WITH SOME CURVATURE PROPERTIES

Transcription

1 Khayyam J. Math. DOI:10.034/kjm ON GENERAL (, β)-metrics WITH SOME CURVATURE PROPERTIES BANKTESHWAR TIWARI 1, RANADIP GANGOPADHYAY 1 GHANASHYAM KR. PRAJAPATI AND Communicated by B. Mashayekhy Abstract. In this paper, we study a class of Finsler metric called general (, β) metrics and obtain an equation that characterizes these Finsler metrics of almost vanishing H-curvature. As a consequence of this result, we prove that a general (, β)-metric has almost vanishing H-curvature if and only if it has almost vanishing Ξ-curvature. 1. Introduction One of the finest differential geometer of twentieth century, Chern, used to say Finsler geometry is just Riemannian geometry without the quadratic restriction on its metrics [7]. In the study of Finsler geometry, we often encounter long and complicated calculations. However, when we consider Finsler metrics with certain symmetries, that would make things much easier. In 1996, Rutz [14] introduced a special class of Finsler metrics called spherically symmetric which is invariant under rotation. In general relativity, the solution of vacuum Einstein field equations describing the gravitational field, which is spherically symmetric, we obtain the Schwarzschild metric in four-dimensional space-time [1]. A Finsler metric F on B n (δ) is called spherically symmetric if F (Ax, Ay) = F (x, y), for all n n orthogonal matrix A, x = (x i ) B n (δ) and y = (y i ) T x B n (δ). Here B n (δ) denotes the Euclidean ball of radius δ around the origin and T x B n (δ) denotes the tangent space of B n (δ) at the point x. Zhou [] proved that a Date: Received: 4 September 018; Revised: 19 December 018; Accepted: January 019. Corresponding author. 010 Mathematics Subject Classification. Primary 53B40; Secondary 53B50. Key words and phrases. Finsler space, General (, β)-metric, Ξ-curvature, H-curvature. 1

2 B. TIWARI, R. GANGOPADHYAY, G.K. PRAJAPATI Finsler metric F on B n (δ) is spherically symmetric if and only if there exists a function φ : [0, δ) R R such that ( ) x, y F (x, y) = y φ x,, (1.1) y where. denotes the Euclidean norm and, denotes the Euclidean inner product on R n. The concept of (, β)-metrics was introduced by Matsumoto in 197 [9] as a generalization of Randers metrics, and the Randers metrics was introduced by Randers [13]. The (, β)-metrics are of the form F = φ(s), where φ is a C positive function and s = β. In 01, Yu and Zhu [0] introduced a new class of Finsler metrics, called general (, β)-finsler metrics given by F = φ(b, s), where φ = φ(b, s) is a C positive function and b := β. This class of Finsler metrics not only generalize (, β)-metrics in a natural way, but also includes spherically symmetric Finsler metrics. It is interesting to note that the general (, β) metric includes an interesting family of Finsler metric constructed by Bryant [5, 4, 3]. Bryant metrics are rectilinear Finsler metrics on the unit sphere S n with flag curvature K = 1 defined by { } Q(X, X)Q(Y, Y ) Q(X, Y ) F (X, Y ) = R i Q(X, Y ), Q(X, X) Q(X, X) where Q(X, Y ) = x 0 y 0 +e ip 1 x 1 y 1 +e ip x y + +e ipn x n y n is a complex quadratic form on R n+1 for n with the parameters satisfying 0 p 1 p p n < π and X = (x 0,..., x n ) S n, Y = (y 0,..., y n ) T X S n. Shen et al. showed that a Randers metric is locally projectively flat and of constant flag curvature if and only if is locally projectively flat and β is closed and homothetic with respect to [, 15]. The Beltrami theorem says that a Riemannian metric is locally projectively flat if and only if it has a constant sectional curvature. Thus in this case and β have to satisfy the following: R i j = µ( δ i j y i y j ), b i j = c(x)a ij, (1.) where R i j denotes the Riemann curvature of the Riemannian metric and µ is the Ricci constant. Shen [18] also showed that a general (, β)-metric satisfying (1.) will be projectively flat if and only if φ satisfies φ ss = (φ b sφ b s). The spherically symmetric metric F given by (1.1) satisfies the property (1.) automatically. The Cartan torsion, the S-curvature, the Ξ-curvature and the H-curvature are the examples of few non-riemannian quantities in Finsler geometry as they vanish for Riemannian metrics. The S-curvature S(x, y) was introduced by Shen [6, 16] and was defined as follows: S(x, y) = d dt [τ(γ(t), γ (t))] t=0, where τ(x, y) is the distortion of the metric F and γ(t) is the geodesic with γ(0) = x and γ (0) = y on M.

3 ON GENERAL (, β)-metrics WITH SOME CURVATURE PROPERTIES 3 The non-riemannian quantity Ξ-curvature is denoted by Ξ = Ξ j dx j and is defined as Ξ j := S.j i y i S j, where denotes the horizontal covariant derivative and. denotes the vertical covariant derivative of F [16]. The Finsler metric F is said to have almost vanishing Ξ-curvature if ( ) θ Ξ j = (n + 1)F. (1.3) F y j In 1988, Akbar-Zadeh introduced H-curvature which is closely related to the S-curvature [1]. The H-curvature H y = H ij dx i dx j is defined by H ij = 1 4 (Ξ i.j + Ξ j.i ). (1.4) Also F is said to have almost vanishing H-curvature if H ij = n + 1 θf y i yj. (1.5) Several authors studied the H-curvature of different class of Finsler metrics [10, 1]. In [11], Mo proved that all spherically symmetric Finsler metrics of almost vanishing H-curvature are of almost vanishing Ξ-curvature and corresponding one forms are exact, generalizing a result previously only known in the case of metrics with vanishing H-curvature. In general, it is difficult to find the Riemann curvature tensor for general (, β)-metrics. In this paper, we further generalize Mo s result for general (, β)-metrics under the assumption (1.) and prove the following results. Theorem 1.1. The general (, β)-metric F = φ(b, s) satisfying (1.) has almost vanishing H-curvature if and only if [ s (n + 1) R 1 s + 3(b s ) R ] s + (n + 1)R 3 = 3(n+1)θ(φ sφ s ), θ = θ j (x)y j, where R 1, R, and R 3 are given in (.6), (.9), and (.8), respectively. As an application of Theorem 1.1, we have the following corollary. (1.6) Corollary 1.. For the general (, β)-metric F = φ(b, s) satisfying (1.) the H-curvature almost vanishes if and only if the Ξ-curvature almost vanishes. In this case, the corresponding 1-form θ is an exact form. As a consequence of Corollary 1., for θ = 0, we get the following corollary. Corollary 1.3. For the general (, β)-metric F = φ(b, s) satisfying (1.) the H-curvature vanishes if and only if the Ξ-curvature vanishes. A Finsler metric is said to be R-quadratic if its Riemann curvature R y is quadratic in y T x M. These R-quadratic Finsler metrics always have vanishing H-curvature [10]. Together with Corollary 1.3, we have the following. Corollary 1.4. The Ξ-curvature of a R-quadratic general (, β)-metric always vanishes.

4 4 B. TIWARI, R. GANGOPADHYAY, G.K. PRAJAPATI. preliminaries Let M be an n-dimensional smooth manifold. T x M denotes the tangent space of M at x. The tangent bundle of M is the union of tangent spaces T M := x M T xm. We denote the elements of T M by (x, y) where y T x M and T M 0 := T M \ {0}. Definition.1 (see [8]). A Finsler metric on M is a function F : T M [0, ) satisfying the following conditions: (i) F is C on T M 0. (ii) F is a positively 1-homogeneous on the fibers of tangent bundle T M. (iii) The Hessian of F with element g ij = 1 F is positive definite on T M y i y j 0. The pair (M, F ) is called a Finsler space. The metric F is called the fundamental function and g ij is called the fundamental tensor. The spray coefficients of the Finsler metric F is defined by where g ij = (g ij ) 1. G i = 1 4 gil { [F ] x m y lym [F ] x l}, (.1) Definition. (see[0]). A Finsler metric F on a manifold M is called a general (, β)-metric, if it can be expressed in the form F = φ ( b, s ), (.) for some C function φ(b, s), where is a Riemannian metric and β is a 1-form. Also F is called an (, β)-metric, if F can be expressed as F = φ (s) for some C function φ(s), Riemannian metric, and 1-form β. You and Zhu [0] have proved that a general (, β)-metric F = φ (b, s) satisfies or φ sφ s > 0, φ sφ s + (b s )φ ss > 0, for, n 3, φ sφ s + (b s )φ ss > 0, for n =, where s and b are arbitrary numbers with s b < b 0. Here φ s denotes the differentiation of φ with respect to s. For a general (, β)-metric F = φ (b, s), the fundamental tensor g ij is given by [0] g ij = ρa ij + ρ 0 b i b j + ρ 1 ( bi y j + b j y i) sρ1 y i y j, (.3) where ρ = φ (φ sφ s ), ρ 0 = φφ ss + φ s φ s, ρ 1 = (φ sφ s ) φ s sφφ ss. Moreover, det (g ij ) = φ n+1 (φ sφ s ) n (φ sφ s + (b s ) φ ss ) det (a ij ), and the inverse metric tensor g ij is given by g ij = ρ 1 { a ij + ηb i b j + η 0 1 ( b i y j + b j y i) + η 1 y i y j}, (.4)

5 ON GENERAL (, β)-metrics WITH SOME CURVATURE PROPERTIES 5 where (a ij ) = (a ij ) 1, b i = a ij b j, φ ss η =, η φ sφ s + (b s 0 = (φ sφ s) φ s sφφ ss ) φ ss φ (φ sφ s + (b s ) φ ss ), η 1 = (sφ + (b s ) φ s ) ((φ sφ s ) φ s sφφ ss ). φ (φ sφ s + (b s ) φ ss ) For any x M and y T x (M) \ {0} the Riemann curvature R y = Rk i dx k x i of F is defined by R i k = Gi x k G i x m y k ym + G m G i x k y Gi G m m y m y. k The Riemann curvature tensor R i j of the general (, β)-metric under the assumption (1.) is given by [19] where R i j = R 1 δ i j sr y j b i + R b j b i + R 3 b j y i + R 4 y j y i, (.5) R 1 = µ(1 + sψ) + c [ψ sψ b ψ + χ(1 + sψ + uψ s )], (.6) R = µ(χ sχ s )+c [(ψ b sψ b s) χ ss +χ(χ sχ s )+u(χχ ss χ s)] (.7) R 3 = µ(ψ sψ s ) + c [(ψ b sψ b s) ψψ s ψ ss +χ(ψ sψ s + uψ ss ) χ s (1 + ssψ + uψ s )], R 4 = µ[1 s(ψ sψ s )] + c [ψ s + sψ ss ψ(ψ sψ s ) s(ψ b sψ b s) with χ = χ(1 + sψ + uψ s ) + sχ (1 + sψ + uψ s ) sχ(ψ sψ s ψ ss )], φ ss (φ 1 sφ b s), ψ = φ s + sφ b (φ sφ s ) + (b s )φ ss φ One can observe that here χ φ [sφ + (b s )φ s ]. (.8) (.9) c = k µb and R 1 + R 4 + sr 3 = 0 (.10) for some constant k. Therefore, using (.10) in (.5), we have R i j = R 1 ( δ i j y j y i ) + R (b j sy j )b i + R 3 (b j sy j )y i. (.11) 3. The Ξ-curvature and H-curvature of a general (, β)-metric In this section, we find the expressions of non-riemannian quantities Ξ and H of general (, β) Finsler metrics. The Ricci curvature Ric is defined by Ric = R i i, and for a general (, β)-metrics, Ric can be obtained as Ric = R i i = R 1 ( δ i i y i y i ) + R (b i sy i )b i + R 3 (b i sy i )y i where = [(n 1)R 1 + (b s )R ] = R, (3.1) R = (n 1)R 1 + (b s )R. (3.)

6 6 B. TIWARI, R. GANGOPADHYAY, G.K. PRAJAPATI By some simple calculations, we can obtain the following results: and Using (3.) and (3.3), we have y j Ric = y j = y j, s y j = b j sy j (3.3) s y jy j = 0, s y jb j = b s. (3.4) y j R = R s b j + (R sr s )y j, (3.5) where R s = R s. Now differentiating (.11) with respect to y i and using (3.3) and (3.4) and then taking the summation over i, we have i Let and R i j y i = [ R 1s + sr + (b s )R s + (n + 1)R 3 ] bj + [ (1 n)r 1 sr 1s b R s(b s )R s (n + 1)sR 3 ] yj. M = R 1s + sr + (b s )R s + (n + 1)R 3 N = (1 n)r 1 sr 1s b R s(b s )R s (n + 1)sR 3. Therefore, (3.6) becomes i (3.6) R i j y i = Mb j + Ny j. (3.7) We will use the following lemma to calculate the Ξ-curvature. Lemma 3.1 (see[10, 16]). The Ξ-curvature of a Finsler metric F is given by ( Ξ j = 1 ) Rj i + 3 y i y Ric. (3.8) j i Plugging (3.5) and (3.7) into (3.8), we obtain From (3.), we have Then we can have and Ξ j = 1 3 [(M + R s)b j + (N + R sr s )y j ]. (3.9) R s = (n 1)R 1s + (b s )R s sr s. (M + R s ) = (n + 1)R 1s + 3(b s )R s + (n + 1)R 3 := κ. (3.10) N + R sr s = sκ. (3.11)

7 ON GENERAL (, β)-metrics WITH SOME CURVATURE PROPERTIES 7 Substituting (3.10) and (3.11) into (3.9), we obtain the following formula for Ξ as: Ξ j = κ 3 (b j sy j ), (3.1) where κ is given as in (3.10). Now differentiating (3.1) with respect to y i and using (3.3), we have Ξ j.i = κ s 3 (b i sy i )(b j sy j ) κ ( bj y i b i y j + s ) 3 y iy j sa ij, where κ s := κ s. Therefore, from (1.4), we have H ij = κ s 6 (b i sy i )(b j sy j ) sκ 6 = 1 6 ( yi y j a ij [ sκa ij κ s b i b j + sκ s (b jy i + b i y j ) s ] (κ + sκ s)y i y j. 4. Almost vanishing H-curvature In this section, we prove Theorem 1.1 and Corollary 1.. Using (3.3), we obtain ) y i y j = a ij y i y j 3. (4.1) s y i y j = 3sy iy j b i y j b j y i s a ij 4. (4.) Proof of Theorem 1.1. Differentiating (.) with respect to y i, we have Differentiating again (4.3) with respect to y j yields F y i = y iφ + φ s s y i. (4.3) F y i y j = y i y jφ + ( y is y j + y js y i)φ s + s y is y jφ ss + s y i y jφ s. (4.4) Plugging (3.3),(4.1),(4.) into (4.4) yields F y i y j = 1 3 [ (φ sφs ) a ij + φ ss b i b j sφ ss (b i y j + b j y i ) (φ sφ s s φ ss )y i y j ]. In the view of (1.5), the general (, β)-metric is of almost vanishing H-curvature if and only if sκa ij κ s b i b j + sκ s (b jy i + b i y j ) s (κ + sκ s)y i y j 3(n + 1)θ [ = (φ sφs ) a 3 ij + φ ss b i b j sφ ss (b i y j + b j y i ) ] (φ sφ s s φ ss )y i y j. (4.5) Now equating the similar coefficients of both sides of (4.5), we have the following equations: 3(n + 1)θ sκ = (φ sφ s ), (4.6) 3(n + 1)θ κ s = φ ss, (4.7)

8 8 B. TIWARI, R. GANGOPADHYAY, G.K. PRAJAPATI 3(n + 1)θ sκ s = sφ ss, (4.8) 3(n + 1)θ s(κ + sκ s ) = (φ sφ s s φ ss ). (4.9) At first we show that (4.6) implies (4.7), (4.8), and (4.9). Suppose (4.6) holds. Since F is a Finsler metric, we have φ sφ s > 0. Since s = β, the 1-form θ can be expressed by κ θ = β. (4.10) 3(n + 1)(φ sφ s ) Furthermore, κ is independent of y. In fact, it depends only on 3(n+1)(φ sφ b s). Let κ 3(n + 1)(φ sφ s ) := σ ( b ). (4.11) Therefore, from (4.10) and (4.11), we have ( ) b θ = σ β. (4.1) As s = β, we have θ ( b = sσ By using (4.11) and (4.13), we obtain [ ( ) ] b κ s = 3(n + 1)σ (φ sφ s ) ). (4.13) s = 3(n + 1) θ φ ss. Thus, we get (4.7). Now multiplying (4.7) by s yields (4.8). Equation (4.9) can be obtained easily from (4.6) and (4.7). Now substitute the value of κ from (3.10) into (4.6), we get (1.6), which proves the theorem. Proof of Corollary 1.. It is sufficient to show that Ξ-curvature almost vanishes if the H-curvature almost vanishes and in this case corresponding 1-form is exact. Suppose that a general (, β)-metric F = φ(b, β ) has almost vanishing H- curvature. Then (4.6), (4.11), and (4.1) hold. By using (4.1), we have [ ( )] ( ) ( ) ( ) b b d f = f b b d = σ b j db j = θ, where f(t) := σ(t)dt. Hence θ is an exact form. Using (3.3) into (4.3) yields j Using (4.1), we get F y j ( ) θ = F y j = φ s b j + φ φ s y j. σ ( b ) F (φ sφ s )(b j sy j ). (4.14)

9 Now we have ON GENERAL (, β)-metrics WITH SOME CURVATURE PROPERTIES 9 Ξ j = κ 3 (b j sy j ) (from (3.1)) = (n + 1)(φ sφ s )σ( b )(b j sy j ) (from (4.11)) ( ) θ = (n + 1)F. (using (4.14)) F y j Hence we have the proof of Corollary 1.. Acknowledgement. The second author expresses his sincere thanks to the University Grant Commission for funding the joint CSIR-UGC Junior Research Fellowship Ref. No: 3/1/01 (ii) EU-V, Sr. No References 1. H. Akbar-Zadeh, Sur les espaces de Finsler à courbures sectionnelles constantes, (French) [On Finsler spaces with constant sectional curvature] Bull. Acad. Roy. Bel. Bull. Cl. Sci. (5) 74 (1988), no. 10, D. Bao, C. Robbles, Z. Shen, Zermelo navigation on Riemannian manifolds, J. Differential Geom. 66 (004) R. Bryant, Finsler structures on the -sphere satisfying K = 1, in: Finsler Geometry (Seattle, WA, 1995), pp. 7 41, Contemp. Math. 196, Amer. Math. Soc. Providence, RI, R. Bryant, Projectively flat Finsler -spheres of constant flag curvature, Selecta Math. (N.S.) 3 (1997) R. Bryant, Some remarks on Finsler manifolds with constant flag curvature, Houston J. Math. 8 (00), no., X. Cheng, Z. Shen, Finsler Geometry. An Approach via Randers Spaces, Science Press Beijing, Beijing; Springer, Heidelberg, S.S. Chern, Finsler geometry is just Riemannian geometry without the quadratic restriction, Notices Amer. Math. Soc. 43 (1996) S.S. Chern, Z. Shen, Riemannian-Finsler Geometry, World Scientific, Singapore, M. Matsumoto, On C-reducible Finsler spaces, Tensor (N.S.) 4 (197) X. Mo, On the non-riemannian quantity H of a Finsler metric, Differential Geom. Appl. 7 (009), X. Mo, A class of Finsler metrics with almost vanishing H-curvature, Balkan J. Geom. Appl. 1 (016), no. 1, B. Najafi, Z. Shen, A. Tayebi, Finsler metrics of scalar flag curvature with special non- Riemannian curvature properties, Geom. Dedicata 131 (008) G. Randers, On an asymmetrical metric in the fourspace of general relativity, Phys. Rev. 59 (1941) S.F. Rutz, Symmetry in Finsler spaces, in: Finsler Geometry (Seattle, WA, 1995), pp , Contemp. Math. 196, Amer. Math. Soc. Providence, RI, Z. Shen, Projectively flat Randers metrics with constant flag curvature, Math. Ann. 35 (003) Z. Shen, On some non-riemannian quantities in Finsler geometry, Canad. Math. Bull. 56 (013) Z. Shen, C. Yu, On a class of Einstein Finsler metrics, Internat. J. Math. 5 (014), no. 4, 18 pp. 18. D. Tang, On the non-riemannian quantity H in Finsler geometry, Differential Geom. appl. 9 (011),

10 10 B. TIWARI, R. GANGOPADHYAY, G.K. PRAJAPATI 19. Q. Xia On a Class of Finsler Metrics of Scalar Flag Curvature, Results Math. 71 (017), C. Yu, H. Zhu, On a new class of Finsler metrics, Differential Geom. Appl. 9 (011), Y.Q. Yun, An Introduction to General Relativity (Chinese), Peking University Press, L. Zhou, Spherically symmetric Finsler metrics in R n, Publ. Math. Debrecen 80 (01), no. 1-, DST-CIMS, Institute of Science, Banaras Hindu University, Varanasi-1005, India address: banktesht@gmail.com; gangulyranadip@gmail.com Loknayak Jai Prakash Institute of Technology, Chhapra-84130, India address: gspbhu@gmail.com