On Randers-Conformal Change of Finsler Space with Special (α, β)-metrics

Transcription

1 International Journal of Contemporary Mathematical Sciences Vol no HIKARI Ltd On Randers-Conformal Change of Finsler Space with Special ( β)-metrics Sruthy Asha Baby Department of Mathematics and Statistics Banasthali University Banasthali Rajasthan India Gauree Shanker Centre for Mathematics and Statistics Central University of Punjab Bathinda Bathinda Punjab India Copyright c 2016 Sruthy Asha Baby and Gauree Shanker. This article is distributed under the Creative Commons Attribution License which permits unrestricted use distribution and reproduction in any medium provided the original work is properly cited. Abstract In the present paper we first find the fundamental metrics tensor of the Randers conformally transformed special ( β)-metrics F = + ɛβ + k β2 (ɛ and k 0 are constants) and then we find a condition under which the Randers conformally transformed special ( β)-metrics is locally dually flat. Mathematics Subject Classification: 53B40 53C60 Keywords: Dually flat Projective flatness Finsler space with ( β)- metrics Conformal transformation 1 Introduction Let = a ij y i y j be a Riemannian metrics and β = b i y i be a differential 1- form of an n-dimensional differentiable manifold M n. The Finsler metrics F

2 416 Sruthy Asha Baby and Gauree Shanker = + β was originated by G. Randers in his unified field theory and named the Randers space by R. S. Ingarden [8]. The Kropina metrics F = 2 a kind β of ( β)-metrics was investigated by C. Shibata [13] and treated by Ingarden in his paper concerned with thermodynamics [9]. In 1976 Hashiguchi [7] introduced the conformal change of Finsler metrics given by F = e σ(x) F. In particular he studied the special conformal transformation called as C-conformal. This transformation has also been studied by many authors ([10] [15]). In 1984 Shibata [12] extended the notion of β-change to a general case in Finsler geometry i.e. F = f(f β). Under this transformation he also studied the change of torsion tensor and curvature tensor and also dealt with some special Finsler spaces corresponding to specific forms of this function. In 2008 Abed ([1] [2]) generalized the conformal Randers and generalized Randers changes by a new transformation F = e σ(x) F + β. In addition to this he also obtained the correlation between some relevant tensors associated with (M F ) and the corresponding tensors associated with (M F ). Further he discussed the invariant and σ-invariant properties and determined an association between the Cartan connection associated with (M F ) and the transformed Cartan connection associated with (M F ). The main purpose of the current paper is to investigate the special ( β)- metrics F = e σ( ) +ɛβ +k β2 +β which is obtained by the Randers conformal change of the special ( β)- metrics F = + ɛβ + k β2 (ɛ and k 0). The paper is organized as follows: Starting with literature survey in section one we find fundamental metrics tensor ḡ ij and its inverse ḡ ij in section two [see Theorem 2.1 and 2.2 ] for the Randers conformally transformed metrics. In section three first we find the spray coefficients for the transformed metrics (see lemma 3 ) and then we find the necessary and sufficient conditions for this metrics to be locally dually flat (see Theorem 3.1 ). 2 Fundamental metrics tensor of Randers conformal change of special ( β) metrics A Finsler metrics (1) where = a ij y i y j is a Riemannian metrics and β = b i (x)y i is a differential one form which was introduced by physicst Randers in 1941 from view point of general theory of relativity. Further Matsumoto Ingarden ([5] studied this metrics as a Finsler metrics and investigated its properties. In 2008 S. Abed introduced a Finsler metrics F = e σ(x) F + β where F is a Finsler metrics σ(x) is a conformal factor and β is a one form. This metrics is called Randers conformal change of Finsler metrics.

3 On Randers-conformal change of Finsler space with special ( β)-metrics 417 Here we consider the Randers conformal change of the special ( β)-metrics F = + ɛβ + k β2 given by F = e σ + (e σ ɛ + 1)β + ke σ β2. (1) The differentiation of (1) with respect to y i yields the normalized supporting element l i given by l i = e σ l i + b i In view of (2) we have l i = eσ y i + (ɛeσ + 1)b i + e σ kβ ( 2b i βy ) i (2) 3 Differentiation of (2) with respect to y j yields lij = 1 ( e σ β2 ) aij + 1 ( 3β eσ kβ 3 From (2) and (3) we obtain h ij = F l ij where eσ) y i y j + 2eσ k b ib j y j b i 2β 3 y ib j (3) = p 1 a ij + p 2 y i y j + p 3 b i b j + p 4 (y j b i + y i b j ) (4) p 1 = 1 p 2 = 1 ( 3β 2 3 ( e σ β2 )( e σ + (e σ ɛ + 1)β + eσ kβ 2 ) 2 2 eσ)( e σ + (e σ ɛ + 1)β + eσ kβ 2 p 3 = 2e σ k [ e σ + (e σ ɛ + 1) β p 4 = 2eσ kβ ( e σ + (eσ ɛ + 1)β 2 + eσ kβ2 2 ] ) (5) + eσ kβ 2 ). 2 From (2) and (4) the fundamental metrics tensor ḡ ij of Finsler space (M F ) is given by: where ḡ ij = h ij + l i lj = ρa ij + ρ 0 b i b j + ρ 1 (b i j + b j i ) + ρ 2 ( i j ) (6) ρ = 1 ( e σ β2 )( e σ + (e σ ɛ + 1)β + eσ kβ 2 ) 2 ρ 0 = 2e2σ k(3kβ ) + 4e σ k(ɛe σ + 1)β (7) 2 ρ 1 = 4e2σ k 2 β 3 + e σ (ɛe σ + 1)( 2 3kβ 2 ) 5 ρ 2 = eσ (2 3e σ k)β 2 e σ (ɛe σ + 1)β 6.

4 418 Sruthy Asha Baby and Gauree Shanker Theorem 2.1 For the Randers conformal transformed ( β)- metrics F = e σ F + β where F = + ɛβ + k β2 the fundamental metrics tensor ḡ ij is given by equations (6) and (7). The contravariant metrics tensor ḡ ij of Finsler space (M F ) is given by where ḡ ij = ρ 1 {a ij τb i b j ηy i Y j } = ρ 1{ a ij ( δ ) b i b j µ 1 + δb { + (λ + ξ)β + λξb 2 )}µ ( y i + y j λbi)( + λbj)} (8) ρ = 1 ( e σ β2 )( e σ + (e σ ɛ + 1)β + eσ kβ 2 ) 3 2 δ = l l 2 5 β + l 3 4 β 2 + l 4 3 β 3 + l 5 2 β 4 + l 6 β 5 16e 4σ k 4 β 6 l 7 11 β + l 8 10 β 2 + l 9 9 β 3 + l 10 8 β 4 + l 11 7 β 5 + l 12 6 β 6 e σ (2 3e σ k)β 2 e σ (ɛe σ + 1)β µ = ( )( ) (9) 2 eσ 2 β 2 eσ 2 + (e σ ɛ + 1)β + e σ kβ 2 λ = ρ 1ρ (ρ 2 ρ 0 ρ 2 1)β ρρ 2 + (ρ 2 ρ 0 ρ 2 1)b 2 = [ 9 q βq β 2 q β 3 q β 4 q β 5 q β 6 q β 7 q 8 6 βq 9 5 β 2 q 10 4 β 3 q 11 3 β 4 q 12 2 β 5 q 13 β 6 q 14 q 15 ][ 11 β 2 q β 3 q β 4 q β 5 q β 6 q 5 + b 2 ( 7 q βq β 2 q β 3 q β 4 q β 5 q 14 + β 5 q 15 ) ] ξ = 4eσ k 2 β 3 + (ɛe σ + 1) 2 ( 2 3kβ 2 ) (ɛe σ + 1)β + (2 3e σ k)β 2 in which l 1 = e 2σ (ɛe σ + 1) 2 l 2 = 2e 3σ k(ɛe σ + 1) l 3 = 3e 3σ k 2 (4 6ɛ) 9k 2 (1 + 2k)e 4σ 9k 2 e 2σ l 4 = (ɛe σ + 1)(2e 3σ k(2k 3) + 8e 2σ k) l 5 = 6k 2 e 3σ (2 3e σ k) 9k 2 e 2σ (ɛe σ + 1) 2 l 6 = 24k 3 e 3σ (ɛe σ + 1) l 7 = e 3σ (ɛe σ + 1) l 8 = 2e 3σ (1 ɛ) e 4σ (3k + ɛ 2 ) e 2σ l 9 = (e σ ɛ + 1)(3e 2σ 4e 3σ k)

5 On Randers-conformal change of Finsler space with special ( β)-metrics 419 l 10 = (5k + ɛ 2 )e 3σ + 2(ɛ 1)e 2σ 3e 4σ k 2 + e σ l 11 = (ɛe σ + 1)(4e 2σ k 2e σ ) l 12 = e 2σ k(2 3e σ k) q 1 = e 3σ (ɛe σ + 1) q 2 = e 2σ (ɛe σ + 1) 2 q 3 = e 2σ (ɛe σ + 1)(1 + 2e σ k) q 4 = e 4σ (4k + 2ɛ 2 )k + e 3σ (ɛ 2 6ɛk) + e 2σ (2ɛ 3k) + e σ q 5 = (2 7e σ k)(ɛe σ + 1)e 2σ k q 6 = 4e 4σ k + e 3σ (3ɛ 2 k + 4) + 6ɛke 2σ q 7 = e 2σ (ɛe σ + 1)(4 + 3k 2 ) q 8 = 4e 3σ k 3 q 9 = 2ke 4σ (k 5ɛ 2 ) + 4e 3σ k(1 5ɛ) 6ke 2σ 4e σ k q 10 = (ɛe σ + 1){(2k 3)2e 3σ k + 8e 2σ k} q 11 = e 2σ (ɛe σ + 1) 2 q 12 = 2e 3σ k(ɛe σ + 1) q 13 = (ɛ 2 + 2k)9e 4σ k 2 + (2 3ɛ)6k 2 e 3σ + 9k 2 e 2σ q 14 = 24e 3σ (ɛe σ + 1)k 3 q 15 = 16e 4σ k 4 β 7. q 1 = 2e 3σ (1 ɛ) e 4σ (3k + ɛ 2 ) e 2σ q 2 = (ɛe σ + 1)(3e 2σ 4ke 3σ ) q 3 = e 3σ (ɛe σ + 1) q 4 = e 2σ (ke σ 1)(2 3e σ k) + e σ (ɛe σ + 1)k q 5 = e 2σ k(2 3e σ k). Theorem 2.2 For the Randers conformal transformed ( β)- metrics F = e σ F + β where F = + ɛβ + k β2 the contravariant metrics tensor ḡij of F is given by equations (8) and (9). 3 Locally dually flatness of transformed ( β)- metrics. The notion of dually flat Riemannian metrics was introduced by S. I. Amari and H. Nagaoka ([3][4]) when they studied the information geometry on Riemannian manifolds. In Finsler geometry Shen extended the notion of locally dually flatness for Finsler metrics. Dually flat Finsler metrics form a special and valuable class of Finsler metrics in Finsler information geometry which plays a very important role in studying flat Finsler information structure.

6 420 Sruthy Asha Baby and Gauree Shanker Information geometry has been emerged from investigating the geometricsal structure of the family of probability distributions. A Finsler space F n = (M F(xy)) is called the ( β)-metrics if there exists a 2-homogenous function L of two variables such that the Finsler metrics F: TM R is given by: F 2 (x y) = L((x y) β(x y)) (10) where 2 (x y) = a ij (x)y i y j is Riemannian metrics on M and β(x y) = b i (x)y i is a 1-form on M. A Finsler metrics F = F (x y) on a manifold M n is said to be locally dually flat if at any point there is a standard coordinate system (x i y i ) in TM such that [F 2 ] x k y lyk = 2[F 2 ] x l. (11) In this case the coordinate (x i ) is called an adapted local coordinate system. Every locally Minkowskian metrics is locally dually flat. Consider the Randers conformally transformed metrics F = e σ(x) F + β where F is a special ( β)-metrics. If φ(s) = e σ (1 + ɛs + ks 2 ) + s; s = β then ( β)-metrics is exactly the metrics of the form of Randers conformally transformed special ( β)-metrics. Let Ḡi (x y) and G i (x y) denote the spray coefficients of F and respectively. To express formulae for the spray coefficients Ḡi of F in terms of and β we need to introduce some notations. Let b i:j be a covariant derivative of b i with respect to y j. Denote r ij = 1 2 (b i j + b j i ) s ij = 1 2 (b i j b j i ) s i j = a ih s hj s j = b i s i j = s ij b i r j = r ij b i r 0 = r j y j s 0 = s j y j r 00 = r ij y i y j. The spray coefficients Ḡi of the Randers conformally transformed special ( β)- metrics are related to G i by where Ḡ i = G i + Qs i 0 + Θ( 2Qs 0 + r 00 ) yi + ψ( 2Qs 0 + r 00 )b i (12) Q = Θ = φ φ sφ = eσ (ɛ + 2ks) + 1 e σ (1 ks 2 ) (φ sφ )φ 2φ[(φ sφ ) + (b 2 s 2 )φ ]

7 On Randers-conformal change of Finsler space with special ( β)-metrics 421 = ψ = = 2e 2σ k (e 2σ ɛ 2 + 1)s + (2e 2σ k 2 6ɛke 2σ 6e σ k)s 2 (6k 2 e 2σ )s 3 [4k 2 e σ (1 + e σ )]s 4 [2k(e σ ɛ + 1)(e σ + 1)]s 3 [4e 2σ k(1 b 2 k)]s 2 + [2e σ (ɛe σ + 1)]s + 2e σ (1 + 2b 2 k) φ 2{(φ sφ ) + (b 2 s 2 )φ } e σk 1 + k(2b 2 3s 2 ). Here b i = a ij b j and b 2 = a ij b i b j = b j b j. From (11) we have the following Theorem 3.1 An ( β)-metrics F = φ(s) = (e σ + {e σ ɛ + 1}s + e σ ks 2 ) where s = β is dually flat on an open subset U Rn if and only if 2{ 5 e 2σ + e σ (ɛe σ + 1) 4 β + e σ (ɛe σ + 1)k 2 β 3 e 2σ k 2 β 4 } 2 a ml G m + [2k 2 e σ 2 β 3 +{2ke σ (ɛ + k) + kɛ 2 e 2σ + k} 3 β β{2e σ ɛ + 2e σ k + ɛ 2 e 2σ + 1} + e σ (ɛe σ + 1) 5 ] [ (3s l0 r l0 ) 3 3 (e σ 2 + {ɛe σ + 1}β + e σ kβ 2 ) e σ ( 2 kβ 2 G m )y m y l + {2ke σ β +(ɛe σ + 1) 2 G m ] [ }b m y l (r b m G m )y l + {4k 2 e σ β 3 + 2(2ke σ ɛ + 2ke σ k + kɛ 2 e 2σ +k)β β(4e σ ɛ + 4e σ k + 2ɛ 2 e 2σ + 2) + β 3 2e σ (ɛe σ + 1)}y m G m + [e 2σ (2k + ɛ 2 ) 3 ] +(2e σ ɛ + 1) 3 + {6ke σ (ɛe σ + 1)} 3 β + 6e 2σ k 2 β 2 ]{r ( 2 b m βy m )G m } ( 2 b l βy l ) = 0. By direct computation F is locally dually flat on U if and only if [ F 2 ] x k y lyk = 2[ F 2 ] x l which implies On the other hand φ 2 ( x k y lyk 2 x l) + 2 φφ (s x k y lyk 2s x l) + 2φφ ( y ls x ky k +s y l x ky k ) + 2 (φ 2 + φφ )(s x ky k )s y l = 0 x l = 1 G m y y l m x ky k = 2 Gm y m l = y l (13) s x l = 1 b m;ly m (b m sy m ) Gm y l s y l = b l sy l 2 (14) s x ky k = r (b 2 m sy m )G m (15) x k y lyk 2 x l = 2 (a ml 2 y 3 m y l )G m 1 G m y y l m (16) s x k y lyk 2s x l = r 00 3 y l + 2 s l0 4y l 4 (b m sy m )G m ( y l b m b i sy l 2 y m sa ml ) G m 1 b m;ly m 1 2 (b m sy m ) Gm y l. (17)

8 422 Sruthy Asha Baby and Gauree Shanker Putting (13) (14) (15) (16) and (17) into (13) and noting b m;l y m = r 0l + s 0l we get which implies 2φ(φ sφ ) 2 a ml G m + φφ (3s l0 r l0 ) 3 2 φ [ (φ sφ )y m G m y l +φ b m G m y l ] + φφ (r b m G m )y l + [2φφ y m G m + (φ 2 + φφ ) (r (b m sy m )G m )](b l sy m ) = 0 2{e 2σ + e σ (ɛe σ + 1)s + e σ (ɛe σ + 1)ks 3 e 2σ k 2 s 4 } 2 a ml G m + [2k 2 e σ s 3 + {2ke σ (ɛ + k) + kɛ 2 e 2σ + k}s 2 + s{2e σ ɛ + 2e σ k + ɛ 2 e 2σ + 1} + e σ (ɛe σ + 1)] (3s l0 r l0 ) 3 2 (e σ + {ɛe σ + 1}s + e σ ks 2 ) [ e σ (1 ks 2 )y m G m y l + {2ke σ s + (ɛe σ + 1)}b m G m y l ] (r00 + 2b m G m )y l + [ {4k 2 e σ s 3 + 2(2ke σ ɛ + 2ke σ k + kɛ 2 e 2σ + k)s 2 + s(4e σ ɛ + 4e σ k + 2ɛ 2 e 2σ + 2) + 2e σ (ɛe σ + 1)}y m G m + [e 2σ (2k + ɛ 2 ) + 2e σ ɛ {6ke σ (ɛe σ + 1)}s + 6e 2σ k 2 s 2 ]{r (b m sy m )G m } ] (b l sy l ) = 0. This completes the proof. References [1] S. H. Abed Conformal β-changes in Finsler Spaces Proc. Math. Phys. Soc. Egypt 86 (2008) [2] S. H. Abed Cartan connection associated with a β-conformal change in Finsler geometry Tensor N. S. 70 (2008) [3] S. I. Amari Differential Geometricsal Methods in Statistics Springer Lecture Notes in Statistics Springer-Verlag [4] S. I. Amari and H. Nagaoka Method of Information Geometry AMS Translation of Math. Monographs 191 Oxford University Press [5] P. L. Antonelli R. S. Ingarden and M. Matsumoto The Theory of Sprays and Finsler Spaces with Applications in Physics and Biology Kluwer Academic Publishers London [6] S. S. Chern and Z. Shen Riemann- Finsler Geometry World Scientific Singapore

9 On Randers-conformal change of Finsler space with special ( β)-metrics 423 [7] M. Hashiguchi On conformal transformations of Finsler metrics J. Math. Kyoto Univ. 16 (1976) [8] R. S. Ingarden Trav. Soc. Sci. Lettr Wroclaw B 45 (1957). [9] R. S. Ingarden Geometry of thermodynamics Proc. XV Intern. Conf. Diff. Geom. Methods in Theor. Phys. (1987) [10] H. Izumi Conformal transformations of Finsler spaces Tensor N. S. 31 (1977) [11] Z. Shen and G. C. Yildirim On a class of projectively flat metrics with constant flag curvature Canadian Journal of Mathematics 60 (2008) no [12] C. Shibata On invariant tensors of β-changes of Finsler metrics J. Math. Kyoto Univ. 24 (1984) [13] C. Shibata On Finsler spaces with Kropina metrics Rep. on Math. Phys. 13 (1978) [14] Q. Xia On a class of locally dually flat Finsler metrics of isotropic flag curvature Publ. Math. Debreccan 78 (2011) no [15] N. L. Youssef S. H. Abed and A. Soleiman A global theory of conformal Finsler geometry Tensor N. S. 69 (2008) Received: August ; Published: October