma 2.1 we have˙ ∂ i φ = L − ( e σ + ρ ) m i , (2.7)where m i = b i − ( L − β ) l i (2.8)Differentiating (2.4) with respect to y k and using (2.3), (2.4), (2.7) and therelation ˙ ∂ k h ij = 2 C ijk − L − ( l i h jk + l j h ik ), the Cartan covariant tensor ¯ C ijk is given by ¯ C ijk = φC ijk + ( e σ + ρ )2 L ( h ij m k + h jk m i + h ki m j ) , (2.9)where C ijk is (h)hv-torsion tensor of Cartan’s connection C Γ of Finsler space F n .From the definition of m i , it is evident that( a ) m i l i = 0 , ( b ) m i b i = b − β L = b i m i , (2.10)( c ) g ij m i = h ij m i = m j , ( d ) C ihj m h = L − ρh ij From (2.1), (2.6), (2.9) and (2.10), we get¯ C hij = C hij + 12 ¯ L ( h ij m h + h hj m i + h hi m j ) − L [ { ρ + L L ( b − (2.11) β L ) } h ij + L ¯ L m i m j ] l h roposition 2.1. Let ¯ F n = ( M n , ¯ L ) be an n-dimensional Finsler spaceobtained from the h-Randers conformal change of the Finsler space F n =( M n , L ) , then the normalized supporting element ¯ l i , angular metric tensor ¯ h ij , fundamental metric tensor ¯ g ij and (h)hv-torsion tensor ¯ C ijk of ¯ F n aregiven by (2.3), (2.4), (2.5) and (2.9) respectively. ¯ F n Let s be the arc-length of a curve x i = x i ( t ) on a differentiable manifold M n , then the equation of a geodesic [5] of F n = ( M n , L ) is written in thewell-known form: d x i ds + 2 G i ( x, dxds ) = 0 , (3.1)where functions G i ( x, y ) are the geodesic spray coefficients given by2 G i = g ir ( y j ˙ ∂ r ∂ j F − ∂ r F ) , F = L . Now suppose ¯ s is the arc-length of a curve ¯ x i = ¯ x i ( t ) on a differentiable man-ifold M n in the Finsler space ¯ F n = ( M n , ¯ L ), then the equation of geodesicin ¯ F n can be written as d x i d ¯ s + 2 ¯ G i ( x, dxd ¯ s ) = 0 , (3.2)where functions ¯ G i ( x, y ) are given by2 ¯ G i = ¯ g ir ( y j ˙ ∂ r ∂ j ¯ F − ∂ r ¯ F ) , ¯ F = ¯ L . Since d ¯ s = ¯ L ( x, dx ), this is also written as d ¯ s = e σ ( x ) L ( x, dx ) + b i ( x, y ) dx i = e σ ( x ) ds + b i ( x, y ) dx i Since ds = L ( x, dx ), we have dx i ds = dx i d ¯ s [ e σ ( x ) + b i ( x, y ) dx i ds ] (3.3)Differentiating (3.3) with respect to s , we have d x i ds = d x i d ¯ s [ e σ ( x ) + b i dx i ds ] + dx i d ¯ s ( de σ ( x ) ds + db i ds dx i ds + b i d x i ds )5ubstituting the value of dx i d ¯ s from (3.3), the above equation becomes d x i ds = d x i d ¯ s [ e σ ( x ) + b i dx i ds ] + dx i ds [ e σ ( x ) + b i dx i ds ] ( de σ ( x ) ds + (3.4) db i ds dx i ds + b i d x i ds )Now differentiating equation (1.1) with respect to x i we have ∂ i ¯ L = e σ A i + B i , (3.5)where A i = L∂ i σ + ∂ i L and B i = ∂ i { b r ( x, y ) } y r .Differentiating above equation with respect to y j we have˙ ∂ j ∂ i ¯ L = e σ ˙ ∂ j A i + ˙ ∂ j B i , (3.6)where ˙ ∂ j A i = l j ∂ i σ + ˙ ∂ j ∂ i L and ˙ ∂ j B i = ˙ ∂ j { ∂ i b r ( x, y ) } y r + ∂ i b r ( x, y ) δ rj .Since 2 ¯ G r = y j (¯ l r ∂ j ¯ L + ¯ L ˙ ∂ r ∂ j ¯ L ) − ¯ L∂ r ¯ L therefore using equations (2.3), (3.5) and (3.6) we have2 ¯ G r = 2 e σ G r + y j { e σ l r L∂ j σ + e σ ( l r B j + b r A j ) + b r B j + (3.7) e σ L ˙ ∂ r B j + βe σ ˙ ∂ r A j + β ˙ ∂ r B j } − ( e σ L ∂ r σ + e σ LB r + βe σ A r + βB r ) , where G r = y j { l r ∂ j L + L ˙ ∂ r ∂ j L }− L∂ r L is the spray coefficients for the Finslerspace F n .Using equations (2.6) and (3.7) we have¯ G i = J G i + M i , (3.8)where G i = g ir G r , J = φ , and M i = e σ G r {− µl i l r − φ − ( e σ + ρ )( l i b r + l r b i ) } + [ φ − g ir − µl i l r − φ − ( e σ + ρ )( l i b r + l r b i )][ y j { e σ l r L∂ j σ + e σ ( l r B j + b r A j ) + b r B j + e σ L ˙ ∂ r B j + βe σ ˙ ∂ r A j + β ˙ ∂ r B j } − ( e σ L ∂ r σ + e σ LB r + βe σ A r + βB r )]. Theorem 3.1.
Let ¯ F n = ( M n , ¯ L ) be an n-dimensional Finsler space obtainedfrom the h-Randers conformal change of the Finsler space F n = ( M n , L ) ,then the the geodesic spray coefficients ¯ G i for the Finsler space ¯ F n are givenby (3.8) in the terms of the geodesic spray coefficients G i of the Finsler space F n . orollary 3.1. Let ¯ F n = ( M n , ¯ L ) be an n-dimensional Finsler space ob-tained from the h-Randers conformal change of the Finsler space F n =( M n , L ) , then the equation of geodesic of ¯ F n is given by (3.2), where d x i d ¯ s and ¯ G i are given by (3.4) and (3.8) respectively. ¯ F n Following Matsumoto [8], in this section we shall investigate special casesof the Finsler space with h-Randers conformally changed Finsler space ¯ F n . Definition 4.1.
A Finsler space ( M n , L ) with dimension n ≥ is said to bequasi-C-reducible if the Cartan tensor C ijk satisfies C ijk = Q ij C k + Q jk C i + Q ki C j , (4.1) where Q ij is a symmetric indicatory tensor. Substituting h = j in equation (2.11) we get¯ C i = C i + ( n + 1)2 ¯ L m i (4.2)Using equations (2.9) and (4.2), we have¯ C ijk = φC ijk + φ ( n +1) π ( ijk ) { h ij ( ¯ C k − C k ) } ,where π ( ijk ) represents cyclic permutation and sum over the indices i, j and k .The above equation can be written as¯ C ijk = φC ijk + φ ( n +1) π ( ijk ) ( h ij ¯ C k ) − φ ( n +1) π ( ijk ) ( h ij C k )Thus Lemma 4.1.
In an h-Randers conformally changed Finsler space ¯ F n , theCartan’s tensor can be written in the form ¯ C ijk = π ( ijk ) ( ¯ H ij ¯ C k ) + V ijk , (4.3) where ¯ H ij = ¯ h ij ( n +1) and V ijk = φC ijk − φ ( n +1) π ( ijk ) ( h ij C k ) . Since ¯ H ij is a symmetric and indicatory tensor, so from the above lemmaand (4.1) we get 7 heorem 4.1. An h-Randers conformally changed Finsler space ¯ F n is quasi-C-reducible if the tensor V ijk of equation (4.3) vanishes identically. We obtain a generalized form of Matsumoto’s result known [8] as a corol-lary of the above theorem
Corollary 4.1. If F n is Reimannian then an h-Randers conformally changedFinsler space ¯ F n is always a quasi-C-reducible Finsler space. Definition 4.2.
A Finsler space ( M n , L ) of dimension n ≥ is called C-reducible if the Cartan tensor C ijk is written in the form C ijk = 1( n + 1) ( h ij C k + h ki C j + h jk C i ) (4.4)Now from equation (2.9) and definition of C-reducibility we have φC ijk = π ( ijk ) (¯ h ij N k ) , (4.5)where N k = n +1) ¯ C k − L m k . Conversely, if (4.5) is satisfied for certaincovariant vector N k then from (2.9) we have¯ C ijk = 1( n + 1) π ( ijk ) (¯ h ij ¯ C k ) (4.6)Thus we have Theorem 4.2.
An h-Randers conformally changed Finsler space ¯ F n is C-reducible iff equation (4.5) holds good. Corollary 4.2.
If the Finsler space F n is C-reducible Finsler space, thenan h-Randers conformally changed Finsler space ¯ F n is always a C-reducibleFinsler space. ¯ F n The v -curvature tensor [8] of Finsler space with fundamental function L is given by S hijk = C ijr C rhk − C ikr C rhj Therefore the v -curvature tensor of an h-Randers conformally changed Finslerspace ¯ F n will be given by¯ S hijk = ¯ C ijr ¯ C rhk − ¯ C ikr ¯ C rhj (5.1)8rom equations (2.9) and (2.11) we have¯ C ijr ¯ C rhk = φ [ C ijr C rhk + ( ρL ¯ L − m L ) h hk h ij + 12 ¯ L ( C ijk m h + (5.2) C ijh m k + C ihk m j + C hjk m i ) + 14 ¯ L ( h hj m i m k + h hi m j m k + h jk m i m h + h ik m h m j )] , where h jr C rhk = C jhk = h rj C rhk , m i m i = m .Using equations (5.1) and (5.2) we have¯ S hijk = φ [ S hijk + ( ρL ¯ L − m L ) { h hk h ij − h hj h ik } + 14 ¯ L { h hj m i m k (5.3) − h hk m i m j + h ik m h m j − h ij m h m k } ] Proposition 5.1.
In an h-Randers conformally changed Finsler space ¯ F n the v-curvature tensor ¯ S hijk is given by (5.3). It is well known[8] that the v -curvature tensor of any three-dimensionalFinsler space is of the form L S hijk = S ( h hj h ik − h hk h ij ) , (5.4)where scalar S in (5.4) is a function of x alone.Owing to this fact M. Matsumoto defined the S3-like Finsler space as Definition 5.1.
A Finsler space F n ( n ≥ is said to be S3-like Finslerspace if the v -curvature tensor is of the form (5.4). The v -curvature tensor of any four-dimensional Finsler space may bewritten as [8]: L S hijk = Θ ( jk ) { h hj K ki + h ik K hj } , (5.5)where K ij is a (0, 2) type symmetric Finsler tensor field which is such that K ij y j = 0 and the symbol Θ ( jk ) { ... } denotes the interchange of j, k andsubtraction. The definition of S4-like Finsler space is given as Definition 5.2.
A Finsler space F n ( n ≥ is said to be S4-like Finsler spaceif the v -curvature tensor is of the form (5.5). From equation (5.3) we have 9 emma 5.1.
The v-curvature tensor ¯ S hijk of a h-Randers conformally changedFinsler space can be written as ¯ S hijk = ¯ S (¯ h hj ¯ h ik − ¯ h hk ¯ h ij ) + U hijk , (5.6) where ¯ S = − φ ( ρL ¯ L − m L ) and U hijk = φ [ S hijk + L { h hj m i m k − h hk m i m j + h ik m h m j − h ij m h m k } ]From lemma (5.1) and definition of S3-like Finsler space we have Theorem 5.1.
An h-Randers conformally changed Finsler space ¯ F n is S3-like if the tensor U hijk of equation (5.6) vanishes identically. From equation (5.3) we have
Lemma 5.2.
The v-curvature tensor ¯ S hijk of an h-Randers conformallychanged Finsler space can also be written as ¯ S hijk = Θ ( jk ) (¯ h ij K ij + ¯ h ik K hj ) + φS hijk , (5.7) where K ij = L m i m j − ( ρL ¯ L − m L ) h ij . Thus from lemma(5.2) and definition of S4-like Finsler space we have
Theorem 5.2.
If the v -curvature tensor of Finsler space F n vanishes iden-tically then an h-Randers conformally changed Finsler space ¯ F n is S4-likeFinsler space. Now we are concerned with ( v ) hv -torsion tensor P ijk . With respect tothe Cartan’s connection C Γ , L | i = 0 , l i | j = 0 , h ij | k = 0 hold good [8].Taking h-covariant derivative of the equation (2.9) we have¯ C ijk | h = L − ¯ L ( e σ σ | h + ρ | h ) { C ijk + 12 ¯ L ( h ij m k + h jk m i + h ki m j ) } (5.8)+ φ { C ijk | h + 12 ¯ L ( h ij m k | h + h jk m i | h + h ki m j | h ) } , where m i | h = b i | h − L − l i b r | h y r . Lemma 5.3.
The h -covariant derivative of the Cartan tensor ¯ C ijk of anh-Randers conformally changed Finsler space ¯ F n can be written as ¯ C ijk | h = φC ijk | h + V ijkh , (5.9) where V ijkh = L − ¯ L ( e σ σ | h + ρ | h ) { C ijk + L ( h ij m k + h jk m i + h ki m j ) } + φ L ( h ij m k | h + h jk m i | h + h ki m j | h ) . v ) hv -torsion tensor P ijk of the Cartan connection C Γ is written inthe form P ijk = C ijk | ,where the subscript ’0’ means the contraction with respect to the supportingelement y i .From the equation (5.8), the ( v ) hv -torsion tensor ¯ P ijk is given by¯ P ijk = φP ijk + L − ¯ L ( e σ σ | + ρ | ) { C ijk + 12 ¯ L ( h ij m k + h jk m i + (5.10) h ki m j ) } + φ L { h ij m k | + h jk m i | + h ki m j | } Thus we have
Proposition 5.2.
The ( v ) hv -torsion tensor ¯ P ijk of an h-Randers confor-mally changed Finsler space can be written in the form of (5.10). From the equation (5.10) we have
Lemma 5.4.
The ( v ) hv -torsion tensor ¯ P ijk of an h-Randers conformallychanged Finsler space can also be written as ¯ P ijk = φP ijk + W ijk , (5.11) where W ijk = L − ¯ L ( e σ σ | + ρ | ) { C ijk + L ( h ij m k + h jk m i + h ki m j ) } + φ L { h ij m k | + h jk m i | + h ki m j | } . We have
Definition 5.3.
A Finsler space is called a Berwald space if C ijk | h = 0 holdsgood. Definition 5.4.
A Finsler space is called a Landsberg space if P ijk = 0 holdsgood. In view of above definition (5.3) and the lemma (5.3) we have
Theorem 5.3.
If a Finsler space F n is a Berwald space and the tensor V ijkh of equation (5.9) vanishes identically then an h-Randers conformally changedFinsler space ¯ F n is a Berwald space. In view of above definition (5.4) and the lemma (5.4) we have
Theorem 5.4.
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