AnyclosedFinslermanifoldofnegativesectionalflagcurvature manifold( 3 )ofnonzeroscalarflagcurvaturemustbeRiemannian[ ],the flagcurvature.Firstwerecallthatin1975NumataprovedthatanyLandsberg dependsnotonlyonthesectionbutalsoontheflagpole.AFinslermetricis of

Tomus 46 (2010), 99–104

SOME RIGIDITY THEOREMS FOR FINSLER MANIFOLDS OF SECTIONAL FLAG CURVATURE

Bing Ye Wu

Abstract. In this paper we study some rigidity properties for Finsler mani- folds of sectional flag curvature. We prove that any Landsberg manifold of non-zero sectional flag curvature and any closed Finsler manifold of negative sectional flag curvature must be Riemannian.

1. Introduction

The flag curvature, a natural extension of the sectional curvature in Riemannian geometry, plays the central role in Finsler geometry. Generally, the flag curvature depends not only on the section but also on the flagpole. A Finsler metric isof scalar flag curvatureif its flag curvature depends only on the flagpole. Contrast to it, Professor Zhongmin Shen suggests a parallel notion: a metric isof sectional flag curvature if its flag curvature depends only on the section (see also [2]).

In this paper we shall study the rigidity properties for Finsler metrics of sectional flag curvature. First we recall that in 1975 Numata proved that any Landsberg manifold (dim≥3) of nonzero scalar flag curvature must be Riemannian [7], the following theorem can be viewed as the analogous result for sectional flag curvature.

Theorem 1.1. Any Landsberg manifold of nonzero sectional flag curvature must be Riemannian.

For general Finsler metrics, the most important rigidity result is the Akbar-Zadeh’s theorem: any closed Finsler manifold of negative constant flag curvature must be Rie- mannian [1]. Our second result can be viewed as the generalization of Akbar-Zadeh’s theorem.

Theorem 1.2. Any closed Finsler manifold of negative sectional flag curvature must be Riemannian.

2000Mathematics Subject Classification: primary 53C60; secondary 53B40.

Key words and phrases: Finsler manifold, Landsberg manifold, scalar flag curvature, sectional flag curvature, Cartan tensor.

This project was supported by the Fund of the Education Department of Fujian Province of China (No JA09191).

Received June 1, 2009. Editor J. Slovák.

2. Preliminaries

In this section, we give a brief description of basic quantities and fundamental formulas in Finsler geometry, for more details one is referred to see [5]. Throughout this paper, we shall use the Einstein convention, that is, repeated indices with one upper index and one lower index denotes summation over their range.

Let (M, F) be a Finslern-manifold with Finsler metricF:T M →[0,∞). Let (x, y) = (xⁱ, yⁱ) be the local coordinates onT M, andπ:T Mg =T M\0→M the natural projection. Unlike in the Riemannian case, most Finsler quantities are functions of T M rather thanM. Some fundamental quantities and relations:

gij(x, y) := 1 2

∂²F²(x, y)

∂yⁱ∂y^j , (positive definite fundamental tensor) C_ijk(x, y) := 1

4

∂³F²(x, y)

∂yⁱ∂y^j∂y^k , (Cartan tensor) (g^ij) := (gij)⁻¹, C_jkⁱ =g^ilCljk,

γ_ij^k :=1 2g^km

∂g_mj

∂xⁱ +∂g_im

∂x^j − ∂g_ij

∂x^m

, N_jⁱ:=γ_jkⁱ y^k−C_jkⁱ γ_rs^ky^ry^s.

According to [4], the pulled-back bundleπ^∗T M admits a unique linear connection, called theChern connection. Its connection forms are characterized by the structure equations:

• Torsion freeness:

(2.1) dωⁱ=ω^j∧ωⁱ_j,

• Almostg-compatibility:

dg_ij=g_ikω_j^k+g_kjω^k_i + 2C_ijkω^n+k, where

(2.2) ωⁱ :=dxⁱ, ω^n+k:=dy^k+y^jω^k_j.

It is easy to know that torsion freeness is equivalent to the absence ofdy^k terms in ωⁱ_j; namely,

ω_jⁱ = Γⁱ_jkdx^k, together with the symmetry

Γⁱ_jk= Γⁱ_kj.

Thefirst Chern curvature tensorR_{j kl}ⁱ and thesecond Chern curvature tensorP_{j kl}^,i are defined by the following structure equation:

(2.3) dω_jⁱ =ω^k_j ∧ω_kⁱ +1

2R_{j kl}ⁱ ω^k∧ω^l+P_{j kl}ⁱ ω^k∧ω^n+l,

whereR_{j kl}ⁱ =−R_{j lk}ⁱ . The local expressions ofR_{j kl}ⁱ andP_{j kl}ⁱ are R_{j kl}ⁱ = δΓⁱ_jl

δx^k −δΓⁱ_jk

δx^l + Γⁱ_ksΓ^s_jl−Γ^s_jkΓⁱ_ls, and

P_{j kl}ⁱ =−∂Γⁱ_jk

∂y^l , respectively, where

δ δxⁱ := ∂

∂xⁱ −N_i^j ∂

∂y^j . From (2.2) and (2.3) we have

(2.4) dωⁿ⁺ⁱ =ω^n+j∧ωⁱ_j+1

2Rⁱ_klω^k∧ω^l−Lⁱ_klω^k∧ω^n+l, where

Rⁱ_kl:=y^jR_{j kl}ⁱ , Lⁱ_kl:=−y^jP_{j kl}ⁱ .

Lⁱ_jk is called theLandsberg curvature, and (M, F) is called aLandsberg manifold ifLⁱ_jk= 0. LetL_ijk=g_ilL^l_jk, then bothC_ijk andL_ijk are symmetric on all their indices, and by Euler’s Lemma we have

(2.5) yⁱCijk=yⁱLijk = 0.

LetRijkl :=gjsR_{i kl}^s , Rⁱ_j:=y^kRⁱ_jk andRij :=gikR^k_j, then

(2.6) Rij =Rji.

Letgy:=gij(x, y)dxⁱ⊗dx^j,Cy :=Cijk(x, y)dxⁱ⊗dx^j⊗dx^k, andRy :=Rij(x, y)dxⁱ

⊗dx^j, they are all symmetric. For a tangent planeP ⊂TxM containingy, let (2.7) K(P, y) =K(y;u) := Ry(u, u)

g_y(y, y)g_y(u, u)−[g_y(y, u)]²,

whereu∈P such thatP = span{y, u}.Kis called theflag curvature. In general K(P, y) depends both on the sectionP and the flagpoley. We say that (M, F) is of sectional flag curvature ifK(P, y) depends only on the sectionP. For tensors on slit tangent bundleT Mg, one can define the horizontal covariant derivative and the vertical covariant derivative. For example, ifT =T_jⁱdx^j⊗_∂x^∂i, thenthe horizontal covariant derivative T_j|kⁱ andthe vertical covariant derivativeT_j·kⁱ are related by

T_j|kⁱ ω^k+T_j·kⁱ ω^n+k :=dT_jⁱ+T_j^kωⁱ_k−T_kⁱω^k_j, and thus

T_j|kⁱ = δT_jⁱ

δx^k −T_sⁱΓ^s_jk+T_j^sΓⁱ_sk, T_j·kⁱ = ∂T_jⁱ

∂y^k .

In term of horizontal covariant, thegeodesic differentiation T˙ ofT is defined by T˙_jⁱ=T_j|kⁱ y^k. The horizontal and vertical covariant derivative satisfies the product rule, and

(2.8) g_ij|k= 0, yⁱ_|k = 0.

In term of geodesic differentiation the Landsberg curvature and the Cartan tensor are related byLijk=C_ijk|ly^l= ˙Cijk.

3. The Proof of Theorems

In this section we shall complete the proof of Theorems 1.1 and 1.2. For this purpose, Let us first prove some lemmas.

Lemma 3.1. Let R·y := Rij·kdxⁱ ⊗dx^j ⊗dx^k. If (M, F) is of sectional flag curvature, then for any y, u∈T_xM with y6= 0, the following holds:

(3.1) R_·y(u, u, u) = 2K(y;u)F²(y)C_y(u, u, u).

Proof. . Express yanduasy=y^{i ∂}_∂xi andu=u^{i ∂}_∂xi, respectively, then (2.7) can be rewritten as

(3.2) R_ij(x, y)uⁱu^j =K(y;u) g_ij(x, y)yⁱy^jg_kl(x, y)u^ku^l− g_ij(x, y)yⁱu^j2 . Let y(t) = y+tu, then y(0) = y, y⁰(0) = u. Since (M, F) is of sectional flag curvature, and span{y, u}= span{y(t), u} for anyt, we conclude thatK y(t);u is constant for anyt. Hence, replaceybyy(t) in (3.2) and calculate the derivative with respect toton the two sides att= 0, and use (2.5), one can reach at (3.1)

easily.

For y∈TxM\{0}, let{y^⊥}={u∈TxM :gy(y, u) = 0}, Sx={(y, u) :y, u∈ TxM, u∈ {y^⊥}, F(y) =gy(u, u) = 1}, and S = S

x∈M

Sx. Note that Sx is always closed for anyx∈M, and S is also closed ifM is closed. We have

Lemma 3.2. Let y₀, u₀∈T_xM be two vectors such that (3.3) Cy₀(u₀, u₀, u₀) = max

(y,u)∈Sx

Cy(u, u, u),

then Cy₀(u0, u0, v0) = 0 for any v0 ∈ {y₀^⊥} with gy₀(u0, v0) = 0. Consequently, Cy0(u0, u0, v) =gy0(v, u0)·Cy0(u0, u0, u0)for anyv∈TxM.

Proof. Let y₀, u₀ be two vectors such that (3.3) holds, and v₀ ∈ {y^⊥₀} with g_y0(u₀, v₀) = 0. Without loss of generality, we may assume thatg_y0(v₀, v₀) = 1.

Letu(t) =u0cost+v0sint, thenu(0) =u0, u⁰(0) =v0, and y0, u(t)

∈Sxfor any t. It is clear that the functionf(t) =C_y0 u(t), u(t), u(t)

attains its maximum at t= 0, and thus

0 = df dt

_t=0= 3Cy₀(u0, u0, v0).

Notice that Cy0(y₀,·,·) = 0, one has Cy0(u₀, u₀, v) = gy0(v, u₀)·Cy0(u₀, u₀, u₀)

for anyv∈T_xM.

In order to prove Theorems 1.1 and 1.2, we need the following fundamental identity for Cartan tensor [3, 6]:

C_ijk|p|qy^py^q+C_ijmR^m_k =−1

3g_imR^m_k·j−1

3g_jmR^m_k·i

−1

6g_imR^m_j·k−1

6g_jmR^m_i·k. (3.4)

(3.4) can be rewritten as C¨_ijk=1

3(C_ijmR^m_k+C_jkmR^m_i +C_kimR^m_j)

−1

3(R_ij·k+R_jk·i+R_ki·j), (3.5)

and consequently,

(3.6) C¨ijkuⁱu^ju^k=Cy

u, u, u^kR^m_k ∂

∂x^m

−R·y(u, u, u), ∀u=uⁱ ∂

∂xⁱ. Now we are ready to prove Theorems 1.1 and 1.2.

Proof of Theorem 1.1. Suppose that (M, F) be a Landsberg manifold of nonzero sectional flag curvature. For fixedx∈M, lety₀, u₀∈TxM be two vectors such that (3.3) holds. Then by Lemma 3.2, we have

Cy₀

u₀, , u₀, u^k₀R^m_k ∂

∂x^m

=gy₀

u^k₀R^m_k ∂

∂x^m, u₀

·Cy₀(u₀, , u₀, u₀)

=Ry₀(u0, u0)·Cy₀(u0, , u0, u0). (3.7)

Since (M, F) is Landsberg, ¨Cijk= ˙Lijk= 0, which together with (2.7), (3.1), (3.6) and (3.7) yields

0 =Ry₀(u0, u0)·Cy₀(u0, , u0, u0)−R_·y0(u0, u0, u0)

=−K(y₀;u₀)·C_y0(u₀, , u₀, u₀).

Notice thatK(y₀;u₀)6= 0, we getCy0(u₀, , u₀, u₀) = 0, namely, max

(y,u)∈Sx

Cy(u, u, u) = 0. Asx∈M is arbitrary, we finally obtainCy(u, u, u) = 0 for any (y, u)∈S, Hence

(M, F) is Riemannian.

Proof of Theorem 1.2. Let (M, F) be a closed Finsler manifold of negative sectional curvature. Since M is closed, so isS, and there exist two vectorsy0, u0∈ T_x0M such thatCy0(u₀, u₀, u₀) = max

(y,u)∈SCy(u, u, u). Letc: (−, )→M be the normal geodesic with the initial conditionc(0) =x0,c(0) =˙ y0, andU =U(t) be the parallel vector field alongc such thatU(0) =u0. Then c(t), U(t)˙

∈Sc(t)for any t∈(−, ), and the functionf(t) =C_c(t)˙ U(t), U(t), U(t)

attains its maximum at t= 0. By maximum principle and (3.6) we have

0≥d²f dt²

_t=0=Cy₀

u0, u0, u^k₀R^m_k(x0, y0) ∂

∂x^m

−R_·y0(u0, u0, u0), which together with Lemma 3.1 and (3.7) yields

0≥ −K(y0;u₀)·C_y0(u₀, , u₀, u₀).

SinceK(y₀;u₀)<0, we conclude thatCy0(u₀, , u₀, u₀) = 0, i.e., max

(y,u)∈SCy(u, u, u) = 0. Consequently, (M, F) is Riemannian, and the theorem is proved.

References

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Belg. Bull. Cl. Sci. (6)74(1988), 281–322.

[2] Chen, B., Zhao, L. L.,Randers metrics of sectional flag curvature, Houston J. Math., to appear.

[3] Chen, X., Mo, X., Shen, Z.,On the flag curvature of Finsler metrics of scalar curvature, J.

London Math. Soc.68(2003), 762–780.

[4] Chern, S. S.,Local equivalence and Euclidean connections in Finsler spaces, Sci. Rep. Nat.

Tsing Hua Univ. Ser. A5 (1948), 95–121, or Selected Papers, II, 194-212, Springer 1989.

[5] Chern, S. S., Shen, Z.,Riemannian-Finsler geometry, World Sci., Singapore, 2005.

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[7] Numata, S.,On Landsberg spaces of scalar curvature, J. Korean Math. Soc.12(1975), 97–100.

Department of Mathematics, Minjiang University Fuzhou, Fujiang, 350108, China

E-mail:[email protected]