On Einstein Kropina change of m-th root Finsler metrics

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1 On Einstein Kropina change of m-th root insler metrics Bankteshwar Tiwari, Ghanashyam Kr. Prajapati Abstract. In the present paper, we consider Kropina change of m-th root metric and prove that if it is an Einstein metric or weak Einstein metric), then it is Ricci-flat. M.S.C. 2010: 53B40, 53C60. Key words: insler space; Kropina metrics; m-th root metrics; Einstein metrics. 1 Introduction The theory of m-th root insler metrics has been developed by Shimada 11] in 1979, applied to ecology by Antonelli 1] and studied by several authors 12],14],11]). It is regarded as a generalization of Riemannian metric in the sense that for m = 2, 3 and 4, it is called Riemannian metric, cubic metric and quartic metric respectively 7]. In the four-dimension, a special fourth root metric of the form = 4 y 1 y 2 y 3 y 4 is called the Berwald-Moór metric 4], which is considered by physicists as an important subject for a possible model of space time. Recent studies show that m-th root insler metrics plays a very important role in physics, space-time and general relativity as well as in unified gauge field theory 3],2]). Z. Shen and B. Li have studied the geometric properties of locally projectively flat fourth root metrics in the form = 4 a ijkl x)y i y j y k y l and generalized fourth root metrics in the form aijkl = x)y i y j y k y l + b ij x)y i y j 7]. Recently, B. Tiwari and M. Kumar 13] have studied Randers change of a insler space with m-th root metric. Also, A. Tayebi, T. Tabatabaeifar and E. Peyghan introduced the Kropina change of m-th root metric and established conditions on Kropina change of m-th root metric, to be locally dually flat and locally projectively flat. Let M, ) = n be an n-dimensional insler manifold. or a 1-form x, y) = b i x)y i on M, define a insler change as follows x, y) x, y) = f, ), Differential Geometry - Dynamical Systems, Vol.18, 2016, pp c Balkan Society of Geometers, Geometry Balkan Press 2016.

2 140 Bankteshwar Tiwari, Ghanashyam Kr. Prajapati where f, ) is a positively homogeneous function of and. A insler change is called Kropina change if f, ) = 2. The purpose of the present paper is to investigate Kropina change of m-th root metrics, defined by 1.1) = 2, where = m A is an m-th root metric on the manifold M, for which we shall restrict our consideration to the domain where = b i x)y i > 0. The Einstein metrics are solutions to Einstein field equation in General Relativity, which closely connect Riemannian geometry with gravitation in General Relativity. C. Robles studied a special class of Einstein insler metrics, that is, Einstein Randers metrics, and proved that for a Randers metric on a 3-dimensional manifold, it is Einstein if and only if it has constant flag curvature. E. Guo, X. Mo and X. Zhang have explicitely constructed an Einstein insler metrics of non-constant flag curvature in terms of navigation representation 6]. Recently, Z. Shen and C. Yu, using certain transformation, have constructed a large class of Einstein metrics 9]. In this paper, we establish following theorems Theorem 1.1. Let = 2 insler metric on a manifold of dimension n 2, with m 3. If is Einstein metric, then it is Ricci-flat. Theorem 1.2. Let = 2 insler metric on a manifold of dimension n 2, with m 3. If is a weak Einstein metric, then it is Ricci-flat. Theorem 1.3. Let = 2 insler metric on a manifold of dimension n 2, with m 3. If is of scalar flag curvature Kx, y) and isotropic S-curvature, then K = 0. Throughout the paper we call the insler metric as Kropina change of m-th root metric and n = M, ) as Kropina transformed insler space. We restrict ourselves for m 3 throughout the paper and also the quantities corresponding to the Kropina transformed insler space n will be denoted by putting bar on the top of that quantity. 2 Preliminaries Let M be an n-dimensional C -manifold. Denote by T x M the tangent space at x M and by T M := x M T xm the tangent bundle of M. Each element of T M has the form x, y), where x M and y T x M. Let T M 0 = T M \ {0}. Definition. A insler metric on M is a function : T M 0, ) with the following properties: i) is C on T M 0, ii) is positively 1-homogeneous on the fibers of tangent bundle T M, and iii) the Hessian of 2 2 with components g ij = y i y is positive definite on T M j 0.

3 On Einstein Kropina change of m-th root insler metrics 141 The pair n = M, ) is called a insler space of dimension n. is called fundamental function and the tensor g with components g ij is called the fundamental tensor of the insler space n. The normalized supporting element l i and the angular metric tensor h ij of are defined, respectively as l i = y, and h i ij = 2 y i y. The S-curvature j S = Sx, y) in insler geometry has been introduced by Shen 8] as a non-riemannian quantity, defined as Sx, y) = d dt τσt), σt))] t=0, where τ = τx, y) is a scalar function on T x M\{0}, called distortion of and σ = σt) be the geodesic with σ0) = x and σ0) = y. A insler metric is called of isotropic S-curvature if S = n + 1)c, for some scalar function c = cx), on M. Let be a insler metric defined by = m A, where A is given by A = a i1 i 2...i m x)y i 1 y i 2...y i m with a i1...i m symmetric in all its indices 11]. Then is called an m-th root insler metric. Clearly, A is homogeneous of degree m in y. Let A i = A y i, A ij = 2 A y i y j, A x i = A x i, A 0 = A x iy i. Then the following relations hold g ij = A 2 m 2 m 2 maa ij + 2 m)a i A j ], y i A i = ma, y i A ij = m 1)A j, y i = 1 m A 2 m 1 A i, A ij A jk = δ i k, Aij A i = 1 m 1 yj, A i A j A ij = m m 1 A. 3 undamental metric tensors and geodesic sprays of Kropina changed m-th root metrics The differentiation of equation1.1) with respect to y i yields the normalized supporting element l i given by 3.1) li = 2Ai ma b ) i and the angular metric tensor h ij given by 3.2) hij = 2 2 ma A 22 m) ij + m 2 A 2 A i A j 2 ma A ib j + A j b i ) + 2 ] 2 b ib j. Also the fundamental metric tensor ḡ ij of insler space n is given by ḡ ij = h ij + l i lj. Therefore, by using the equations 3.1) and 3.2), we obtain the metric tensor ḡ ij as g ij = 2 2 ma A 24 m) ij + m 2 A 2 A i A j 4 ma A ib j + A j b i ) + 3 ] 2 b ib j. This equation can be rewritten as 3.3) ) )] g ij = 2 2 ma A 24 3m) ij + A 3m 2 A 2 i A j + A i 3mA b i A j 3mA b j.

4 142 Bankteshwar Tiwari, Ghanashyam Kr. Prajapati In order to calculate the components of inverse metric tensor g ij, we use the following Proposition twice. Proposition. 8] Let G = g ij ) and H = h ij ) be symmetric n n matrices and C = c i ) be an n-vector. Assume that H is invertible with H 1 = h ij ) and g ij = h ij + δc i c j. Then detg ij ) = 1 + δc 2 )deth ij ), where c = h ij c i c j. If 1 + δc 2 0, then G is invertible. The inverse matrix G 1 = g ij ) is given by Let g ij = h ij δci c j 1 + δc 2, where ci = h ij c j. 3.4) H ij = 2 ma A 24 3m) ij + 3m 2 A 2 A i A j. By using the Proposition from above, we infer H ij = ma 2 Aij 4 3m 2m 1) yi y j. Thus, in view of equations 3.3) and 3.4), g ij can be written as g ij = 2 H ij + K i K j ], where 4 K i = 3mA A i ) 3. By direct computation, we have b i g ij = 1 2 a0 A ij + a 1 y i y j + a 2 B i B j + a 3 y i B j + y j B i ) ], where, 3.5) { } a 0 = ma 2, a 1 = 2m 4) 2 +m3m 4)AB 2 2{m 2) 2 +mm 1)AB 2 }, a 2 = a 3 = m 2 A 2{m 2) 2 +mm 1)AB 2 }, Bi = A ij b j, B 2 = B i b i. m 2 1 m)a 2 2{m 2) 2 +mm 1)AB 2 }, Thus we have Proposition 3.1. The covariant metric tensor ḡ ij and the contravariant metric tensor ḡ ij of Kropina trasformed insler space n are given as g ij = 2 2 ma A 24 m) ij + m 2 A 2 A i A j 4 ma A ib j + A j b i ) + 3 ] 2 b ib j and g ij = 1 2 a0 A ij + a 1 y i y j + a 2 B i B j + a 3 y i B j + y j B i ) ], where a 0, a 1, a 2, a 3, B i and B 2 are given by equation 3.5). In local coordinates, the geodesics of a given insler metric = x, y) are characterized by the equations d 2 x i ) x, dt 2 + 2Gi dxi = 0, dt where 3.6) G i = 1 4 gil { 2 ] xk y lyk 2 ] x l}

5 On Einstein Kropina change of m-th root insler metrics 143 are called spray coefficients. To calculate the spray coefficients Ḡi, we use the relations 3.7) 2 ) = 2 4Ax k x k ma 2 ) x k and 3.8) 2 ) y k = 2 4Al ) 0 x k y l ma m)A la 0 2b l) 0 + 6b l 0 8A ) l 0 + A 0b l ). m 2 A 2 2 ma In view of equations 3.6),3.7),3.8) and Proposition 3.1, we have 3.9) Ḡ i = 1 a0 A il + a 1 y i y l + a 2 B i B l + a 3 y i B l + y l B i ) ] 4 4Al ) 0 ma m)A la 0 m 2 A 2 2b l) 0 + 6b l 0 2 8A ] l 0 + A 0 b l ) ma Proposition 3.2 Let = 2 insler metric on a manifold of dimension n 2, with m 3. Then the spray coefficients Ḡi of n are given by equation 3.9). Remark 3.1 It is remarkable to note that the metric tensors ḡ ij and ḡ ij of n are not necessarily rational functions of y, but the spray coefficients Ḡi of n are rational functions of y. 4 Einstein metrics or a insler metric = x, y), its Riemann curvature R y = Rk i x dx k is defined i by Rk i = 2 Gi x k 2 G i yj x j x k + 2 G i 2Gj y j y k Gi G j y j y k. The insler metric = x, y) is said to be of scalar flag curvature if there is a scalar function K = Kx, y) such that Rk i = Kx, y) 2 δk i ky i ). The Ricci curvature is the trace of the Riemann curvature, Ric = Rk k. In view of the definition of Riemann curvature, Ricci curvature and Remark 3.1, we have Lemma 4.1. Let = 2 be a non-riemannian Kropina change of an m-th root insler metric on a manifolds of dimension n 2, with m 3. Then Rk i and Ric = Rk k are rational functions of y. A insler metric = x, y) on an n-dimensional manifold M is called an Einstein metric if there is a scalar function λ = λx) on M such that Ric = n 1)λ 2. is said to be Ricci constant resp. flat) if λ = constant resp. zero). By definition, every 2-dimensional Riemann metric is an Einstein metric, but generally not of Ricci constant. In dimension n 3, the second Schur Lemma ensures that

6 144 Bankteshwar Tiwari, Ghanashyam Kr. Prajapati every Riemannian Einstein metric must be Ricci constant. In particular, in dimension n = 3, a Riemann metric is Einstein if and only if it is of constant sectional curvature. Proof of Theorem 1.1. By Lemma 4.1, Ric is a rational function of y. Suppose is an Einstein metric, that is Ric = n 1)λ 2 and 2 is not a rational function. Therefore λ = 0. In insler geometry, the flag curvature is an analogue of the sectional curvature from Riemannian geometry. A natural problem is to study and characterize insler metrics of constant flag curvature. There are only three local Riemannian metrics of constant sectional curvature, up to a scaling. However there are lots of non- Riemannian insler metrics of constant flag curvature. or example, the unk metric is positively complete and non-reversible with K = 1 4 and the Hilbert-Klein metric is complete and reversible with K = 1. Clearly, if a insler metric is of constant flag curvature, then it is an Einstein metric. We obtain Corollary 4.2. Let = 2 insler metric on a manifold of dimension n 2, where m 3. If is of constant flag curvature K, then K = 0. Example 1. Let = n y i ) 4 i=1 y j, for any fixed j, 1 j n. By direct computation, we get Ḡi = 0 and Rk i = 0. Thus the flag curvature of is zero. n x i ) 2 y i ) 4 i=1 Example 2. Let = x j y, for any fixed j, 1 j n. By direct j computation, we get Ḡi = yi ) 2 4x and R i i k = 0. Thus the flag curvature of is zero. It is known that every Berwald metric with K = 0 is locally Minkowskian. So is locally Minkowskian. 5 Weak Einstein metrics A weakly Einstein metric is the generalization of the Einstein metric. A insler metric is called a weakly Einstein metric if its Ricci curvature Ric is of the form Ric = n 1) 3θ + λ) 2, where θ is a 1-form and λ = λx) is a scalar function. In general, a weak Einstein metric is not necessarily an Einstein metric and vice versa. Proof of Theorem 1.2. Suppose is a weak Einstein metric, then Ric = n 1)3θ + λ 2 ), where θ is an 1-form and λ = λx) is a scalar function. By Lemma 4.1 Ric is rational function of y. If λ 0, we get = 3n 1)θ± 9n 1) 2 θ 2 +4n 1)λRic 2n 1)λ. On the other hand, = a i 1 i 2...im x)yi 1 y i 2...y im ) 2 m, so we get

7 On Einstein Kropina change of m-th root insler metrics 145 ai1 i 2...i m x)y i 1 y i 2...y m) ) i 2 m 3n 1)θ± 9n 1) = 2 θ 2 +4n 1)λRic 2n 1)λ. Here the left hand side is purely irrational for m 3. Then the right hand side will be irrational if and only if θ = 0. Thus we have that is an Einstein metric. Using Theorem 1.1, we obtain Ric = 0. 6 Scalar flag curvature or a tangent plane P = spany, u), y and u are linearly independent vectors of tangent space T x M of M at point x M, the flag curvature K = KP, u) depends on plane P as well as vector u P. a) A insler metric is of scalar flag curvature if for any non-zero vector y T x M, K = Kx, y) is independent of P containing y T x M. b) is called of almost isotropic flag curvature if K = 3c x m ym + λ, where c = cx) and λ = λx) are some scalar functions on M. c) is of weakly isotropic flag curvature if K = 3θ + λ, where θ is an 1-form and λ = λx) is a scalar function. Clearly, if a insler metric is of weakly isotropic flag curvature, then it is a weak Einstein metric. Lemma 6.1. Let = 2 insler metric on a manifold of dimension n 2, where m 3. If is of almost isotropic flag curvature K, then K = 0. The S-curvature S = Sx, y) in insler geometry was introduced by Shen 8] as a non-riemannian quantity, defined as Sx, y) = d dt τσt), σt))] t=0 where τ = τx, y) is a scalar function on T x M\{0}, called distortion of and σ = σt) is the geodesic with σ0) = x and σ0) = y. A insler metric is called of isotropic S-curvature if S = n + 1)c, for some scalar function c = cx), on M. Theorem ] Let M, ) be an n-dimensional insler manifold of scalar flag curvature Kx, y). Suppose that the S-curvature is isotropic, S = n + 1)cx), then there is a scalar function λx) on M such that K = 3c x m ym + λ. In particular, cx) = c is a constant if and only if K = Kx) is a scalar function on M. In dimension n 3, a insler metric is of isotropic flag curvature if and only if is of constant flag curvature by Schur s Lemma. In general, a insler metric of weakly isotropic flag curvature and that of isotropic flag curvature are not equivalent. Proof of Theorem 1.3. Lemma 6.1 and Theorem 6.2 yield the claimed result.

8 146 Bankteshwar Tiwari, Ghanashyam Kr. Prajapati References 1] P. L. Antonelli, R. S. Ingarden and M. Matsumoto, The Theory of Sprays and insler spaces with Applications in Physics and Biology, Kluwer Academic Publishers, Netherlands, ] G. S. Asanov, inslerian Extension of General Relativity, Reidel, Dordrecht, ] V. Balan and N. Brinzei, Einstein equations for h, v) - Berwald-Moór relativistic models, Balkan. J. Geom. Appl., 11 2) 2006), ] V. Balan, Notable submanifolds in Berwald-Moór spaces, BSG Proc. 17, Geometry Balkan Press, 2010), ] X. Cheng, X. Mo and Z. Shen, On the flag curvature of insler metrics of scalar curvature, J. London Math. Soc ), ] E. Guo, X. Mo, X. Zhang, The explicit construction of Einstein insler metrics with non-constant flag curvature, SIGMA Symmetry Integrability Geom. Methods Appl., ] B. Li and Z. Shen, Projectively flat fourth root insler metrics, Canad. Math. Bull ), ] Z. Shen and S. S. Chern, Riemann-insler Geometry, Nankai Tracts in Mat. 6, World Scientific ] Z. Shen and C. Yu, On a class of Einstein finsler metrics, Int. J. Math ), article Id pp). 10] C. Shibata, On invariant tensors of -changes of insler metrics, J. Math. Kyoto Univ ), ] H. Shimada, On insler spaces with the metric L = m a i1...i m y i 1...y i m, Tensor, N. S ), ] A. Tayebi and B. Najafi, On m-th root insler metrics, J. Geom. Phys ), ] B. Tiwari and M. Kumar,On Randers change of a insler space with m-th root metric, Int. J. Geom. Meth. Mod. Phys ), ). 14] Y. Yu and Y. You, On Einstein m-th root metrics, Diff. Geometry Appl ), Authors address: Bankteshwar Tiwari and Ghanashyam Kr. Prajapati corresponding author) DST-CIMS, aculty of Science, Banaras Hindu University, Varanasi , India. s: banktesht@gmail.com, gspbhu@gmail.com