3 Mean Berwald curvature of homogeneous Randers metrics

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Shaoqiang Deng and Xiaoyang Wang

Abstract. In this paper we give a formula of the S-curvature of homo- geneous (α, β)-metrics. Then we use this formula to deduce a formula of the mean Berwald curvatureEij of Randers metrics.

M.S.C. 2010: 22E46, 53C30.

Key words: (α, β) metric; S-curvature; mean Berwald curvature.

1 Introduction

The purpose of this paper is to give an explicit formula for theS-curvature of homogeneous (α, β)-metrics in Finsler geometry, for fundamental theory of Finsler geometry we refer to [1] and [4]. LetM be a connected smooth manifold andαbe a Rieman- nian metric on M. Then a Randers metric on M with the underlying Riemannian metric α is a Finsler metric of the form F = α+β, where β is a smooth 1-form on M satisfying ||βx||α < 1, ∀x ∈ M, here ||β||α denote the length of the 1-form under the Riemannian metricα. This kind of spaces was first studied by G. Randers in [11] and was then named after him. Randers metrics are most closely related to Riemannian metrics among the class of Finsler metrics. The Finsler metrics of the formF=αφ(s) are called (α, β)-metric, heres= ^β_α, andφ(s) is a function ofs. It is clear that (α, β)-metrics are the generalization of Randers metrics, in fact, if we set φ(s) = 1 +s, thenF =α(1 +_α^β) =α+β.

There are several interesting curvatures in Finsler geometry, among them the flag curvature is the most important one, which is the natural generalization of sectional curvature in Riemannian geometry. On the other hand, Z. Shen introduced the notion of S-curvature of a Finsler space in [12]. It is a quantity to measure the rate of change of the volume form of a Finsler space along the geodesics. S-curvature is a non-Riemannian quantity, i.e., any Riemann manifold has vanishingS-curvature.

It is a very interesting fact that theS-curvature and the flag curvature are subtly related with each other. We now recall the definition ofS-curvature. Let V be an n-dimensional real vector space andF be a Minkowski norm onV. For a basis{ei} ofV, let

σF = Vol(Bⁿ)

Vol{(yⁱ)∈Rⁿ|F(yⁱei)<1},

Balkan Journal of Geometry and Its Applications, Vol.15, No.2, 2010, pp. 47-56 (electronic version);∗

pp. 39-48 (printed version).

°c Balkan Society of Geometers, Geometry Balkan Press 2010.

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where Vol means the volume of a subset in the standard Euclidean spaceRⁿ andBⁿ is the open ball of radius 1. This quantity is generally dependent on the choice of the basis{ei}. But it is easily seen that

τ(y) = ln

pdet(gij(y))

σF , y∈V\{0}

is independent of the choice of the basis. τ=τ(y) is calledthe distortion of (V,F).

Now let (M, F) be a Finsler space. Letτ(x, y) be the distortion of the Minkowski normFx on TxM. For y ∈TxM − {0}, let τ(t) be the geodesic with τ(0) = xand

˙

τ(0) =y. Then the quantity

S(x, y) = d

dt[τ(σ(t),σ(t))]|˙ t=0

is calledtheS-curvatureof the Finsler space (M,F). A Finsler space (M,F) is said to havealmost isotropicS-curvature if there exists a smooth functionc(x) on M and a closed 1-formη such that:

S(x, y) = (n+ 1)(c(x)F(y) +η(y)), x∈M, y∈TxM.

If in the above equationη= 0, then (M,F) is said to haveisotropicS-curvature. If η= 0 andc(x) is a constant, then (M, F) is said to haveconstant S-curvature.

In this paper we will obtain an explicit formula of theS-curvature of homogeneous (α, β)-metrics, without using local coordinate systems. As an application, we prove that a homogeneous (α, β)-metric has isotropicS-curvature if and only if it has van- ishingS-curvature. We also give an explicit formula of the mean Berwald curvature Eij of homogeneous Randers metrics.

2 The S-curvature

In this section we will compute theS-curvature of aG-invariant homogeneous (α, β) metricF = αφ(s) on the coset space G/H of a Lie group G, where s = _α^β. Since (G/H, F) is homogeneous, we only need to compute this at the origino=H. By [2], in a local coordinate system, theS-curvature of the (α, β) metricF =αφ(s) with the underlying Riemann metricαcan be expressed as

S = (2Ψ−f⁰(b)

bf(b))(r0+s0)−α⁻¹ Φ

2∆²(r00−2αQs0), where

Q= _φ−sφ^φ⁰ 0, ∆ = 1 +sQ+ (b²+s²)Q⁰, Ψ = _2∆^Q⁰, Φ =−(Q−sQ⁰){n∆ + 1 +sQ} −(b²−s²)(1 +sQ)Q⁰⁰, rij =¹₂(bi;j+bj;i), sij =¹₂(bi;j−bj;i), bⁱ =bja^ji, rj=bⁱrij, sj=bⁱsij, r00=rijyⁱy^j, s0=siyⁱ, r0=riyⁱ,

and the functionf(b) in the formula is defined as follows. TheBusemann-Hausdorff volume formdVBH =σBH(x)dxis defined by

σBH(x) = ωn

V ol{(yⁱ)∈R:F(x, y^{i ∂}_∂xi)<1},

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and theHolmes-Thompson volume form,dVHT =σHT(x)dxis defined by σHT(x) = 1

ωn

Z

{(yⁱ)∈Rⁿ|F(x,yⁱ ^∂

∂xi)<1}

det(gij)dy,

where Vol denotes the Euclidean volume,gij =_∂y^∂i∂y² ^j[F²], and ωn=V ol(Bⁿ(1)) = 1

nV ol(Sⁿ⁻¹) = 1

nV ol(Sⁿ⁻²) Z _π

0

sinⁿ⁻²(t)dt.

WhenF = p

gij(x)yⁱy^j is a Riemannian metric, both volume forms reduce to the same Riemannian volume formdVHT =dVBH =p

det(gij(x))dx. Now for the (α, β) metricF =αφ(s),s= ^β_α,b=||βx||α, letdV =dVBH ordVHT. Then

f(b) =









 R_π

0 sinⁿ⁻²tdt R_π

0 sinⁿ⁻²t

φ(bcost)ⁿdt, if dV =dVBH, R_π

0 (sinⁿ⁻²t)T(bcost)dt R_π

0 sinⁿ⁻²tdt , if dV =dVHT,

whereT(s) =φ(φ−sφ⁰)ⁿ⁻²{(φ−sφ⁰) + (b²−s²)φ⁰⁰}. Then the volume formdV is given bydV =f(b)dVα, wheredVα=p

det(αij)dx, denote the Riemannian volume form ofα.

In [2] the authors showed that ifb=||βx||αis a constant, thenr0+s0= 0. So in this case we have

S=−α⁻¹ Φ

2∆²(r00−2αQs0).

Now we will deduce a formula of the S-curvature of homogeneous (α, β)-metrics.

Recall that the groupI(M, F) of isometries of a Finsler space (M, F) is a Lie transformation group ofM ([7]). IfI(M, F) acts transitively onM, then (M, F) is called homogeneous. Let (G/H, F) be a homogeneous (α, β) metric of the formF =αφ(s), wheres= _α^β with αa G-invariant Riemannian metric onG/H andβ a G-invariant vector field on G/H. As pointed out in [6], to β corresponds a unique vector u in To(G/H) which is fixed under the linear isotropy representation of H onTo(G/H) ando=H is the origin ofG/H. It is clear thatb=||βx||α is a constant. Therefore theS-curvature can be expressed as

S=−α⁻¹ Φ

2∆²(r00−2αQs0). (∗)

Also,G/His a reductive homogeneous manifold in the sense of Nomizu ([10], see also [9]), i.e, the Lie algebra ofGhas a decomposition:

g=h+m, (direct sum of subspaces) (2.1)

such thatAd(m)⊂m, ∀h∈ H. Then we can identifym with the tangent space of (G/H) at the origin o = H and in this way β corresponds a vector inm which is

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invariant under the adjoint action of H on m. In the following we still denote this vector byu.

Leth,ibe the corresponding inner product on m . We now deduce some results concerning the Levi-Civita connection of (G/H, α) which will be useful to compute the S-curvature. In literature, there are several versions of the formula of the connection for Killing vector fields. Since we are interested in the differential of (left) invariant vector fields on G/H, we adopt the formula in [9]. Given v ∈ g , we can define a one-parameter transformation groupϕt ,t∈RofG/H by

φt(gH) = (exp(tv)g)H, g∈G.

Thenϕtgenerates a vector field onG/H which is a Killing vector field (this is called the fundamental vector field generated byv in [9]). We denote this vector field by ¯v.

The following formula is a direct consequence of the formula in [9, Vol.2, page 201]

(see also [13]):

(2.2) h(∇v¯1¯v2)|o, wi= 1

2(−h[v1, v2]m, wi+h[w, v1]m, v2i+h[w, v2]m, v1i), wherev1, v2, w∈m, and [v1, v2]m denote the protection of [v1, v2] tom with respect to the decomposition (2.1).

To apply the formula (2.2) to our study, we need to deduce some formula for the connection in a local coordinate system. Letu1· · ·um be an orthonormal basis ofm with respect toh ·,· i. Then by [8] there exists a neighborhood U ofo inG/H such that the mapping

(expx¹u1expx²u2· · ·expx^mum)H7→(x¹, x²· · ·x^m) defines a local coordinate system onU.

In the following we adopt the computation ofr00 ands0 on (G/H, F) in [5]. Let (u1, u2,· · · , um) be an orthonormal basis of (m,h ·, · i) such that um = _|u|^u . Recall that uis the vector in m corresponding to the 1-from β. Let (U,(x¹, x²· · ·xⁿ)) be the local coordinate system defined as above. Then in [5] it was shown that

s0(y)|o=y^lsl(o) =cy^lsnl(o) = 1

2c²y^lh[un, ul]m, uni

= 1

2h[cun, y^lul]m, cuni=1

2h[u, y]m, ui, and

rij(o) =−c

2(h[un, ui]m, uji+h[un, uj]m, uii).

Moreover,

r00|o=rij(o)yⁱy^j=−c

2(h[un, ui]m, uji+h[un, uj]m, uii)yⁱy^j

=−c

2(h[un, yⁱui]m, y^juji+h[un, y^juj]m, yⁱuii)

=−c

2(h[un, y]m, yi+h[un, y]m, yi) =−ch[un, y]m, yi.

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Substituting the above into the formula (∗) we obtain the formula ofS-curvature:

S(o, y) =− 1 α(y)

Φ

2∆²(rij(o)yⁱy^j−2α(y)Qs0(y))

=− 1 α(y)

Φ

2∆²(−ch[u, y]m, yi −α(y)Qh[u, y]m, ui).

Summarizing, we get

Theorem 2.1 Let F =αφ(s) be aG-invariant (α, β)-metric on the reductive ho- mogeneous manifoldG/H with a decomposition of the Lie algebra

g=h+m.

Then theS-curvature of F has the form

S(o, y) =− 1 α(y)

Φ

2∆²(−ch[u, y]m, yi −α(y)Qh[u, y]m, ui), y∈m,

whereu is the vector in m corresponding to the 1-form β, and we have identified m with the tangent space ofG/H at the origin o=H.

As a direct application of the formula we have

Corollary 2.2 Let (G/H, F) be as in Theorem 2.1. Then F has isotropic S- curvature if and only ifF has vanishingS-curvature.

Proof. We only need to prove the direct implication. Suppose F has isotropic S-curvature:

S(x, y) = (n+ 1)c(x)F(y), x∈G/H, y∈Tx(G/H).

Setting x = o and y = uand using the formula in Theorem 2.1, we get c(o) = 0.

HenceS(o, y) = 0, ∀y ∈To(G/H). SinceF is a homogeneous metric, we must have

S= 0 everywhere. ¤

3 Mean Berwald curvature of homogeneous Randers metrics

In this section we apply the results in Section 2 to give a formula of mean Berwald curvature of homogeneous Randers metrics.

The mean Berwald curvature (E-curvature) is an important non-Riemannian quantity defined by (see [3])

Eij = 1 2

∂²

∂yⁱ∂y^j µ∂G^m

∂y^m

¶ ,

whereG^m=G^m(x, y) are the spray coefficients. We know that S=∂G^m

∂y^m −(lnσ(x))_x^ky^k

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here (lnσ(x))_x^k is the function ofxbecause lnσ(x) is the function ofx. Hence 0 = ∂²

∂yⁱ∂y^j[(lnσ(x))_x^ky^k] This means that

∂²S

∂yⁱ∂y^j = ∂²

∂yⁱ∂y^j

·∂G^m

∂y^m −(lnσ(x))_x^ky^k

¸

= ∂²

∂yⁱ∂y^j µ∂G^m

∂y^m

¶

= 2Eij

Now we compute

∂²S

∂yⁱ∂y^j(o, y) = ∂²S(o, y)

∂yⁱ∂y^j = 2Eij(o, y).

By Theorem 2.1 we have

∂²S(o, y)

∂yⁱ∂y^j = ∂²

∂yⁱ∂y^j

µ cΦ

2∆²α(y)h[u, y]m, yi

¶ + ∂²

∂yⁱ∂y^j µΦQ

2∆²h[u, y]m, ui

¶ .

Before the computation, we recall that

∂s

∂y = 1 α

³

bm−sym

α

´ , ∂α

∂y^m =ym

α ,

where ym = amjy^j. Since u1, u2,· · · , um in Section 2 is an orthonormal basis, we have amj|o =δ^m_j . Therefore at the origin we have ym =y^m. Now we consider the special case of the Randers metricφ= 1 +s. Letψ=φ−sφ⁰. Then we have

Q= φ⁰

ψ = 1, Q⁰ = 0, Ψ = Q⁰ 2∆ = 0,

∆ = 1 +sQ+ (b²−s²)Q⁰ = 1 +s, Θ = Q−sQ⁰

2∆ = 1

2∆= 1

2(1 +s), Φ =−(Q−sQ⁰){n∆ + 1 +sQ} −(b²−s²)(1 +sQ)Q⁰⁰=−(n+ 1)(1 +s).

Lettingc= 1 we get S(o, y) = 1

α(y) Φ

2∆²(−ch[u, y]m, yi −α(y)Qh[u, y]m, ui)

=− n+ 1

2(1 +s)α(y)h[u, y]m, yi − n+ 1

2(1 +s)h[u, y]m, ui.

Therefore

2Eij(o, y) =∂²S(o, y)

∂yⁱ∂y^j = ∂²

∂yⁱ∂y^j µ

− n+ 1

2(1 +s)α(y)h[u, y]m, yi − n+ 1

2(1 +s)h[u, y]m, ui

¶

= ∂²

∂yⁱ∂y^j µ

− n+ 1

2(1 +s)α(y)h[u, y]m, yi

¶

− ∂²

∂yⁱ∂y^j

µ n+ 1

2(1 +s)h[u, y]m, ui

¶ .

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Now

∂²

∂yⁱ∂y^j µ

− n+ 1

2(1 +s)α(y)h[u, y]m, yi

¶

=−n+ 1 2

½

h[u, y]m, yi ∂²

∂yⁱ∂y^j

µ 1

(1 +s)α(y)

¶

+ 1

(1 +s)α(y)

∂²h[u, y]_m, yi

∂yⁱ∂y^j +∂h[u, y]_m, yi

∂y^j

∂

∂yⁱ

µ Φ

(1 +s)α(y)

¶

+∂h[u, y]_m, yi

∂yⁱ

∂

∂y^j

µ 1

(1 +s)α(y)

¶¾ ,

where

∂h[u, y]_m, yi

∂yⁱ =h[u, ui]m, yi+h[u, y]m, uii,

∂²h[u, y]_m, yi

∂yⁱ∂y^j =h[u, ui]m, uji+h[u, uj]m, uii, and

∂

∂y^j

µ 1

(1 +s)α(y)

¶

=−

∂s

∂y^jα(y) + (1 +s)_∂y^∂α_j (1 +s)²α²(y)

=−

1

α(y)(bj−s_α(y)^y^j )α(y) + (1 +s)_α(y)^y^j (1 +s)α(y)

=−(bj−s_α(y)^y^j ) + (1 +s)_α(y)^y^j

(1 +s)²α²(y) =− bj+_α(y)^y^j (1 +s)²α²(y). Hence

∂²

∂yⁱ∂y^j

µ 1

(1 +s)α(y)

¶

= ∂

∂yⁱ



− bj+_α(y)^y^j (1 +s)²α²(y)





=−

∂

∂yⁱ(bj+_α(y)^y^j )(1 +s)²α²(y)−(bj+_α(y)^y^j )_∂y^∂i[(1 +s)²α²(y)]

(1 +s)⁴α⁴(y)

=−

∂

∂yⁱ(_α(y)^y^j )(1 +s)²α²(y)−(bj+_α(y)^y^j )2(1 +s)α(y)[_∂y^∂siα²(y) +s^∂α(y)_∂yi ] (1 +s)⁴α⁴(y)

=− 1

(1 +s)²α²(y)

∂y^j

∂yⁱα(y)−y^{j ∂α(y)}_∂yi

α²(y)

+ 2

(1 +s)α³(y) µ

bj+ y^j α(y)

¶ · 1

α(y)(bj−s yⁱ

α(y))α(y) +s yⁱ α(y)

¸

=− 1

(1 +s)²α⁴(y)

·

δ_i^jα(y)−y^j yⁱ α(y)

¸

+ 2

(1 +s)α³(y) µ

bj+ y^j α(y)

¶ bi

=− δ_i^j

(1 +s)²α³(y) + yⁱy^j

(1 +s)²α⁵(y)+2bi(bj+_α(y)^y^j ) (1 +s)α³(y) .

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Therefore we have

∂²

∂yⁱ∂y^j µ

− n+ 1

2(1 +s)α(y)h[u, y]m, yi

¶

=



− δ_i^j

(1 +s)²α³(y)+ yⁱy^j

(1 +s)²α⁵(y)+2bi(bj+_α(y)^y^j ) (1 +s)α³(y)



h[u, y]m, yi

+ 1

(1 +s)α(y)h[u, ui]m, uji+h[u, uj]m, uii

− bj+_α(y)^y^j

(1 +s)²α²(y)h[u, ui]m, yi+h[u, y]m, uii − bi+_α(y)^yⁱ

(1 +s)²α²(y)h[u, uj]m, yi+h[u, y]m, uji.

Note that

∂²

∂yⁱ∂y^j µ

− n+ 1

2(1 +s)h[u, y]m, ui

¶

=−n+ 1 2

∂²

∂yⁱ∂y^j

µh[u, y]_m, yi (1 +s)

¶

=−n+ 1 2

( 1 1 +s

∂²h[u, y]_m, ui

∂yⁱ∂y^j +h[u, y]m, ui∂^{2 1}_1+s

∂yⁱ∂y^j

+∂_1+s¹

∂yⁱ

∂h[u, y]_m, ui

∂y^j +∂_1+s¹

∂y^j

∂h[u, y]_m, ui

∂yⁱ )

,

where

∂h[u, y]_m, ui

∂y^j =h[u, uj]m, ui, ∂²h[u, y]_m, ui

∂yⁱ∂y^j = ∂

∂yⁱh[u, uj]m, ui= 0,

∂_1+s¹

∂y^j =−

1

α(y)(bj−s_α(y)^y^j ) (1 +s)² . Hence

∂^{2 1}_1+s

∂yⁱ∂y^j = ∂

∂yⁱ



−

1

α(y)(bj−s_α(y)^y^j ) (1 +s)²





=−

∂

∂yⁱ[_α(y)¹ (bj−s_α(y)^y^j )](1 +s)²−_α(y)¹ (bj−s_α(y)^y^j )^∂(1+s)_∂yi

(1 +s)⁴

=−α(y)_∂y^∂i(bj−s_α(y)^y^j )−(bj−s_α(y)^y^j )^∂α(y)_∂yi

(1 +s)²α²(y) + 2(1 +s) (1 +s)⁴α(y)

µ

bj−s y^j α(y)

¶ ∂s

∂yⁱ

=− 1

(1 +s)²



 1 α(y)



−∂s

∂yⁱ yⁱ

α(y)−s∂_α(y)^y^j

∂yⁱ



− 1 α²(y)

µ

bj−s y^j α(y)

¶ yⁱ α(y)





+ 2

(1 +s)³α(y) µ

bj−s y^j α(y)

¶ 1 α(y)

µ

bi−s yⁱ α(y)

¶

=− 1

(1 +s)²



 1 α(y)



− 1 α(y)

µ

bi−s yⁱ α(y)

¶ y^j α(y) −s

∂y^j

∂yⁱα(y)−y^{j ∂α}_∂y_i α²(y)

− 1 α²(y)

µ

bj−s y^j α(y)

¶ yⁱ α(y)

¸

+ 2

(1 +s)α²(y) µ

bj−s y^j α(y)

¶ µ

bi−s yⁱ α(y)

¶¾

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=− 1 (1 +s)²

·

− 1 α³(y)

µ

bi−s yⁱ α(y)

¶

y^j− s α³(y)

µ

α(y)δ_jⁱ−y^j yⁱ α(y)

¶

− yⁱ α³(y)

µ

bj−s y^j α(y)

¶

+ 2

(1 +s)α²(y) µ

bj−s y^j α(y)

¶ µ

bi−s yⁱ α(y)

¶¸

.

So,

∂²

∂yⁱ∂y^j µ

− n+ 1

2(1 +s)h[u, y]m, ui

¶

=−h[u, y]_m, ui (1 +s)²

·

− 1 α³(y)

µ

bi−s yⁱ α(y)

¶

y^j− s α³(y)

µ

α(y)δjⁱ−y^j yⁱ α(y)

¶

− yⁱ α³(y)

µ

bj−s y^j α(y)

¶

+ 2

(1 +s)α²(y) µ

bj−s y^j α(y)

¶ µ

bi−s yⁱ α(y)

¶¸

−

1

α(y)(bj−s_α(y)^y^j )

(1 +s)² h[u, ui]m, ui −

1

α(y)(bi−s_α(y)^yⁱ )

(1 +s)² h[u, uj]m, ui.

Finally we have 2Eij(o, y) =∂²S(o, y)

∂yⁱ∂y^j = ∂²

∂yⁱ∂y^j µ

− n+ 1

2(1 +s)α(y)h[u, y]m, yi

¶

− ∂²

∂yⁱ∂y^j

µ n+ 1

2(1 +s)h[u, y]m, ui

¶

=−h[u, y]_m, ui (1 +s)²

·

− 1 α³(y)

µ

bi−s yⁱ α(y)

¶

y^j− s α³(y)

µ

α(y)δⁱj−y^j yⁱ α(y)

¶

− yⁱ α³(y)

µ

bj−s y^j α(y)

¶

+ 2

(1 +s)α²(y) µ

bj−s y^j α(y)

¶ µ

bi−s yⁱ α(y)

¶¸

−

1

α(y)(bj−s_α(y)^y^j )

(1 +s)² h[u, ui]m, ui −

1

α(y)(bi−s_α(y)^yⁱ )

(1 +s)² h[u, uj]m, ui

+ 1

(1 +s)α(y)(h[u, ui]m, uji+h[u, uj]m, uii)

− bj+_α(y)^y^j

(1 +s)²α²(y)(h[u, ui]m, yi+h[u, y]m, uii)

− bi+_α(y)^yⁱ

(1 +s)²α²(y)(h[u, uj]m, yi+h[u, y]m, uji) +



− δ^j_i

(1 +s)²α³(y)+ yⁱy^j

(1 +s)²α⁵(y) +2bi(bj+_α(y)^y^j ) (1 +s)α³(y)



h[u, y]m, yi.

Acknowledgement. The present work was supported by NSFC (no. 10971104) of China.

References

[1] D. Bao, S. S. Chern and Z. Shen,An Introduction to Riemann-Finsler Geometry, Springer-Verlag 2000.

[2] X. Cheng and Z. Shen, A class of Finsler metrics with isotropic S-curvature, Israel J. Math. 169 (2009), 317-340.

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[3] X. Cheng and Z. Shen, On Douglas metrics, Publ. Math. Debrecen 66 (2005), 503-512.

[4] S. S. Chern and Z. Shen,Riemann-Finsler Geometry, World Scientific Publishers 2004.

[5] S. Deng, The S-curvature of homogeneous Randers spaces, Differ. Geom. Appl.

27 (2009), 75-84.

[6] S. Deng and Z. Hou, Invariant Randers metrics on Homogeneous Riemannian manifolds, J. Phys. A: Math. Gen. 37 (2004), 4353-4360; Corrigendum, ibid, 39 (2006), 5249-5250.

[7] S. Deng and Z. Hou,The group of isometries of a Finsler space, Pacific J. Math.

207 (2002), 149-155.

[8] S. Helgason,Differential Geometry, Lie groups and Symmetric Spaces, 2nd ed., Academic Press 1978.

[9] S. Kobayashi, K. Nomizu,Foundations of Differential Geomtry, Interscience Pub- lishers, Vol. 1, 1963; Vol. 2, 1969.

[10] K. Nomizu,invariant affine connections on homogeneous spaces, Amer. J. Math.

76 (1954), 33-65.

[11] G. Randers, On an asymmetric metric in the four-space of General Relativity, Physics Rev. 59 (1941), 195-199.

[12] Z. Shen,Volume comparison and its applications in Riemann-Finsler geometry, Advances in Mathematics 128 (1997), 306-328.

[13] N. Wallach,Compact homogeneous Riemannian manifolds with strictly positive curvature, Annals of Mathematics (2), 96 (1972), 277-295.

Authors’ address:

Shaoqiang Deng and Xiaoyang Wang

School of Mathematical Sciences and LPMC, Nankai University, Tianjin 300071, P.R.China.

E-mail: [email protected]