© Hindawi Publishing Corp. RANDERS MANIFOLDS OF POSITIVE CONSTANT

We prove that any simply connected and complete Riemannian 
manifold, on which a Randers metric of positive constant flag curvature exists, must be diffeomorphic to an odd-dimensional sphere, provided a certain 1-form vanishes on it.


Introduction.
The geometry of Finsler manifolds of constant flag curvature is one of the fundamental subjects in Finsler geometry.Akbar-Zadeh [1] proved that, under some conditions on the growth of the Cartan tensor, a Finsler manifold of constant flag curvature K is locally Minkowskian if K = 0 and Riemannian if K = −1.So far, the case K > 0 is the least understood.Bryant [9] has constructed interesting Finsler metrics of positive constant flag curvature on the sphere S 2 .Recently, Bao and Shen [5] constructed nonprojectively flat Randers metrics of constant flag curvature K > 1 on the sphere S 3 .The present authors have extended the Bao-Shen result to higher dimensions (cf.Bejancu and Farran [7]).We proved that, for any constant K > 0, there exists a Randers metric on the tangent bundle of the unit sphere S 2n+1 , n ≥ 1, such that the Finsler manifold F 2n+1 = (S 2n+1 ,F) has constant flag curvature K and is not projectively flat.Recently, Shen [13,14] constructed interesting examples of Randers manifolds of constant curvature, and Bao and Robles [4] found necessary and sufficient conditions for a Randers manifold to have constant flag curvature.The purpose of the present paper is to show that, subject to some natural conditions, Randers manifolds of positive constant flag curvature are diffeomorphic to odd-dimensional spheres.More precisely, we prove Theorem 2.2.
The proof we give to this theorem reveals a surprising relationship between Randers manifolds of positive constant flag curvature and Sasakian space forms.

Finsler manifolds of constant flag curvature.
In the first part of this section, we present the concept of Finsler manifold of constant flag curvature.Then, we consider Randers manifolds and present the Yasuda-Shimada theorem [17] on Randers manifolds of positive constant curvature.Finally, we state the main result of the paper.
Throughout the paper, we denote by Ᏺ(M) the algebra of differentiable functions on M and by Γ (E) the Ᏺ(M)-module of the sections of a vector bundle E over M. Also, we make use of Einstein convention, that is, repeated indices with one upper index and one lower index denote summation over their range.
Let F m = (M, F ) be a Finsler manifold, where M is an m-dimensional C ∞ manifold and F is the Finsler metric of F m .Here, F is supposed to be a C ∞ function on the slit tangent bundle T M 0 = T M \ {0} satisfying the following conditions: (i) F(x,ky) = kF (x, y), for any x ∈ M, y ∈ T x M, and k > 0; (ii) the m × m Hessian matrix is positive definite at every point (x, y) of T M 0 .We denote by (x i ,y i ) the coordinates on T M 0 , where (x i ) are the coordinates on M. The local frame field on T M 0 is {∂/∂x i ,∂/∂y i }.Then, the Liouville vector field L = y i (∂/∂y i ) is a global section of the vertical vector bundle V T M 0 .Moreover, = (1/F )L is a unit Finsler vector field, that is, we have A complementary vector bundle to V T M 0 in T T M 0 is called a nonlinear connection.The canonical nonlinear connection of F m is the distribution GT M 0 whose local frame field is given by (see Bejancu and Farran [6,page 37]) where we set (2.4) The local coefficients G j i are used to define the following Finsler tensor fields: (2.5) Next, we consider a flag ∧V at x ∈ M determined by and the tangent vector V = V i (∂/∂x i ).Then, according to Bao et al. [3, page 69], the flag curvature for the flag ∧ V is the number where h ij are the local components of the angular metric on F m given by (2.8) A special Finsler metric was considered by Randers [12].To define it, we suppose that M is an m-dimensional manifold endowed with a Riemannian metric a = (a ij (x)) and a nowhere zero 1-form b = (b i (x)).Then, we define on T M 0 the function (2.9) where b i (x) = a ij (x)b j (x), and [a ij (x)] is the inverse matrix of [a ij (x)].A Finsler metric given by (2.9) is called a Randers metric, and F m = (M,F,a ij ,b i ) is called a Randers manifold.Next, we consider the 1-form where the covariant derivative is taken with respect to Levi-Civita connection on M. In dimensions 2 and 3, Shen [13,14] constructed examples of Randers manifolds whose flag curvature is constant and β ≠ 0 on M.This motivated Bao and Robles [4] to determine necessary and sufficient conditions for a Randers manifold to have constant flag curvature.Also they proved that Yasuda-Shimada theorem [17] is true with the additional condition β = 0 on M. From these papers, we need the following result. (2.13) We should note that the above local components R hijk are taken as follows where R is the curvature tensor of Levi-Civita connection ∇ on (M, a), and it is given by for any of the vector fields X, Y , Z on M. As any Randers manifold of dimension m = 1 is a Riemannian manifold, from now on we consider m > 1.
Apart from the conditions we put in Theorem 2.1, we find in Matsumoto [11] the condition (2.16) We show here that (2.16) is a consequence of conditions (i), (ii), and (iii).First, from (2.12) we deduce that B = b i (x)(∂/∂x i ) is a Killing vector field on M. Thus, we have (cf.Yano and Kon [16, page 268]) which, in local coordinates, is expressed as follows: Next, from (2.10) and taking into account (i), we deduce that Then, contracting R hijk by b j and taking into account (2.12) and (2.19), we obtain (2.20) Thus, from (2.18) and (2.20), it follows (2.16).
We make use of (2.16) in the proof of Lemma 4.2, which is crucial for proving our main result which is stated as follows.
Theorem 2.2.Let F m = (M,F,a ij ,b i ) be an m-dimensional Randers manifold of positive constant flag curvature with β = 0 on M. Then m must be an odd number 2n+1.Moreover, M is a Sasakian space form that is isomorphic to the sphere S 2n+1 , provided it is a simply connected and complete manifold with respect to the Riemannian metric a = (a ij ).

Sasakian space forms.
Let M be a (2n + 1)-dimensional differentiable manifold and ϕ, ξ, and η be a tensor field of type (1, 1), a vector field, and a 1-form, respectively, on M, satisfying where I is the identity map on Γ (T M).Then, we say that M has a (ϕ,ξ,η)structure.It is proved that we have (cf.Blair [8, pages 20, 21]) ) Also, there exists a Riemannian metric a on M such that Taking Y = ξ in (3.3) and using (3.1b) and (3.2a), we obtain Similarly, replace Y by ϕX in (3.3), and using (3.1a), (3.2b), and (3.4), we deduce that The manifold M endowed with a (ϕ,ξ,η,a)-structure is a Sasakian manifold if and only if the above tensor fields satisfy (cf.Blair [8, page 73]) where ∇ is the Levi-Civita connection with respect to the Riemannian metric a.
The following result on the existence of Sasakian structures on Riemannian manifolds will be used later in the paper.Theorem 3.1 (Hatakeyama et al. [10]).Let (M, a) be a (2n +1)-dimensional Riemannian manifold admitting a unit Killing vector field ξ such that where R is the curvature tensor of the Levi-Civita connection on M.Then, M is a Sasakian manifold.
We need the local expression of (3.7).To this end, we take X = ∂/∂x j , Y = ∂/∂x i , and ξ = ξ k (∂/∂x k ).Then, using (2.15) and (3.4), we deduce that (3.7) is equivalent to where Next, we denote by {ξ} the line distribution spanned by ξ on M.Then, the orthogonal complementary distribution to {ξ} is denoted by {ξ} ⊥ and is called the contact distribution on M. A plane section in T x M is called a ϕ-section if there exists a vector X ∈ {ξ} ⊥ x such that {X, ϕX} is an orthonormal basis of the plane section.The sectional curvature H(X), determined by the ϕ-section span{X, ϕX}, is called a ϕ-sectional curvature.Thus, we have for any unit vector X in {ξ} ⊥ x .A Sasakian manifold M of constant ϕ-sectional curvature c is called a Sasakian space form, and it is denoted by M(c).There are many examples of Sasakian space forms in the literature (see Blair [8, page 99]).However, here we are interested in examples of Sasakian space forms M(c) with c > −3.It was proved by Tanno [15] that, for any ε > 0, there exists on the unit sphere S 2n+1 a structure of Sasakian space form of constant ϕsectional curvature c = −3+4/ε.We denote this Sasakian space form structure by S 2n+1 (c).The same author proved the following theorem.Theorem 3.2 (Tanno [15]).Let M(c) be a (2n + 1)-dimensional simply connected and complete Sasakian manifold with constant ϕ-sectional curvature c > −3.Then, M is isomorphic to S 2n+1 (c).
Here, "M isomorphic to S 2n+1 (c)" means that M is diffeomorphic to S 2n+1 , and the diffeomorphism maps the structure tensors on M(c) into the corresponding structure tensors on S 2n+1 (c).

Proof of the main result.
In the present section, we prove Theorem 2.2.The proof is based on a striking similitude we discovered between Randers manifolds of positive constant flag curvature and a special class of Sasakian space forms.First, we prove the following lemma.Proof.First, we define on M the Riemannian metric and the 1-form Then, the function is a new Randers metric on T M 0 .Also, (4.3) and (2.1) imply that Next, using (2.4), (4.3), and (4.4), we deduce that F and F * define the same canonical nonlinear connection, that is, we have As a consequence, (2.5), (4.3), and (4.4a) yield Moreover, using (2.8) for both F and F * and taking into account (4.3) and (4.4a), we infer that Finally, since F * is a Randers metric of positive constant flag curvature K * , by (2.7) we have Thus, (4.5), (4.6), and (4.7) imply that Next, we consider a Randers manifold F m = (M,F,a ij ,b i ) of constant flag curvature K = 1 and β = 0. Our purpose is to prove that M is a Sasakian space form.To this end, we first define on M a new 1-form η = (1/ b )b.Clearly, η is a unit 1-form, that is, we have Also, from (2.12), (2.13), and (2.16), we obtain η i|j + η j|i = 0, (4.9) (4.10) respectively, since b is a constant on M. Now, we define on M the unit vector field ξ = ξ i (∂/∂x i ), where we set Then, using (4.12), (4.9), and (4.11), we deduce that The distribution {ξ} ⊥ that is complementary orthogonal to the line distribution {ξ} on M is called, as in the case of Sasakian manifolds, the contact distribution on M. It is easy to see that X ∈ Γ ({ξ} ⊥ ) if and only if η(X) = 0. Now, we prove the following important lemmas.Proof.Using the Levi-Civita connection on M with respect to the Riemannian metric a = (a ij ) and the vector field ξ, we define on M a tensor field ϕ = (ϕ i j ) where we set Then, (4.15), (4.12), and (4.9) yield Next, from (4.8), it follows that a js η j|h η s = 0. (4.17) Taking the covariant derivative of (4.17) and using (4.11) and (4.8), we deduce that Taking account of (4.18) in (4.16), we obtain Finally, consider X = X k (∂/∂x k ) from the contact distribution of M, and infer that since η k X k = 0. Hence, the restriction of ϕ to {ξ} ⊥ is an almost complex structure.Thus, the fibers of {ξ} ⊥ must be of even dimension 2n.This implies that m = 2n + 1 with n > 0.
Proof.First, from (4.8) and (4.13), we deduce that there exists on M a Killing vector field ξ.Then from (4.14), we obtain . Hence, by Theorem 3.1, we get the assertion of our lemma.
Since M is a Sasakian manifold, we may use the local expressions of some formulas from Section 3. First, we consider a unit vector field X = X i (∂/∂x i ) from the contact distribution of M.Then, from (3.4) and (3.2b), we infer that Also, (3.3) and (3.5) yield Finally, using (4.9), (4.12), (4.15), and (4.20), we obtain Now, we prove the following theorem.Proof.Let X i = X i (∂/∂x i ) be a unit vector field from the contact distribution of M.Then, using (3.9) and (2.14), we deduce that As by Lemma 4.3, M is a Sasakian manifold, we only need to prove that H(X) is a constant on M. To this end, we replace the components of the curvature tensor from (4.10) in (4.25), and using (4.22), (4.23), and (4.24), we obtain Finally, suppose that F m = (M,F,a ij ,b i ) is a Randers manifold satisfying the conditions from Theorem 2.2.Then, by Corollary 4.5 and Theorem 3.2, we deduce that M is isomorphic to an odd-dimensional sphere.This completes the proof of our main result in Theorem 2.2.

Conclusions
. By Theorem 2.2, we classified the simply connected and complete Randers manifolds of positive constant curvature satisfying the "Bao-Robles condition" β = 0. We stress that the 1-form b = (b i (x)) that defines our Randers metric is nowhere zero on the manifold.Examples of Randers metrics of positive constant curvature for which b vanishes at some points of the manifold are given by Shen [14], and Bao and Robles [4].Finally, we conjecture that Randers metrics of positive constant curvature whose β is nowhere zero on the manifold live only on open sets of R m .

Call for Papers
Thinking about nonlinearity in engineering areas, up to the 70s, was focused on intentionally built nonlinear parts in order to improve the operational characteristics of a device or system.Keying, saturation, hysteretic phenomena, and dead zones were added to existing devices increasing their behavior diversity and precision.In this context, an intrinsic nonlinearity was treated just as a linear approximation, around equilibrium points.
Inspired on the rediscovering of the richness of nonlinear and chaotic phenomena, engineers started using analytical tools from "Qualitative Theory of Differential Equations," allowing more precise analysis and synthesis, in order to produce new vital products and services.Bifurcation theory, dynamical systems and chaos started to be part of the mandatory set of tools for design engineers.
This proposed special edition of the Mathematical Problems in Engineering aims to provide a picture of the importance of the bifurcation theory, relating it with nonlinear and chaotic dynamics for natural and engineered systems.Ideas of how this dynamics can be captured through precisely tailored real and numerical experiments and understanding by the combination of specific tools that associate dynamical system theory and geometric tools in a very clever, sophisticated, and at the same time simple and unique analytical environment are the subject of this issue, allowing new methods to design high-precision devices and equipment.
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Theorem 2 . 1 (
Yasuda and Shimada [17], Bao and Robles[4]).Let F m = (M,F,a ij ,b i ) be a Randers manifold such that β = 0 on M. Then F m is of positive constant curvature K if and only if the following conditions are satisfied: (i) the length b of b is a constant on M; (ii) the covariant derivative of b with respect to Levi-Civita connection defined by a on M satisfies b i|j + b j|i = 0; (2.12) (iii) the curvature tensor of the Riemannian manifold (M, a) is given by

Lemma 4 . 1 .
Let F m = (M, F * ,a * ij ,b *i ) be a Randers manifold of positive constant flag curvature K * .Then, there exists on T M 0 a Randers metric F = (a ij ,b i ) of constant flag curvature K = 1.

Corollary 4 . 5 .
.26) By assertion (i) of Theorem 2.1, b is a constant on M. So, M is a Sasakian space form of constant ϕ-sectional curvature c = 1−4 b 2 .Moreover, we have −3 < c < 1 since 0 < b < 1, which completes the proof of the theorem.Let F m = (M, F * ,a * ij ,b * i ) be a Randers manifold of positive constant flag curvature K * and β = 0.Then, M is a Sasakian space form of constant ϕ-sectional curvature c ∈ (−3, 1).Proof.By Lemma 4.1, there exists on T M 0 a Randers metric F = (a ij ,b i ) of constant flag curvature K = 1.Thus, the assertion of the corollary follows from Theorem 4.4.

First
Round of ReviewsMarch 1, 2009 is proved that F is positive-valued on the whole T M 0 if and only if the length b of b satisfies (see Antonelli et al. [2, page 43])