Khayya J. Math. 5 09, no., 69 78 DOI: 0.03/kj.08.7578 ON RANDERS CHANGE OF GENERALIZED TH ROOT METRIC MANOJ KUMAR Counicated by B. Mashayekhy Abstract. In the present paper, we find a condition under which a Finsler space with Randers change of generalized th root etric is projectively related to an th root etric. Then we find a condition under which Randers change of generalized th root Finsler etric is locally projectively flat and also find a condition for locally dually flatness.. Introduction In 9, Randers 6 introduced a special class of Finsler space defined by F = F + β, F is a Rieannian etric and β = b i xy i is a one-for. Matsuoto studied Randers spaces and generalized Randers spaces in which F is Finslerian. The theory of th root etrics has been developed by Matsuoto and Shiada see 5 and 7 and applied to ecology by Antonelli and studied by any authors see 0,,, 3. Recent studies show that th root Finsler etrics plays a very iportant role in physics, space-tie, general relativity as well as in unified gauge field theory see. In 9, Tayebi and Najafi characterized locally dually flat and Antonelli th root etrics. Recently, Tayebi, Peyghan, and Shahbazi Nia 0 has introduced generalized th root Finsler etrics in which they studied locally dually flatness of generalized th root Finsler etrics and found a condition under which a generalized th root etric is projectively related to an th root etric. Throughout this paper, we call the Finsler etric F as transfored th root etric and M n, F = F n as transfored Finsler space. We restrict ourselves to >, throughout the paper and also the quantities corresponding to the Date: Received: 9 July 08; Revised: Septeber 08; Accepted: 0 October 08. 00 Matheatics Subject Classification. Priary 53B0; Secondary 53C60. Key words and phrases. Finsler space; Randers change of generalized th root etrics; projectively related etrics; locally projectively flat etric; locally dually flat etric. 69
70 MANOJ KUMAR transfored Finsler space F n will be denoted by putting bar on the top of that quantity. In this paper, we obtain the condition under which the transfored Finsler space is projectively related with the given Finsler space. Also we find the condition under which the transfored Finsler space is locally projectively flat and locally dually flat.. Preliinaries Let M n be an n-diensional C -anifold, and let T x M denote the tangent space of M n at x. The tangent bundle T M is the union of tangent spaces, T M := x M T xm. We denote the eleents of T M by x, y, x = x i is a point of M n and y T x M, called supporting eleent. We denote T M 0 = T M \ {0}. Definition.. A Finsler etric on M n is a function F : T M 0, with the following properties: i F is C on T M 0, ii F is positively -hoogeneous on the fibers of tangent bundle T M, and iii the Hessian of F with eleent g ij = F is positive definite on T M y i y j 0. The pair M n, F = F n is called a Finsler space. F is called the fundaental function, and g ij is called the fundaental etric tensor of the Finsler space F n. The noralized supporting eleent l i and angular etric tensor h ij of F n are defined, respectively, as: l i = F y i and h ij = F F y i y j.. Let F = A be an th root Finsler etric, A is given by A := a i i...i x y i y i... y i, with a i...i is syetric in all its indices 7. The generalized th root etric is defined as F = F + B. Here F is called an associated th root etric of the generalized th root etric F 8. Consider the transfored Finsler etric F = F + B + C,. B = b ij xy i y j and C = c i xy i is a one-for on the anifold M n. This transfored etric F is called Randers change of generalized th root etric. Clearly F is also a Finsler etric on M n. Let, A i = A y i, A ij = A y i y j, B i = B y i, C i = C y i, A x i = A x i, A 0 = A x iy i, B x i = B x i, B 0 = B x iy i, C x i = C x i, C 0 = C x iy i.
ON RANDERS CHANGE OF GENERALIZED TH ROOT METRIC 7 3. Fundaental etric tensor with Randers change of generalized th root etric Fro., we have F = A + B + C + C A + B. 3. Differentiating equation 3. with respect to y i yields F = A y i A A A i + b ij y j + CC i + C i A i + b ij y j C + B +. A + B 3. Again, differentiating equation 3. with respect to y j yields F = y i y j A A i A j + A A ij + b ij + C i C j C i A A j + b jk y k + A + B { } A + B A A i A j + A A ij + b ij C + A + B C j A + B A A i + b ij y j + A + B C A A i + b ij y j A A j + b ij y i. A + B 3 Therefore, g ij = A A i A j + A A ij + b ij + C i C j C i A A j + b jk y k + A + B { +A + B A A i A j + A A ij + b ij C } + C j A A i + b ij y j CA + B 3 A A i + b ij y j A A j + b ij y i. The etric tensor of Finsler space with Randers change of generalized th root etric is given by C ḡ ij = gˆ ij + + C i C j + C id j + C j D i D i D j C, 3.3 A + B A + B A + B 3/
7 MANOJ KUMAR and gˆ ij = g ij + b ij, D i = A A i + b ij y j, D j = A A j + b ij y i g ij = A AA ij + A i A j. Proposition 3.. The covariant etric tensor ḡ ij of Randers change of generalized th root etric is given as: C ḡ ij = gˆ ij + + C i C j + C id j + C j D i D i D j C A + B A + B A + B. 3/. Spray coefficients of Randers change of generalized th root etric The geodesics of a Finsler space F n are given by the following syste of equations: d x i dt + Gi x, dx = 0, i =,,..., n, dt G i = gil { F x k y l y k F x l }. are called the spray coefficients of F n. Two Finsler etrics F and F on a anifold M n are called projectively related if there is a scalar function P x, y defined on T M 0 such that Ḡi = G i + P y i, Ḡ i and G i are the geodesic spray coefficients of F n and F n, respectively. In other words, two etrics F and F are called projectively related if any geodesic of the first is also geodesic for the second and vice versa. The spray coefficients of Randers change of generalized th root Finsler space F n is given by Ḡ i = ḡil { x k y l y k x l }.. Lea.. 0. Let A = A ij be an n n invertible and syetric atrix, and let D = D i and E = E i be two nonzero n and n vectors, respectively, such that D i E j = D j E i. Suppose that + A pq D p E q 0. Then, the atrix B = B ij defined by B ij := A ij + D i E j is invertible and B ij := B ij = A ij + A pq D p E q A ki A lj D k E l, atrix A ij denotes the inverse of atrix A ij. Theore.. Let F = A + B + C and F = A be Randers change of generalized th root and an th root Finsler etric on an open subset U R n,
ON RANDERS CHANGE OF GENERALIZED TH ROOT METRIC 73 respectively, A := a i i...i xy i y i... y i, B = b ij xy i y j, and C = c i xy i with. Suppose that the following holds: A A il ζ l + J l + Lb il KC i C l + I il ζ l + J l γ + Lbil I il A i = 0,.3 KC i C l ζ l = B 0l B x l + C l C 0 + CC 0l CC x l, C A + B J l = y k C A + B, x k y l x l γ = KCl A i = { F x ky j y k F x j }, A A ip C p, C l = g lk C k and K, L are constants. Then F is projectively related to F. Proof: By equation., we have F = F + B + C + C Then using Proposition 3., we obtain ḡ ij = gˆ ij + C + C i C j + C id j + C j D i H H gˆ ij = g ij + b ij, A + B.. C D id j H 3,.5 g ij = A AA ij + A i A j,.6 and H = A + B. Fro Lea., we get ḡ ij = ˆ g ij KC i C j + I ij,.7 K is a constant, I ij will be the inverse of the last two ter in equation.5, which easily ay not be calculate explicitly, and gˆ ij = g ij Lb ij = A AAij + yi y j Lb ij,.8 L is a constant and g ij is the inverse of g ij. Then by equations.,., and.7, we have Ḡ i = ˆ gil KC i C l + I il F + B + C + C A + B x k y l y k F + B + C + C A + B x l.
7 MANOJ KUMAR Then fro equation.8, we can write Ḡ i = g il Lb il KC i C l + I il F x k y l yk F x l + g il Lb il KC i C l + I il B x k y l yk B x l + g il Lb il KC i C l + I il C x k y l yk C x l + g il Lb il KC i C l + I il C A + B x k y l y k C A + B x l. That is, Ḡ i = + + g il F x k y l yk F x l KCi C l + Lb il I il F x k y l yk F x l g il Lb il KC i C l + I il B 0l B x l + C l C 0 + CC 0l CC x l g il Lb il KC i C l + I il J l, J l = C A + B x k y l y k C A + B x l. Therefore, Ḡ i = G i γ + Lbil I il KC i C l C i + g il Lb il KC i C l + I il ζ l + g il Lb il KC i C l + I il J l,.9 Putting we have γ = KCl { F x ky y k F }, l x l ζ l = B 0l B x l + C l C 0 + CC 0l CC x l..0 Φ := C i = g ip C p = A A y p C p, A i := A A ip C p,. AAip + yi y p C p = A i + Φy i.. By.8,.9, and., we get Ḡ i = G i + A yl ζ l + J l γ + Lbil I il KC i C l γ + Lbil I il KC i C l A i + A A il ζ l + J l + Φ y i Lb il KC i C l + I il ζ l + J l..3
ON RANDERS CHANGE OF GENERALIZED TH ROOT METRIC 75 If the relation.3 holds, then by.3 the Finsler etric F is projectively related to F. 5. Locally projectively flatness of Randers change of generalized th root etric A Finsler etric is called locally projectively flat if, for any point, there is a local coordinate syste in which the geodesics are straight lines as point sets. It is known that a Finsler etric F x, y on an open doain U R is locally projectively flat if and only if its geodesic coefficients G i take the for G i x, y = P x, yy i, P : T U = U R R is positively hoogeneous with degree one and P x, λy = λp x, y, λ > 0 3. We say that P x, y is the projective factor of F. In other words, a Finsler etric F = F x, y on a anifold M n is said to be locally projectively flat, if and only if F x k y lyk = F x l. Using equation., we have x l = A A x l + B x l A + B + C x l. 5. Fro 5., we get and x k y l = x k = A + B A + B 3 A A x k + B x k + C x k A + B A A y la x k + A A x k y l + B x k y l A + B A Ax k + B x k A Ay l + B y l + C x k y l.5. Let the Finsler etric F be locally projectively flat. Then we have x k y l y k x l = 0. 5.3 Therefore, fro 5., 5., and 5.3, we obtain y x k A + B F k y l x l = A A l A 0 + A A 0l + B 0l A + B A + B 3 A A0 + B 0 A Al + B l + C 0l A A x l + B x l C x l = 0. A + B
76 MANOJ KUMAR Hence, F is locally projectively flat etric if and only if A x l = A A la 0 + A 0l + A B0l A + B A Al + B l + C 0l C x l A A 0 + A B0 A + B B xla. 5. Thus, we have the following result. Theore 5.. Let F = A + B + C be a Randers change of generalized th root Finsler etric on a anifold M n. Then, F is a locally projectively flat etric if and only if equation 5. satisfied. 6. Locally dually flatness of Randers change of generalized th root etric In Finsler geoetry, Shen extended the notion of locally dually flatness for Finsler etrics. A Finsler etric F = F x, y on a anifold M n is said to be locally dually flat, if at any point there is a standard coordinate syste x i, y i in T M such that x k y l y k = x l. Every locally Minkowskian etric is locally dually flat. Using equation., we have x l = A Ax l + B x l + CC x l + Fro 6., we get x k = A Ax k + B x k + CC x k + and = x k y l A Ax k y l + x + A + B + B x k y l + CCk y l + C lc x k A + B + C x k y l A + B + C x k { CA A x l + CB x l + C x l A + B. 6. A + B A Al A x k CA A x k + CB x k A + B CA + C x k A + B CA Ax k y l Al A x k + + } A Ax kc l + C l B x k + CB x k y l CA Ax k + CB x k A A + B A Al + B l A l + B l. 6. A + B Let the Finsler etric F be locally dually flat. Then we have x k y l y k x l = 0. 6.3
ON RANDERS CHANGE OF GENERALIZED TH ROOT METRIC 77 Therefore, fro 6., 6., and 6.3, we obtain y k F = x k y l x l A A0l + + B 0l + CC 0l + C l C 0 =0. { + A + B A Al A 0 CA CA A0l Al A 0 + + } A A0 C l + C l B 0 + CB 0l A + B 3 CA A0 + CB 0 + C 0l A + B + C 0 A A l + B l A + B A Ax l B x l CC x l CA A x l + CB x l C x l A + B Therefore, F is locally dually flat etric if and only if A x l = F A 0l A + B + F A A la 0 A + B + F B 0l + CC 0l + C l C 0 A + F CA 0l + CA 0 + CB 0 A + F C 0lA F A 0C l + F C lb 0 + CB 0l A 8 F A + B A + B + C 0 F A Al + B l A + B A + B + A F CA la 0 A Al + B l A l + B l A F B x l + CC xl A A + B C F B xla F C xla A + B. 6. Thus, we have the following results Theore 6.. Let F = A + B + C be a Randers change of generalized th root Finsler etric on a anifold M n. Then, F is a locally dually flat etric if and only if equation 6. satisfied. Corollary 6.. 9. Let F = A be an th root Finsler etric on an open subset U R n. Then F is a locally dually flat etric if and only if the following holds A x l = A A 0A y l + AA 0l. 6.5 Putting C = 0 in equation. and equation 6., we get the following corollary.
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