On Obata theorem in Randers spaces

Differential Geometry and its Applications 49 (2016) 380–387

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Differential Geometry and its Applications www.elsevier.com/locate/difgeo

On Obata theorem in Randers spaces M. Rafie-Rad Department of Mathematics, Faculty of Mathematical Sciences, University of Mazandaran, Babolsar, Iran

a r t i c l e

i n f o

Article history: Received 26 July 2015 Received in revised form 14 April 2016 Available online 12 October 2016 Communicated by Z. Shen

a b s t r a c t The classical Obata theorem in Riemannian space is generalized to the Randers spaces. It is proved that, if the generalized Obata equation on a closed Douglas Randers spaces admits a nontrivial solution, then M is weakly isometric to the Euclidean sphere (Sn (1), h), where, h denotes the standard Riemannian metric of Sn (1). In particular, F is locally projectively flat with positive flag curvature. © 2016 Elsevier B.V. All rights reserved.

MSC: primary 53C60 secondary 53B40 Keywords: Randers metric Weak isometry Douglas space

1. Introduction √ √ The first eigenvalue λ1 of the Laplacian operator on (Sn ( k), h) (the Euclidean sphere of radius 1/ k in Rn+1 ) is nk and each eigenfunction f corresponding to nk satisfies the following system of differential equations (cf. [11]): ∇df + kf h = 0,

k > 0,

(1)

√ where, ∇ denotes the Levi-Civita connection of the induced Riemannian metric h from Rn+1 on Sn ( k). The equation (1) is relevant to the studies on the first eigenvalue of the Laplacian operator on Riemannian manifolds, since it provides that f is an eigenfunction for the first eigenvalue which is valid also in Finsler geometry, see the recent works [15,16]. Lichnérowicz proved in [9] that, given any n-dimensional closed Riemannian manifold with positive constant Ricci curvature k, we have λ1 ≥ nk and Obata proved that, √ the sphere Sn ( k) is unique among all of complete Riemannian manifolds for which the lower bound nk √ is achieved, cf. [11]. It should be noted that the lower bound nk is achieved on Sn ( k) by restricting E-mail address: rafi[email protected]. http://dx.doi.org/10.1016/j.difgeo.2016.09.005 0926-2245/© 2016 Elsevier B.V. All rights reserved.

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√ any linear function on Rn+1 to the unit sphere Sn ( k) (i.e. spherical harmonics of degree one). Any such eigenfunction f , is a solution of (1) and vice-versa, √ cf. [11]; Hence, solvability of (1) can be regarded as a characteristic equation for the Euclidean sphere Sn ( k), (cf. [8]). Assuming a specific curvature condition, Akbar-Zadeh proved also the inequality λ1 ≥ nk for the first eigenvalue of the horizontal Laplacian on the closed and simply connected Finsler manifolds, (cf. [1], p. 95), however, his result yields only homeomorphism to the Euclidean sphere Sn (1). Later, Gallot √ found in [7] several characteristic equations corresponding to n each eigenvalue of the Laplacian on (S ( k), h). Given any nontrivial (i.e. nonconstant) solution f of (1), gradf is a conformal vector field. The characteristic equation (1) can be relaxed on a Riemannian manifold (M, h) to the form ∇df + ϕf h = 0,

(2)

where, ϕ is an appropriate smooth real function. Up to conformal equivalence, Tashiro classified all connected and complete n-dimensional Riemannian manifolds on which the equation (2) has a nontrivial solution into ¯ of an open interval and a complete Riemannian three types: Sn (1), Rn and the warped product J × M ¯ , cf. [14]. Later on, Tanno completed the works of Obata and Tashiro, cf. [13]. A generalization manifold M of the equation (1) on Finsler spaces has been recently studied by Bidabad and Asanjarani in [3] and also in [4] to establish a Finslerian extension of Tashiro’s classification of complete Riemannian spaces as well as Obata theorem. Here, we would like to study the equation (1) in a Randers space. We use the following better – in the sense that it depends only to the geodesic spray coefficient – and equivalent form of the system of PDEs given by (2) on a Finsler space (M, F ), namely: D0 D0 f + ϕf F 2 = 0,

(3)

where, D denotes the Berwald connection and D0 = y i Di stands for the covariant derivative along the canonical geodesic spray. Notice that, the equation (3) is not sensitive to the change of affinely equivalent connections such as Chern, Berwald or Cartan connections; Moreover, equation (3) is equivalent to the following tensorial equation: Di Dj f + ϕf gij = 0.

(4)

Given any Randers metric on M , denote the Levi-Civita connection of α by ∇ and recall the usual symbolic conventions for general (α, β)-metrics, cf. [6]. We give a characterization of the solutions of (3) by following result: Theorem 1. Let us suppose that (M, F = α + β) be a Randers space. The differential equation (3) has a nontrivial solution f if and only if the following three statements hold: (a) F is of isotropic S-curvature S(x, y) = (n + 1)c(x)F (x, y), (b) f is a solution of c∇0 f − ϕf β + si 0 ∇i f = 0, (c) f is a solution of ∇0 ∇0 f + 2∇0 f s0 − 2βsi 0 ∇i f − ϕf (α2 + 3β 2 ) = 0. Two Finsler metrics F and F˜ on a manifold M are said to be weakly isometric if there is a vector field W on M such that the pair (F, W ) solves the Zermelo navigation problem F˜ (x, Fy + Wx ) = 1, x ∈ M, y ∈ ˜ , F˜ ) are said to be weakly isometric if there is diffeomorphism Tx M . Two Finsler manifold (M, F ) and (M ∗˜ φ : M −→ N such that F and φ F are weakly isometric. This terminology was first proposed by Zhongmin Shen and used in [12]. Theorem 1 implies the following result: Theorem 2. Let (M, F = α + β) be a closed and Douglas Randers spaces of dimension n ≥ 2 expressed by the navigation data (h, W ). If the PDE

382

M. Rafie-Rad / Differential Geometry and its Applications 49 (2016) 380–387

where, λ = 1 − h2 (x, Wx ),

D0 D0 f + λf F 2 = 0,

(5)

has a nontrivial solution f then F is (M, F ) is weakly isometric to the Euclidean sphere Sn (1). In particular, F is locally projectively flat. The Randers metrics are supposed to be non-Riemannian in this paper. The exterior differential df of a ∂f function f defined on M is supposed to be the same as its lift D0 f = y i ∂x i as a function defined on the tangent manifold T M . 2. Preliminaries Let M be a smooth and connected manifold of dimension n ≥ 2. Tx M denotes the tangent space of  M at x. The tangent bundle of M is the union of tangent spaces T M := x∈M Tx M . We will denote the elements of T M by (x, y) where y ∈ Tx M . Let us suppose that T M0 = T M \ {0}. The natural projection π : T M0 → M is given by π(x, y) := x. A Finsler metric on M is a function F : T M → [0, ∞) with the following properties: (i) F is C ∞ on T M0 , (ii) F is positively 1-homogeneous on the fibers of tangent bundle T M , and (iii) the Hessian of F 2 with elements gij (x, y) := 12 [F 2 (x, y)]yi yj is positive definite matrix on T M0 . The pair (M, F ) is then called a Finsler space. Throughout this paper, we denote a Riemannian  metric by α = aij (x)y i y j and a 1-form by β = bi (x)y i . A globally defined spray G is induced by F on ∂ ∂ i i T M0 , which in a standard coordinate (xi , y i ) for T M0 is given by G = y i ∂x i − 2G (x, y) ∂y i , where G (x, y) are local functions on T M0 satisfying Gi (x, λy) = λ2 Gi (x, y), λ > 0 and given by Gi =

1 ik h 2 g {y Fxh yk − Fx2k }. 4

(6)

Assume the following conventions: Gi j =

∂Gi , ∂y i

Gi jk =

∂Gi j . ∂y k

Notice that, the local functions Gi jk give rise to a symmetric (or torsion-free) connection on π ∗ T M called the Berwald connection and denoted by D. It is well-known that, any connection on a manifold is naturally associated with a spray whose geodesic spray coefficients Gi are derived from the nonlinear connection coefficient Nji by Gi (x, y) = 12 y j Nji . It is not hard to find out that the associated geodesic spray coefficients of any celebrated connection on a Finsler manifold, e.g. Cartan, Berwald, Chern-Rund, Shen connection, etc, are exactly the same, namely, the natural geodesic spray coefficients given by (6). Now let us suppose that Γkij are the Christoffel symbols of a symmetric connection ∇ with respect to the horizontal local ∂ δ ∂ k j ∂ k i frames δxδ i = ∂x i − Γ ij y ∂y k , i.e. ∇ δ δxj = Γ ij ∂xk ; If w = wi (x)dx is a 1-form of the pull-back bundle i δx

π ∗ T M −→ M , then we have ∇i wj = δxij − wk Γkij = ∂xij − wk Γkij , this formula works also for a 1-form on ∂f M given by wj = ∇j f = ∂x j , where, f is a smooth function on M . Hence, after contracting the two sides i j in y and y , we have δw

∂w

 ∂2f ∂f k  − Γ ∂xi ∂xj ∂xk ij ∂f k i j ∂2f = yi yj i j − Γ yy ∂x ∂x ∂xk ij ∂f k i ∂2f = yi yj i j − N y ∂x ∂x ∂xk i ∂f ∂2f = y i y j i j − 2 k Gk . ∂x ∂x ∂x

∇0 ∇0 f = y i y j ∇i ∇j f = y i y j

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The above argument shows that the term ∇0 ∇0 f depends only to the geodesic spray coefficients Gi and thus, the equation (3) does not depend to the choice of any relevant connection on a Finsler manifold, i.e. Cartan, Chern-Rund, Berwald, connections. Moreover, D0 D0 f is positively homogenous of degree +2 with respect to y as a function of (x, y) ∈ T M . Notice that, any regular connection D should have the property ∇i y j = 0. The interested readers are recommended to see the reference [1] to be introduced with a relevant theory of regular connections on Finsler spaces. Given a Finsler metric F on an n-dimensional manifold M , the Busemann–Hausdorff volume form dVF = σF (x)dx1 · · · dxn is defined by σF (x) :=

Vol(Bn (1)) . ∂ Vol{(y i ) ∈ Rn | F (y i ∂x i |x ) < 1} √ g

Define g = det(gij (x, y)) and τ (x, y) := ln σF (x) . Given a vector y ∈ Tx M , let γ(t), − < t < , denote the d geodesic with γ(0) = x and γ(0) ˙ = y. The function S(x, y) := dt [τ (γ(t), γ(t))] ˙ |t=0 is called the S-curvature with respect to Busemann–Hausdorff volume form. A Finsler space is said to be of isotropic S-curvature if there is a function c = c(x) defined on M such that S = (n + 1)c(x)F . It is called a Finsler space of constant S-curvature when c is constant. Let (M, α) be a Riemannian space and β = bi (x)y i be a 1-form defined on M such that βx := supy∈Tx M β(y)/α(y) < 1. The Finsler metric F = α + β is called a Randers metric on a manifold M . Denote the geodesic spray coefficients of α and F by Giα and Gi , respectively. The Levi-Civita covariant derivative of α is denoted by ∇. Define ∇j bi by (∇j bi )θj := dbi − bj θi j , where ˜ j dxk denote the Levi-Civita connection forms and ∇ denotes its associated covariant θi := dxi and θi j := Γ ik derivation of α. Let us put 1 1 (∇j bi + ∇i bj ), sij := (∇j bi − ∇i bj ), 2 2 := aih shj , sj := bi si j , eij := rij + bi sj + bj si .

rij := si j Then Gi are given by

Gi = Giα +

e

00

2F

 − s0 y i + αsi 0 ,

(7)

where e00 := eij y i y j , s0 := si y i , si 0 := si j y j and Giα denote the geodesic coefficients of α. Let F be a Finsler metric on an n-dimensional manifold and Gi denote the geodesic coefficients of F . ∂ Define Ry = K ik (x, y)dxk ⊗ ∂x i |x : Tx M → Tx M by K ik := 2

2 i 2 i ∂Gi ∂Gi ∂Gj j ∂ G j ∂ G − y + 2G − . ∂xk ∂xj ∂y k ∂y j ∂y k ∂y j ∂y k

The family R := {Ry }y∈T M0 is called the Riemann curvature. The Ricci scalar is denoted by Ric it is defined by Ric := K kk . The Ricci scalar Ric is a generalization of the Ricci tensor in Riemannian geometry. Any Randers metric F = α + β on the manifold M is a solution of the following Zermelo navigation problem: h(x,

y − Wx ) = 1, F

 ∂ where h = hij (x)y i y j is a Riemannian metric and W = W i (x) ∂x i is a vector field with h(x, −Wx ) =  i j hij (x)W (x)W (x) < 1. In fact, α and β are given by √ α=

λh2 + W0 , λ

β=−

W0 , λ

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respectively and moreover, λ = 1 − h(x, −Wx ) and W0 = W i y j hij . Now, F can be written as follows:  F =

λh2 + W02 W0 − . λ λ

(8)

Given a Randers metric F = α + β, the pairs (α, β) and its navigation data (h, W ) are related in the form below: h2 (x, y) = λ(x)(α2 − β 2 ), W0 = −λβ.

(9)

It is well-known that any Randers metric F = α+β on M expressing in terms of a Riemannian metric h and a vector field W (called the navigation data of F ), F is of isotropic S-curvature S = (n +1)c(x)F (x, y) if and only if W is a conformal vector field with LW h = −4c(x)h. We use the following traditional conventions: Rij =

 1 ¯ ¯ i Wj , Sij = 1 ∇j Wi − ∇i Wj , S i = hih Shj , Sj = Sij W i , ∇j Wi + ∇ j 2 2

Rj = Rij W i , R = Rj W j , Ri0 = Rij y j , R00 = Rj y j , Si0 = Sij y j , ¯ denotes the covariant derivative with respect to h. where “∇” Theorem 3. (Obata [11], Theorem A.) In order that a Riemannian manifold (M, h) admits a nontrivial solution f for the system of differential equations √ ∇df + kf h = 0, k > 0, it is necessary and sufficient that √ n M be isometric to an sphere S ( k) of radius 1/ k. 3. Proof of main theorems The technic used here, has been initially appeared in [2] and later on it has been applied in several works. From then, it seems to be such a powerful technic for (α, β)-metrics so that, many interesting results are now proved. This algebraic technic is based on transforming the original equation into the form Rat + αIrrat = 0, where, Rat and Irrat are polynomials in terms of the components of tangent vectors in any given coordinates system. The equation Rat + αIrrat = 0 is itself equivalent to the system of equation  {Rat = 0, Irrat = 0} and this is also equivalent to the following system of equations Rat − βIrrat = 0, Irrat = 0}. Proof of Theorem 1. Let us suppose that (M, F ) is a Randers space and f ∈ C ∞ (M ) is a solution of (3). Taking into account the formula of the geodesic spray of F given in (7), the equation (3) can be re-written as follows: 0 = D0 D0 f + ϕf F 2

e00  − 2s0 ∇0 f − 2αsi 0 ∇i f + ϕf (α + β)2 = ∇0 ∇ 0 f − α+β 1 =− − α∇0 ∇0 f − β∇0 ∇0 f + e00 ∇0 f − 2αs0 ∇0 f − 2βs0 ∇0 f + 2α2 si 0 ∇i f F  + 2αβsi 0 ∇i f − ϕf α3 − 3ϕf α2 β − 3ϕf αβ 2 − ϕf β 3 =−

 1 Rat + αIrrat , F

where, Rat := a0 + a2 α2 , Irrat := a3 α2 + a1 and the terms a0 , ..., a3 are respectively given by:

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385

a3 := −ϕf, a2 := −3ϕf β + 2si 0 ∇i f, a1 := −∇0 ∇0 f − 2s0 ∇0 f + 2βsi 0 ∇i f − 3ϕf β 2 , a0 := −β∇0 ∇0 f + e00 ∇0 f − 2βs0 ∇0 f − ϕf β 3 . Now, the equation (3) is equivalent to the two equations Rat = 0 and Irrat = 0 simultaneously. Moreover, the system of equations Rat = 0 and Irrat = 0 is itself equivalent to the system of equations Rat−βIrrat = 0 and Irrat = 0. After a simple computations, we obtain 0 = Rat − βIrrat = e00 ∇0 f − 2(ϕf β − si 0 ∇i f )(α2 − β 2 ).

(10)

Notice that, the solution f of (3) is supposed to be nontrivial; This is equivalent to ∇0 f = y i fxi = 0, otherwise if ∇0 f = 0, f should be constant due to the connectedness of M . Hence, given any point x ∈ M , the irreducible polynomial α2 − β 2 ∈ R[y 1 , ..., y n ] divides e00 in the form e00 = 2c(x)(α2 − β 2 ), for a function c ∈ C ∞ (M ). Consequently, F is of isotropic S-curvature S = (n + 1)c(x)F . This proves the statement (a) in Theorem 1. Plugging e00 = 2c(x)(α2 − β 2 ) in (10) and eliminating the common term (α2 − β 2 ), (b) in Theorem 1 is proved. The equation (c) is simply as follows: Irrat = a3 α2 + a1 = ∇0 ∇0 f + 2∇0 f s0 − 2βsi 0 ∇i f − ϕf (α2 + 3β 2 ) = 0.  Notice that, (a), (b) and (c) are in fact equivalent to the system of equations Rat −βIrrat = 0, Irrat = 0 and also equivalent conditions for (3) to have a solution f . 2 Remark 4. Theorem 1 follows that if the PDE given by (3) admits a nontrivial solution on a closed and Douglas Randers space (M, F = α + β), then F is of isotropic S-curvature S = (n + 1)c(x)F and also c∇0 f − ϕf β = 0 due to (b). Note that, sij = 0. We may prove that F is not of constant S-curvature; otherwise, closeness of M and constancy of S-curvature yields that S = 0 by Lemma 4.1. in [10] and then it follows that −ϕf β = 0. Using the facts that ϕ = 0 and f = 0, we obtain β = 0 which is impossible. Lemma 5. Let (M, F = α + β) be a n-dimensional Randers spaces of isotropic S-curvature S = (n + 1)c(x)F and with the navigation data of (h, W ). The PDE D0 D0 f + ϕf F 2 = 0,

(11)

has a nontrivial solution f if and only if the following statements hold: ¯ if − 1 S i∇ ¯ i f W0 − c∇ ¯ 0 f − ϕ f W0 = 0 (a) S i0 ∇ λ λ ϕ 1 i ¯ 0∇ ¯ 0f + ( S ∇ ¯ i f + f )h2 = 0, (b) ∇ λ λ ¯ denotes the Levi-Civita connection of h. where, λ(x) = 1 − h2 (x, Wx ) and ∇ Proof. Let us suppose that (M, F = α + β) is a Randers space with the navigation data (h, W ) and λ = 1 − h2 (x, Wx ). Let us assume that F is of isotropic S-curvature S = (n + 1)c(x)F . In this case, the ¯ i of F and h, respectively, are related by [5] (equation (14)) as follows: geodesic spray coefficients Gi and G ¯ i − F S i0 − 1 F 2 S i + cF y i . Gi = G 2

(12)

386

M. Rafie-Rad / Differential Geometry and its Applications 49 (2016) 380–387

Taking into account (12), the equation (11) can be re-written as follows: 0 = D0 D0 f + ϕf F 2 2 i¯ 2 ¯ 0∇ ¯ 0 f + 2F S i ∇ ¯ ¯ =∇ 0 i f + F S ∇i f − 2cF ∇0 f + ϕf F

(13)

= Rat + αIrrat,

(14)

where, i¯ 2 2 i¯ 2 2 ¯ 0∇ ¯ 0 f + 2S i ∇ ¯ ¯ Rat = ∇ 0 i f β + S ∇i f β − 2cβ ∇0 f + ϕf β + S ∇i f α + ϕf α , 

i ¯ i f + βS i ∇ ¯ i f − c∇ ¯ 0 f + ϕf β . Irrat = 2 S ∇ 0

(15)

Hence, the equation (11) has a nontrivial solution f if and only if Rat = 0 and Irrat = 0. Observe that we have 0 = Rat − βIrrat ¯ 0 f + (S i ∇ ¯ i f + ϕf )(α2 − β 2 ). ¯ 0∇ =∇

(16)

The system of equations {Rat = 0, Irrat = 0} is just the equivalent conditions for (11) to have a solution f and it is itself equivalent to the system of equations {Rat − βIrrat = 0, Irrat = 0}. Due to the equation (9), (15) and (16) the equations Irrat = 0 and Rat − βIrrat = 0, can be written as follow: ¯ 0∇ ¯ 0 f + ( 1 S i ∇if ¯ + ϕ f )h2 , 0 = Rat − βIrrat = ∇ λ λ

i  1 ¯ 0 f − ϕ f W0 . ¯ if − S i∇ ¯ i f W0 − c∇ 0 = Irrat = 2 S 0 ∇ λ λ

(17) (18)

This proves Lemma 5. 2 Recall the following proposition in [6]: Proposition 1. Let F = α + β be a Randers metric expressed by the navigation data (h, W ). Then β is closed if and only if  1  Wj Sk − Wk Sj , 1−λ Sk = RWk − (1 − λ)Rk ,

Sjk =

(19) (20)

where, λ(x) = 1 − h2 (x, Wx ). Remark 6. It is known that a Randers metric F = α + β with the navigation data (h, W ) is of Douglas type if and only if β is closed. If moreover, if we assume that F is of isotropic S-curvature S = (n + 1)c(x)F , then Rij = −2chij . Now, we may observe after simple computations that Rk = −2cWk and R = −2c(1 − λ). Plugging these equations in (20) we obtain Sk = 0 and plugging this in (19) it follows that Sjk = 0. As a consequence of Theorem 1, Lemma 5 and Remark 6, it results: Proposition 2. Let (M, F = α + β) be a n-dimensional Douglas Randers spaces expressed by the navigation data (h, W ). The PDE

M. Rafie-Rad / Differential Geometry and its Applications 49 (2016) 380–387

D0 D0 f + ϕf F 2 = 0,

387

(21)

has a nontrivial solution f if and only if the following statements hold: (a) S(x, y) = (n + 1)c(x)F (x, y), ¯ 0 f + ϕf W0 = 0, (b) cλ∇ ¯ 0 f + ϕ f h2 = 0, ¯ 0∇ (c) ∇ λ ¯ denotes the Levi-Civita connection of h. where, λ(x) = 1 − h2 (x, Wx ) and ∇ Proof of Theorem 2. Let us suppose that F is a Douglas metric and (5) has a nontrivial solution f . By Theorem 1, F is of isotropic S-curvature and by Remark 6 we have Sjk = 0 and Sk = 0. Now, from ¯ 0 f + f W0 = 0 and ∇ ¯ 0∇ ¯ 0 f + f h2 = 0, since ϕ = λ. From Theorem 3, it Proposition 2, it results that c∇ follows that (M, h) is isometric to the n-dimensional Euclidean sphere Sn (1). In particular, h is of positive constant sectional curvature +1 and in particular h is locally projectively flat. On the other hand, by equation (12) it follows that F and h are projectively equivalent and this follows immediately that F is also locally projectively flat with isotropic S-curvature. Finally, it results that (M, F ) is weakly isometric to the Euclidean sphere Sn (1). 2 Acknowledgements I would like sincerely to appreciate Dr. Bahman Rezaei (Urmia University) and Dr. Behzad Najafi (Amirkabir University of Technology) for their comments. References [1] H. Akbar-Zadeh, Initiation to Global Finslerian Geometry, North Holland, 2006. [2] D. Bao, C. Robles, On Randers spaces of constant flag curvature, Rep. Math. Phys. 53 (2003). [3] B. Bidabad, A. Asanjarani, Classification of complete Finsler manifolds through a second order differential equation, Differ. Geom. Appl. 26 (2008) 434–444. [4] B. Bidabad, Obata theorem on compact Finsler spaces, Balk. J. Geom. Appl. 17 (2) (2012) 1–5. [5] X. Cheng, Z. Shen, Randers metrics of scalar flag curvature, J. Aust. Math. Soc. 87 (2009) 359–370. [6] X. Cheng, Z. Shen, Finsler Geometry – An Approach Via Randers Spaces, Springer, 2012. [7] S. Gallot, Équations différentielles caratéristiques de la sphère, Ann. Sci. Ec. Norm. Super., 4 Sér. 12 (1979) 235–267. [8] E. García-Río, D.N. Kupeli, B. Ünal, On a differential equation characterizing Euclidean spheres, J. Differ. Equ. 194 (2003) 287–299. [9] A. Lichnérowicz, Géométrie des groupes de Transformations, Dunod, Paris, 1958. [10] X. Mo, A global classification result for Randers metrics of scalar curvature on closed manifolds, Nonlinear Anal. 69 (2008) 2996–3004. [11] M. Obata, Certain conditions for a Riemannian manifold to be isometric with a sphere, J. Math. Soc. Jpn. 14 (3) (1962). [12] M. Rafie-Rad, Weakly conformal Finsler geometry, Math. Nachr. 14–15 (2014) 1745–1755. [13] S. Tanno, Some differential equations on Riemannian manifolds, J. Math. Soc. Jpn. 30 (3) (1978). [14] Y. Tashiro, Complete Riemannian manifolds and some vector fields, Trans. Am. Math. Soc. 117 (1965) 251–275. [15] S. Yin, Q. He, Y. Shen, On the first eigenvalue of Finsler manifolds with nonnegative weighted Ricci curvature, Sci. China Math. 5 (57) (2014) 1057–1070. [16] Q. Xia, A sharp lower bound for the first eigenvalue on Finsler manifolds with nonnegative weighted Ricci curvature, Nonlinear Anal. 117 (2015) 189–199.