A characterization of Finsler metrics of constant flag curvature

A CHARACTERIZATION OF FINSLER METRICS OF CONSTANT FLAG CURVATURE

B. Najafi

Faculty of Science, Department of Mathematics, Shahed University, Tehran, Iran e-mail: najafi@shahed.ac.ir

(Received17April2011;after final revision23May2012;

accepted3August2012)

Using the notion of C-projectively flatness, we give a new characterization of Finsler metrics of constant flag curvature.

Key words: Finsler metrics; flag curvature; Weyl curvature; C-projective transformations.

1. INTRODUCTION

One of the important problems in Finsler geometry is to study and characterize Finsler metrics of constant flag curvature [10]. On the other hand, there are some well-known projective invariants of Finsler metrics namely, Weyl curvature, gener- alized Douglas -Weyl curvature [5] and another C-projective invariantH-curvature [7]. Weyl introduces a projective invariant for Riemannian metrics. Then Douglas extendes Weyl’s projective invariant to Finsler metrics. Finsler metrics with van- ishing projective Weyl curvature are called Weyl metrics. Z. Szab´o proves that Weyl metrics are exactly Finsler metrics of scalar flag curvature.

Recently, the non-Riemannian quantity H which is obtained from the mean Berwald curvature by the covariant horizontal differentiation along geodesics, has been studied extensively (for more details see [6, 7]). Akbar-Zadeh proves that for a Weyl manifold of dimensionn ≥ 3, the flag curvature is constant if and only ifH = 0. Then, it is natural to find other projectively invariant quantity which characterizes Finsler metrics of constant flag curvature [9].

In this paper, we define a projective invariant so calledW-curvature. We show that theW-curvature is another candidate for characterizing Finsler metrics of con- stant flag curvature. More precisely, we prove the following.

Theorem1.1— Let(M, F)be a connected Finsler manifold with dimension n≥3. ThenF is of constant flag curvature if and only if W = 0.

We set the Berwald connection on Finsler manifolds. The vertical and horizon- tal covariant derivatives of Berwald connection are denoted by ’,’ and ’|’, respec- tively.

2. PRELIMINARIES

LetM be ann-dimensionalC^∞ manifold. Denote byTxM the tangent space at x ∈ M, and by T M = ∪_x∈MT_xM the tangent bundle ofM. AFinsler metric onM is a functionF :T M → [0,∞)which has the following properties: (i)F isC^∞onT M0 (ii)F is positively 1-homogeneous on the fibers of tangent bundle T M, and (iii) for each y ∈ T_xM, the following quadratic form g_y on T_xM is positive definite,

g_y(u, v) := 1 2

F²(y+su+tv)

|s,t=0, u, v∈T_xM.

Given a Finsler manifold(M, F), then a global vector fieldGgiven byG = y^{i ∂}_∂xi−2Gⁱ(x, y)_∂y^∂i, whereGⁱ := ¹₄g^il{2^∂g_∂x^jlk−^∂g_∂x^jkl}y^jy^kis called the associated sprayto(M, F). The projection of an integral curve ofGis called ageodesicin M. A diffeomorphismf : (M, F) → (M,F¯) between two Finsler manifolds is

called a projective transformation, iff maps every geodesic ofF to a geodesic of F¯ as a point set.

The Riemannian curvature tensor of Berwald connection are given byKⁱ_hjk= d_jGⁱ_hk +G^m_hkGⁱ_mj −d_kGⁱ_hj −G^m_hjGⁱ_mk, where d_k = ∂_k−G^m_k∂˙_m, ∂_k = _∂x^∂_k,

∂˙_k = _∂y^∂k,Gⁱ_k= ˙∂_kGⁱandGⁱ_jk = ˙∂_jGⁱ_k(for more details, see [7]).

The flag curvature in Finsler geometry is a natural extension of the sectional curvature in Riemannian geometry, which is first introduced by L. Berwald. For a flagP = span{y, u} ⊂ T_xM with flagpoley, K = K(P, y) stands forflag curvature. WhenF is Riemannian,K = K(P)is independent ofy ∈ P, which is just the sectional curvature ofP in Riemannian geometry. We say that a Finsler metric F is of scalar curvature if for any y ∈ T_xM, the flag curvature K = K(x, y)is a scalar function on the slit tangent bundleT M0. IfK=K(x), thenF is said to be ofisotropic flag curvature. IfK=constant, thenF is said to be of constant flag curvature.

3. SPECIALPROJECTIVEWEYLCURVATURE

Letφ : Fⁿ → F¯ⁿ be a projective transformation. Then, there exists a positive homogeneous scalar functionP(x, y)of degree one satisfying

G¯ⁱ =Gⁱ+P yⁱ.

In this case,Pis called the projective factor. Under a projective transformation with projective factorP, the Riemannian curvature tensor of Berwald connection changes as follows

K¯ⁱ_hjk =Kⁱ_hjk+yⁱ∂˙_hQ_jk+δ_hⁱQ_jk+δⁱ_j∂˙_hQ_k−δ_kⁱ∂˙_hQ_j, (1) whereQi =diP−P PiandQij = ˙∂iQj−∂˙jQi. A projective transformation with projective factorP is said to beC-projectiveifQ_ij = 0.

Lemma 3.1 — Every C-projective mapping φ : Fⁿ → F¯ⁿ with projective factorP satisfiesP_ij|sy^s= 0.

PROOF: First, we prove that

P_ik|sy^s=y^s∂˙_iQ_ks. (2) We know that∂˙kdi=di∂˙k−G^r_ik∂˙r. Hence, we get

y^s∂˙_iQ_ks=y^s( ˙∂_id_sP_k−∂˙_id_kP_s)

=y^s(d_sP_ik−G^r_siP_kr−d_kP_is+G^r_ikP_sr)

=y^sdsP_ik−G^s_kPis+G^s_iP_sk

=P_ik|sy^s (3)

By assumptionQ_ij = 0. Then (2) implies the desired result.

LetXbe a projective vector field on a Finsler manifold(M, F). Suppose that the vector fieldX in a local coordinate(xⁱ) on M is written in the form X = Xⁱ(x)∂i. Then the complete lift of X is denoted by Xˆ and locally defined by X=Xˆ ⁱ∂_i+y^j∂_jXⁱ∂˙_i. Suppose that£_Xˆ stands for Lie derivative with respect to the complete lift ofX. Then we have

£_XˆGⁱ =Pyⁱ, (4)

£_XˆG_kⁱ =δ_kⁱP+yⁱP_k, (5)

£_XˆG_jkⁱ =δ_jⁱP_k+δ_kⁱP_j +yⁱP_jk, (6)

£_XˆG_jklⁱ =δⁱ_jP_kl +δⁱ_kP_jl +δ_lⁱP_kj +yⁱP_jkl, (7)

£_XˆK_jklⁱ =δⁱ_j(P_l|k −P_k|l) +δ_lⁱP_j|k−δ_kⁱP_j|l +yⁱ∂˙j(P_l|k −P_k|l). (8) SinceQij =P_i|j −P_j|i, we have

£XˆK_jklⁱ =δⁱ_jQlk+δ_lⁱP_j|k−δ_kⁱP_j|l +yⁱ∂˙_jQlk. (9) We have

∂˙_jP_k|l=P_jk|l−P_rG^r_jkl. (10)

Contractingiandkin (9) and using (2), we get

£_XˆK_jl =P_l|j −nP_j|l +P_jl|sy^s, (11) whereK_jl := K_jrl^r . Now, suppose thatQ_ij = 0. Then by Lemma 3.1, we have P_jl|sy^s= 0. Therefore, (11) reduces to the following

£_XˆK_jl =P_l|j −nP_j|l. (12)

£XˆKlj =P_j|l −nP_l|j. (13)

From (12) and (13), one can obtain P_j|l= 1

1−n²£_Xˆ

K_lj +nK_jl

. (14)

Substituting (14) into (9) and using the assumptionQ_ij = 0, we are led to the following tensor

W_jklⁱ :=K_jklⁱ − 1 1−n²

δⁱ_l(Kkj+nKjk)−δ_kⁱ(Klj+nKjl)

. (15)

If we putWⁱ_k:=W_jklⁱ y^jy^l, then we have Wⁱ_k=Kⁱ_k− 1

1−n²

yⁱ(K_k0+nK0k)−δ_kⁱ(n+ 1)K00

. (16)

The tensorWⁱ_kis said to bespecial projective Weyl curvatureor W-curvature.

Proposition3.2 — LetX be a C-projective vector field of Finsler metricF. Then we have£_XˆW_hjkⁱ =0.

Therefore, special Weyl projective curvature is invariant under C-projective transformations.

4. PROOF OFTHEOREM1.1

First, we prove that the class of Finsler metrics of scalar flag curvature contains the class of Finsler metrics with vanishing W-curvature.

Proposition 4.1 — Let F be a Finsler metric with vanishing W-curvature.

ThenF is of scalar flag curvature.

PROOF: By assumption, we have the following Kⁱ_k− 1

1−n²

yⁱ(Kk0+nK0k)−δ_kⁱ(n+ 1)K⁰⁰

= 0, (17) where by definitionKⁱ_k=K_jklⁱ y^jy^l. It is well known that

Kⁱ_k= 2∂kGⁱ−y^j∂j∂˙kGⁱ+ 2G^j∂˙j∂˙kGⁱ−∂˙jGⁱ∂˙kG^j, (18) which implies that y_iKⁱ_k = 0. Contracting (17) with y_i and using last relation imply that

F²(Kk0+nK0k)−y_k(n+ 1)K⁰⁰= 0. (19) Hence

K_k0+nK0k=F⁻²(n+ 1)ykK00. (20) Plugging (20) into (17), we get

K_kⁱ = 1

n−1K00hⁱ_k, (21) which means thatF is of scalar flag curvature.

To prove Theorem 1.1, we need to find the W-curvature of Finsler metrics of scalar flag curvature.

Proposition4.2 — LetF be a Finsler metric of scalar flag curvatureλ. Then W-curvature is given by

W_kⁱ =a_nF²yⁱλ_k, (22) whereλ_k:= ˙∂_kλanda_n:= ⁿ²₃⁻_n2²₋ⁿ⁻₃¹.

PROOF: By assumption, the Riemannian curvature of Berwald connection is in the following form.

K_jklⁱ = λ(δⁱ_kg_jl−δ_lⁱg_jk) +λ_jF(δⁱ_kF_l−δⁱ_lF_k) +1

3F²(hⁱ_kλ_jl−hⁱ_lλ_jk) + 1

3λ_lF(2δⁱ_kF_j −δⁱ_jF_k−g_jkℓⁱ)

− 1

3F λ_k(2δⁱ_lFj −δⁱ_jF_l−g_jlℓⁱ). (23) whereλ_ij = ˙∂_jλ_i. Hence, we have

K_kⁱ =λF²hⁱ_k. (24) Then, we get the following relations

K_jl= (n−1)(λgjl+F F_lλ_j) + n−2

3 F²λ_jl+2(n−3) 3 F F_jλ_l, K00=λ(n−1)F², K_k0=λ(n−1)F F_k+2n−1

3 F²λ_k, K0k =λ(n−1)F F_k+n−4

3 F²λ_k. (25)

Plugging (24) and (16) into (16), we get the result.

Proof of Theorem 1.1Suppose that (M, F)is a Finsler manifold with dimension n ≥3and vanishing W-curvature. Then, by Proposition 4.1,F is of scalar flag curvatureλ. Then, by Proposition 4.2, we getλ_k= 0. SinceM is connected,λis a function of positions, i.e.,F is of isotropic flag curvature. Now, Schur’s lemma, which asserts that every connected Finsler manifold(M, F)of isotropic flag cur- vature with dimension greater than two is of constant flag curvature, completes the proof.

Conversely, let F be a Finsler metric of constant flag curvature. Then, a straightforward computation shows thatW_hjkⁱ = 0and consequently W_jⁱ = 0.

This completes the proof.

A Finsler metricFis said to be C-projectively flat ifF is C-projectively related to a locally Minkowski metric. It is well known that a Riemannain metricF is C- projectively flat if and only ifF is projectively flat [1, 6].

Beltrami’s theorem establishes an equivalency between projectively flatness and being of constant sectional curvature of Riemannian metrics [3, 4]. There are many projectively flat Finsler metrics which are not of constant flag curvature [8].

Here, we prove that C-projectively flatness implies constancy of the flag curvature.

Proposition4.3 — LetF be a C-projectively flat Finsler metric with dimension n≥3. ThenF is of constant flag curvature.

PROOF : Suppose thatF is C-projectively flat. Thus, by definition, F is C- projectively related to a locally Minkowski metric. It is easy to see that for a locally Minkowski metric W = 0. Therefore, by Proposition 3.2,F has vanishing W-curvature, and, by Theorem 1.1,F is of constant flag curvature.

ACKNOWLEDGMENT

The author expresses his sincere thanks to referees for their valuable suggestions and comments.

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