S-curvature of cohomogeneity one Randers spaces

Accepted Manuscript S-curvature of cohomogeneity one Randers spaces

Jifu Li, Zhiguang Hu, Shaoqiang Deng

PII: DOI: Reference:

S0022-247X(16)30024-5 http://dx.doi.org/10.1016/j.jmaa.2016.03.084 YJMAA 20334

To appear in:

Journal of Mathematical Analysis and Applications

Received date:

28 June 2015

Please cite this article in press as: J. Li et al., S-curvature of cohomogeneity one Randers spaces, J. Math. Anal. Appl. (2016), http://dx.doi.org/10.1016/j.jmaa.2016.03.084

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S-curvature of cohomogeneity one Randers spaces∗ Jifu Li1 , Zhiguang Hu2 and Shaoqiang Deng3† 1

School of Science Tianjin University of Technology Tianjin 300384 People’s Republic of China 2

College of Mathematics Tianjin Normal University Tianjin 300387, P.R. China 3

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

Abstract In this paper, we study S-curvature of cohomogeneity one Randers spaces with orbit space S 1 . In particular, we present an explicit coordinate-free formula of S-curvature in terms of some quantities which is determined by the metric and the structure of the manifold. As an application, we give a sufficient and necessary condition for a cohomogeneity one Randers space to have almost isotropy S-curvature. Furthermore, we give some simple criterions for a cohomogeneity one Randers space to be a Berwald space or a Douglas space. Mathematics Subject Classification (2000): 53C30, 53C60. Key words: S-curvature, Randers metric, cohomogeneity one space, invariant vector field.

1

Introduction

The goal of this paper is to give an explicit coordinate-free formula of S-curvature of cohomogeneity one Randers spaces. S-curvature is a non-Riemannian quantity in Finsler geometry originally introduced by Z. Shen [21]. It is the rate of change of the Busemann-Hausdorff volume form of a Finsler space along a geodesic. Hence it has important application in the study ∗ Supported

by NSFC (no. 11271198, 11221091, 11226053, 11301358) and SRFPD of China author. E-mail: [email protected]

† Corresponding

1

of volume comparison of Finsler spaces. On a standard coordinate system, there is an explicit formula of S-curvature [8]. Although the formula is rather complicated, it has been applied in several occasions and has led to some interesting results. In particular, for a homogeneous Randers space S-curvature have a simple form in terms of Lie algebra structure and an inner product; see [9] or [10]. Using this formula, the third author obtain some interesting results on homogeneous Randers spaces. For example, it is proved that if a homogeneous Randers space has almost isotropic S-curvature, then it has vanishing S-curvature. Moreover, based on this formula, we are able to give a classification of some special homogeneous Randers spaces with almost isotropic S-curvature. Randers metrics are a special kind of Finsler metrics introduced by G. Randers [20] in his study of general relativity, hence they have important applications in physics. Moreover, it has also been found to have significant applications in the study of navigation and biology; see for example [7] and [4]. A Randers metric can be expressed as F = α + β, where α is a Riemannian metric and β is a 1-form on M satisfying βx < 1, ∀x ∈ M . On the other hand, a Finsler space (M, F ) is called cohomogeneity one, if the orbit space of the action of a connected subgroup of the connected full group of isometries I0 (M, F ) is 1 dimensional. This notion is a straightforward generalization of cohomogeneity one Riemannian manifolds. Cohomogeneity one Riemannian manifolds have been studied extensively, and many interesting results have been obtained. For example, many interesting new and significant examples, including Einstein metrics and positively curved metrics, have been constructed; see [1, 2, 3, 5, 12, 13, 14, 15, 16, 19, 22, 23]. There are also some studies on cohomogeneity one action on Alexandrov spaces [11], which is a natural synthetic generalization of Riemannian geometry. This fact indicates that the study of cohomogeneity one Finsler spaces will be interesting and will definitely lead to some important results. However, since the general case seems to be very involved, it is natural to begin with Randers metrics. This paper is a continuation of our previous work [18], where we have established the fundamental structure results of cohomogeneity one Randers spaces. For example, it is proved in [18] that a G-invariant Randers metric F = α + β on the cohomogeneity one space M fall into the following two cases: (1)

(M, α) is a homogeneous Riemannian space and the form β corresponds to a G-invariant  on M . vector field X

(2)

(M, α) is a cohomogeneity one Riemannian space and the form β corresponds to a G on M . invariant vector field X

This fact will be very useful in the study of S-curvature. However, there are many problems concerning cohomogeneity one Radners spaces which need to be handled. For example, although cohomogeneity one Randers metrics have been characterized in [18], it is still very difficult to construct explicit examples, or to classify them in some special cases. A natural idea is to consider those spaces with vanishing or istropic S-curvature, and in this sense, an explicit coordinate-free formula of S-curvature will be very useful. Meanwhile, in a recent paper of Hu-Deng [17], a classification of homogeneous Randers spaces with positive flag curvature and isotropic S-curvature is given. The same problem for cohomogeneity one spaces is very interesting, and the formula of S-curvature will be the start point of this consideration. In Section 2, we recall some fundamental definitions and results about cohomogeneity one Riemannian manifolds. Section 3 is devoted to the calculation of the Levi-Civita connection of cohomogeneity one Riemannia manifolds. In Section 4, we apply the results of Section 3 to give an explicit coordinate-free formula for the S-curvature of cohomogeneity one Randers metrics. Finally, in Section 5, we give some applications of the main results of this paper. The range of indices of this paper is as the following 2

1. The indices i, j, · · · (except n) range from 1 to n. 2. The indices λ, μ, · · · (except n + 1) range from 1 to n + 1.

2

Preliminaries

In this section, we recall some results about cohomogeneity one Riemannian manifolds and G-invariant vector fields. Let (M, α) be a n + 1 dimensional connected Riemannian manifold, and G a connected Lie group which acts smoothly and isometrically on M . Recall that M is called a cohomogeneity one with respect to G if the orbit space I = M/G is 1-dimensional. Then the topological space I is homomorphism to one of the following (see [2, 3, 5, 19, 24]): (i)

I = R;

(ii)

I = [0, +∞);

(iii)

I = S 1 = R/2πZ;

(iv)

I = [0, 1].

In the following, we only consider the case (iii) where the space M is a compact space, and the fundamental group is infinite. The other cases can be considered similarly. In this special case, all the orbits are principal. Let r = r(t), t ∈ [0, 2π] be a normal geodesic orthogonal to all the orbits. Denote K = Gr(π) . Then we have K = Gr(t) , ∀t ∈ [0, 2π] and K preserves the normal geodesic r(t). Hence the principal orbit is diffeomorphic to G/H. By the identification of (0, gH) and (2π, gωH), where ω ∈ NH (H)/H, the space M can be identified with [0, 2π] × G/H ([1], [5]). Then the Riemannian metric can be expressed as α = dt2 + αt , with Rω∗ α0 = α2π , where Rω : gH → gωH is the right action of ω on G/H. Let g = Lie G, h = Lie H, and fix a reductive decomposition of g: g = h + m,

(direct sum of subspaces)

where m is a subspace of g satisfying Ad(h)m ⊂ m,

∀h ∈ H.

Denote V := {u ∈ m| Ad(h)u = u, ∀h ∈ H}, and let T = Let

∂ ∂t

be the geodesic vector field along the normal geodesic r(t), t ∈ [0, 2π]. Tgr(t)

=

u i |gr(t)

=

(dτg )r(t) T, t ∈ [0, 2π], g ∈ G, d   g exp(su) · r(t), 1 ≤ i ≤ n + 1, ds s=0

where τg is the transformation of G/H defined by g  H → gg  H. The following theorem can be found in [18]. Theorem 2.1 Let (M, α) be a cohomogeneity one Riemannian space with respect to a compact Lie group G with M/G = I = S 1 . Then there is a bijection between the set of G-invariant Randers metrics on M with the underlying Riemamnian metric α and the set W = V ∪ {T } = {u ∈ m|Ad(h)u = u, ∀h ∈ H} ∪ {T }, where {T } is a set of an element. 3

 on M can be given by By this theorem, any smooth G-invariant vector field X  = c1 u X 1 |gr(t) + · · · + cs u s |gr(t) + cs+1 T|gr(t) ,

t ∈ [0, 2π],

where u1 , · · · , us is a basis of V , and c1 , · · · , cs+1 are G-invariant smooth functions on M , with ci (ωr(0)) = ci (r(2π)), i = 1, 2, . . . , s + 1.  on M satisfying α(X,  X)  < 1. Denote Selecting X F (x, y) = α(x, y) + β(x, y),

(1)

 y , and , is the inner product defined by α. Then (M, F ) is a where β(x, y) is given by X, cohomogeneous one Randers space under the action of G.

3

The Levi-Civita connection of cohomogeneity one Riemannian manifolds

We first deduce some results concerning the Levi-Civita connection of cohomogeneity one Riemannian space (M, α) with M/G = S 1 = R/2πZ. Let r : [0, 2π] → M be a normal geodesic parameterized by arc length. Fix a bi-invariant metric Q on g = Lie G. Let m = (h)⊥ , where h = Lie H. Then we have a direct sum decomposition g = h ⊕ m, with Ad(h)m ⊂ m, ∀h ∈ H and m can be identified with TH (G/H). Let α = dt2 + αt , be a G-invariant Riemannian metric α on M . ∂ For each t ∈ [0, 2π], the tangent space Tr(t) M is linearly spanned by T = ∂t and we have  be the fundamental vector field generated by X ∈ m, that is, Tr(t) (G · r(t))  TH G/H. Let X d   exp(sX) · gr(t), ∀ t ∈ [0, 2π]. X(gr(t)) =  ds s=0  Then X(r(t)) = X by Tr(t) G/H  TH G/H. Set  Y )r(t) = αt (X, Y ) = Q(Pt X, Y ), ∀ X, Y ∈ m, α(X, where Pt : m → m is a Q symmetric Ad(H)-equivalent endmorphism. Let ∇t be the Levi-civita connection of αt on the orbit G · r(t)  G/H. Then we have ∇X Y = ∇tX Y + αt (St X, Y )T, where St is the operator of G · r(t) at r(t) defined by ([12])  St X = −∇X T = −∇T X. For any X, Y, Z ∈ m, we have αt (∇X Y , Z)

= =

1 (−αt ([X, Y ], Z) + αt ([Z, X], Y ) + αt ([Z, Y ], X)) 2 1 (−Q(Pt [X, Y ], Z) + Q(Pt [Z, X], Y ) + Q(Pt [Z, Y ], X)). 2

4

Let u1 , · · · , un be an orthogonal basis of m with respect to Q. Then for each t ∈ [0, 2π], there exists a neighborhood U of r(t) in G · r(t), such that the map (exp(x1 u1 ) · · · exp(xn un ))r(t) −→ (x1 , · · · , xn ) defines a local coordinate system on U . Since Tr(t) M = span{T } ⊕ Tr(t) (G · r(t)),

(2)

∀ t ∈ [0, 2π],

there is a neighborhood N of r(t) in M such that (x1 , · · · , xn , xn+1 ), is a local coordinate system on N , where x For each fixed t ∈ [0, 2π], let

n+1

(3)

is the local coordinate of r(t)(t ∈ [0, 2π]) in N .

gr(t) = (x1 , · · · , xn , 0) ∈ N. ∂ From [9] or [10] , We know that the coordinate vector field ∂x i , (1  i  n) are given by   1 i−1 ∂  d =   exps(ex adu1 · · · ex adui−1 (ui )) · gr(t) ∂xi gr(t) ds s=0 = νi |gr(t) ,

where νi = ex

1

adu1

· · · ex

i−1

adui−1

(ui ). Furthermore, we have ∂  = νi |r(t) = ui , ∂xi r(t)  ∂  j r(t) , i  j. ∇ ∂ i j r(t) = ∇ui u ∂x ∂x Then we have  ∂  Γγij (r(t)) γ  = ∇ui u j r(t) , i  j, ∂x r(t) where Γγij are the Chistoffel symbols of the Levi-Civita connection under the coordinate system (xi ).   j r(t) = ∇tui u j r(t) + αt (St ui , uj )T , we have Since ∇ui u  j , u k )r(t) = αt (∇tui u j , uk ) α(∇ui u = Moreover

1 (−Q(Pt [ui , uj ], uk ) + Q(Pt [uk , ui ], uj ) + Q(Pt [uk , uj ], ui )). 2

 α(∇ui u j , u k )r(t) = Γlij · αt ( ul , u k ) = Γlij (r(t))Q(Pt ul , uk ), i  j,

whence 1 (−Q(Pt [ui , uj ], uk ) + Q(Pt [uk , ui ], uj ) + Q(Pt uk , uj , ui )). 2 = Q(Pt ul , uk ) and (Qlk ) = (Qlk )−1 . Then it follows From (4) that

Γlij (r(t))Q(Pt ul , uk ) = Denote Qlk

(4)

1 lk Q (−Q(Pt [ui , uj ], uk ) + Q(Pt [uk , ui ], uj ) + Q(Pt [uk , uj ], ui )), i  j. (5) 2 Denote T Q(Pt ui , uj ) = Q(P t ui , uj ). then we have  1 n+1 j , T )r(t) = αt (St ui , uj ) = − Q(Pt ui , uj ), i  j. (6) Γn+1 i u ij (r(t)) = Γij α(T, T ) = α(∇u 2 Similarly, 1 γ Γln+1,i (r(t)) = Qlj Q(Pt ui , uj ), Γn+1 n+1,i (r(t)) = 0, Γn+1,n+1 (r(t)) = 0. 2 Γlij (r(t))

=

5

4

The formula of the S-curvature

From now on, we let u be a nonzero element in V . Select an orthogonal basis u1 , · · · , un of m 1 such that un = u · Q− 2 (u, u) and define the local coordinate system as in (2). Let X=u |gr(t) + b(gr(t))T|gr(t) ,

(7)

with Q(Pt u, u) + b(t)2 < 1, b(gr(t)) = b(gωr(t)), t ∈ [0, 2π], g ∈ G. Then X is a G-invariant vector field on M with α(X, X) < 1. The X defined by (7) and the G−invariant metric α induce a Randers metric F (see (1)). From [9], we know that ∂  , u |gr(t) = c n  ∂x gr(t) 1

where c = Q 2 (u, u). Now we are ready to compute S-curvature of F . Keep all the notation and assumptions as before. Since (M, F ) is G-invariant and G · r(0) and G · r(2π) are the same orbit, we only need to compute along the normal geodesic r(t), t ∈ [0, 2π). Let (N, (x1 , · · · , xn+1 )) be the local coordinate system defined by (3). Then S-curvature of the Randers metric F is given by (see [8])  e 00 − (s0 + ρ0 ) , S = (n + 2) 2F where e00 , s0 and ρ0 are given by: I. e00 = eγμ y γ y μ , where eγμ = rγμ + bγ sμ + bμ sγ , rγμ = 12 (bγ;μ + bμ;γ ) and bγ are defined ∂b ∂bν by β = bγ dxγ . Moreover sγ = bμ sμγ and sγμ = aγτ sτ μ , where sμν = 12 ( ∂xμν − ∂x μ ) and μν −1 (a ) = (aμν ) ; II. s0 = sν y ν ; III. ρ0 = ρxμ y μ , where ρ = ln



1 − ||β||2 , and ||β|| is the length of β with respect to α.

Now we start to compute the quantities defined above. First, since       ∂ ∂ ∂ ∂    bi = β = α X, = α c + bT,    ∂xi r(t) ∂xi r(t) ∂xn ∂xi r(t)   ∂ ∂  = cα , i  = cαt (un , ui ) = Q(Pt u, ui ), n ∂x ∂x r(t)     ∂ ∂ ∂   = α c n + bT, n+1  = b, bn+1 = β  ∂xn+1 r(t) ∂x ∂x r(t) we have ∂bi   ∂xj r(t)

= =

∂bi   ∂xn+1 r(t) ∂bn+1   ∂xi r(t) ∂bn+1 ∂xn+1

∂ α ∂xj  cα ∇ c



 ∂ ∂  ,  ∂xn ∂xi r(t)    ∂ ∂ ∂ ∂   , + cα , ∇ ∂ ∂   ∂xj ∂xn ∂xi ∂xj ∂xi ∂xn r(t) r(t)

=

∂bi = Q(Pt u, ui ), ∂t

=

0,

=

b .

6

By the symmetry of Levi-Civita connection, we obtain      1 1 ∂bi ∂bj ∂ ∂ ∂ ∂  c α ∇ = − α ∇ − , , sij (r(t)) = ∂ ∂  ∂xn ∂xj ∂xn ∂xi ∂xj 2 ∂xj ∂xi 2 ∂xi r(t) 

t  1 t c αt ∇un u = j , u i − αt ∇un u i , u j 2 1 Q(Pt u, [ui , uj ]), = 2 1 sn+1,i (r(t)) = −si,n+1 (r(t) = − Q(Pt u, ui ), 2 and sn+1,n+1 (r(t)) = 0. Notice that at each point r(t), (t ∈ [0, 2π)), we have  Qij (aγμ ) = 

and (aγμ ) =

,

1

Qij 1

.

It follows that sγμ (r(t))

= a (r(t)) · sτ μ (r(t)) =

s n+1,μ (r(t)),

γ = n + 1,

γτ

aiτ · sτ μ (r(t)),

γ = i.

Therefore we have si (r(t)) = = = = sn+1 (r(t)) = = = =

bγ (r(t)) · sγi (r(t)) = bj (r(t)) · sji (r(t)) + bn+1 (r(t)) · sn+1 (r(t)) i kj Q(Pt u, uj ) · ski (r(t)) · Q + bsn+1,i (r(t)) b 1 Q(Pt u, uj )Q(Pt u, [uk , ui ]) · Qkj − Q(Pt u, ui ), 2 2 1 b Q(Pt u, [u, ui ]) − Q(Pt u, ui ), 2 2 bγ (r(t)) · sγn+1 (r(t)) bi (r(t)) · Qij Sj,n+1 1 Q(Pt u, ui )Qij · Q(Pt u, uj ) 2 1  Q(Pt u, u). 2

Thus for Y |r(t) = y i ui + dT = y + dT, we have s0 (Y )

= = =

y i si (r(t)) + dsn+1 (r(t)) b d 1 Q(Pt u, uj )Q(Pt u, [uk , y]) · Qkj − Q(Pt u, y) + Q(Pt u, u) 2 2 2 1 b d  Q(Pt u, [u, y]) − Q(Pt u, y) + Q(Pt u, u). 2 2 2

7

(8)

Now given rμν (r(t)), (t ∈ [0, 2π)), by the symmetry of rμν with respect to μ and ν, we only need to compute the case μ  ν. From the above formulas, we get  1 1 ∂bi ∂bj  (bi;j + bj;i )|r(t) = + − bγ · Γγij rij (r(t)) =  2 2 ∂xj ∂xi r(t) 1 = − Q(Pt u, [ui , uj ]) − bk · Γkij − bn+1 · Γn+1 ij 2 1 1 = − Q(Pt u, [ui , uj ]) − Q(Pt u, uk ) · [ Qkl (−Q(Pt [ui , uj ], ul ) 2 2 b +Q(Pt [ul , ui ], uj ) + Q(Pt [ul , uj ], ui ))] + Q(Pt ui , uj ), 2 1 b = − [Q(Pt [u, ui ], uj ) + Q(Pt [u, uj ], ui ))] + Q(Pt ui , uj ), 2 2 ∂bn+1  rn+1,n+1 (r(t)) = − bγ · Γγn+1,n+1 = b ,  ∂xn+1 r(t)   1 ∂bn+1 ∂bi  rn+1,i (r(t)) = + − bγ · Γγn+1,i  2 ∂xi ∂xn+1 r(t) 1 1 Q(Pt u, ui ) − Q(Pt u, uj )Qjl Q(Pt ui , ul ) = 2 2 = 0.

On the other hand

(bγ sμ + bμ sγ )|r(t)

⎧ Q(Pt u, uγ ) · [ 12 Q(Pt u, [u, uγ ]) − 2b Q(Pt u, uμ )] ⎪ ⎪ ⎪ ⎪ +Q(Pt u, uμ )[ 12 Q(Pt u, [u, uμ ]) − 2b Q(Pt u, uγ )], ⎪ ⎪ ⎪ ⎪ ⎨ 1  = 2 Q(Pt u, ui )Q(Pt u, u)2 ⎪ b b ⎪ ⎪ ⎪ + 2 Q(Pt u, [u, ui ]) − 2 Q(Pt u, ui ), ⎪ ⎪ ⎪ ⎪ ⎩ bQ(Pt u, u),

1  γ, μ  n, 1  γ  n, μ = n + 1, γ = μ = n + 1.

Hence e00 (Y )

=

rij (r(t))y i y j + 2ri,n+1 (r(t))y i y n+1 + rn+1,n+1 (r(t))y n+1 y n+1 +(bi sj + bj si )|r(t) y i y j + 2(bi sn+1 + bn+1 si )|r(t) y i y n+1

=

+2bn+1 sn+1 |r(t) y n+1 y n+1 b −Q(Pt [u, y], y) + Q(Pt y, y) + d2 b 2 +Q(Pt u, y)[Q(Pt u, [u, y]) − bQ(Pt u, y)] +dQ(Pt u, y)Q(Pt u, u) + bdQ(Pt u, [u, y]) −bd2 Q(Pt u, y) + bd2 Q(Pt u, u).

Moreover, since the length of β along each orbit G · γ(t), t ∈ [0.2π) is a constant, we have ρ0 = ργ y γ = dρt . In summarizing, we have proved the following Theorem 4.1 Let (M, α) be a cohomogeneity one Riemannian space with respect to the compact Lie group G with M/G = S 1 . Let F be an invariant Randers metric on M defined by the invariant Riemannian metric α and a G-invariant vector field defined by (7). Then the

8

S-curvature of F is given by  e 00 − (s0 + ρ0 ) S(r(t), Y ) = (n + 2) 2F  1 b [−Q(Pt [u, y], y) + Q(Pt y, y) + Q(Pt u, y)(Q(Pt u, [u, y]) = (n + 2) 2F 2 −bQ(Pt u, y)) + dQ(Pt u, y)Q(Pt u, u) + bdQ(Pt u, [u, y]) − bd2 Q(Pt u, y) 1 +bd2 Q(Pt u, u) + d2 b ] − Q(Pt u, [u, y]) 2  d b  + Q(Pt u, y) − Q(Pt u, u) − dρt , t ∈ [0, 2π), 2 2 Corollary 4.2 Under the same assumption as in Theorem 4.1, if b ≡ 0 in (7) and d ≡ 0 in (8), i.e., if the G-invariant vector field on M is given by u |gr(t) with Q(Pt u, u) < 1 and Y = y, where u is fixed by Ad(H), then we have  1 [−Q(Pt [u, y], y) + Q(Pt u, y)Q(Pt u, [u, y])] S(r(t), Y ) = (n + 2) 2F  1 − Q(Pt u, [u, y]) , t ∈ [0, 2π). 2 Corollary 4.3 Under the same assumption as in Theorem 4.1, if u ≡ 0 in (7), then S(r(t), Y ) = (n + 2)

  1 b [ Q(Pt y, y) + d2 b ] − dρt , (0 < b < 1), t ∈ [0, 2π). 2F 2

Remark Suppose (M, α) is a I0 (M, α) homogeneous space. Let G be a compact subgroup of I0 (M, α) and the action of G on M is cohomogeneity one. Then ρ = ln 1 − ||β|| is a constant on M . Thus ρ0 = ρxγ · y γ ≡ 0.

5

Some applications

Although the formula in Theorem 4.1 is somehow complicated, it is coordinate-free. Hence in many occasions it will be useful and convenient. In this section, we will give some interesting applications of this formula. First we have: Proposition 5.1 Let (M, F ) be a cohomogeneity one Randers metric. Then we have (1)

If F = α + β is generated by α and u ∈ V1 with Q(Pt u, u) < 1, then it is a Berwald metric if and only if Q(Pt [u, v], v) Q(Pt u, v)

(2)

= =

Q(Pt u, [v, w]) = 0, 0, t ∈ [0, 2π).

∀ v, w ∈ m,

If F = α + β is generated by α and bT with 0 < b < 1, then it is a Berwald metric if and only if Q(Pt u, v) = 0, ∀ v ∈ m,

t ∈ [0, 2π).

Proof. We only prove the first assertion, the proof of the other assertion is similar. From [9], we know that F is a Berwald metric if and only if the G-invariant vector field generated by u is parallel with respect to the Riemannian metric α, that is, if and only if Γγni = 0. Since F is a cohomogeneity one Randers metric, we only need to consider the normal geodesic r(t), t ∈ [0, 2π]. By (5) and (6), we have 9

−Q(Pt [ui , uj ], un ) + Q(Pt [un , ui ], uj ) + Q(Pt [un , uj ], ui ) = 0, and

Q(Pt u, v) = 0,

∀ u, v ∈ m, t ∈ [0, 2π).

Then by the invariant of un , we have Q(Pt [ui , uj ], un ) = Q(Pt [un , ui ], uj ), 2

from which the assertion follows.

A Finsler metric F is said to have almost isotropic S-curvature if there exists a closed 1-form η on M and a smooth function c(x) on M such that S(x, Y ) = (n + 2)(c(x)F (Y ) + η(Y )),

∀ Y ∈ T M.

Using the formula in Theorem 4.1 and a similar method as in [9], we can prove Proposition 5.2 Let (M, F ) be a cohomogeneity one Randers space. Then (M, F ) has almost isotropic S-curvature if and only if c(x) = 0, ∀x ∈ M and η(Y ) = 2b Q(Pt u, y) − d2 Q(Pt u, u) − dρt , t ∈ [0, 2π], where b and d are defined by (7) and (8). A Randers metric F = α + β is Douglas metric if and only if the 1-form β is closed, that is, dβ = 0 (see [6]). In the special case that F is a cohomogeneity one Randers metric, this is equivalent to the condition dβ(r(t)) = 0, t ∈ [0, 2π), or sμν (r(t) = 0, t ∈ [0, 2π). In summarizing, we have Proposition 5.3 Let (M, F ) be cohomogeneity one Randers space. Then F is of Douglas type if and only if Q(Pt u, [v, w]) = 0, Q(Pt u, v) = 0, ∀ v, w ∈ m, t ∈ [0, 2π). Example 5.4 Let G be a connected compact Lie group, g = Lie G. The manifold G × S1 is cohomogeneity one with the orbit space S 1 under the action of G. Fix a AdG-invariant inner product Q on g. Choose a normal geodesic γ(t), t ∈ [0, 2π] on G × S1 , then all the G-invariant metrics on G × S1 along γ(t), t ∈ [0, 2π] can be given as g = dt2 + f (t)2 Q, where f (t) is a smooth positive function on [0, 2π] with f (0) = f (2π) and f (t) < 1, t ∈ [0, 2π). For any u ∈ g with Q(u, u) = 1, then X = u |gr(t) , t ∈ [0, 2π), b ≡ 0 by (7) and  F (γ(t), y) =

g(u, y)2 + λg(y, y) g(u, y) − , t ∈ [0, 2π), λ λ

(9)

is a cohomogeneity one Randers metric, where λ = 1 − f 2 . Using adZ, Z ∈ g is skew symmetric with respect to Q and Theorem 4.1, the S-curvature is given S(r(t), Y ) = (n + 2)

 df 3 f  F

Q(u, y) − df f  +

df f   , t ∈ [0, 2π). 1 − f2

If G is semisimple, then the Randers metric F is not Berwald metric by Proposition 5.1. If G is not semisimple, then g = [g, g]. For each u ∈ [g, g]⊥ with Q(u, u) = 1, and f ≡ a < 1, where a is a constant. then the induced Randers metric (9) must be Berwald by Proposition 2 5.1. Moveover, obviously, if we take η = − a2 dt, then the Randers manifold (M, F ) has almost isotropic S-curvature by Proposition 5.2.

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References [1] D.V. Alekseevsky, Riemannian spaces of cohomogeneity one, Colloq. Maht. Soc. J. Bolyai., 56 (1989), 9-22. [2] A.V. Alekseevsky and D.V. Alekseevsky, G-manfolds with one dimensional orbit space, Ad. in Sov.Math., 8 (1992), 1-31. [3] A.V. Alekseevsky and D.V. Alekseevsky, Riemannian G-manfold with One-Dimensional Orbit Space, Ann. Glob. Anal. Geom., 11 (1993), 197-211. [4] P. L. Antonelli, R. S. Ingarden and M. Matsumoto, The Theory of Sprays and Finsler spaces with applications in Physics and Biology, Kluwer Academic Publishers, Dordrecht, 1993. [5] L. B`erard Bergery, Sur de nouvelle varietes riemanniennes d’Einstein, Publ. Inst. E. Cartan., 4 (1982), 1-60. [6] D. Bao, S.S. Chern, Z. Shen, An Introduction to Riemann-Finsler Geometry, SpringerVerlag, New York, 2000. [7] D. Bao, C. Robles, Z. Shen, Zermelo navigation on Riemannian spaces, J. Diff. Geom., 66 (2004), 377-435. [8] S. S. Chern, Z. Shen, Riemannian-Finsler geometry, World Scientific, Singapore, 2004. [9] S. Deng, The S-curvature of Homogeneous Randers Spaces, Diff. Geom. Appl., 27 (2009), 75-84. [10] S. Deng, Homogeneous Finsler Spaces, Springer, New York, 2012. [11] F. Galaz-Garcia, C. Searle, Cohomogeneity one Alexandrov spaces, Trans. Groups. 16 (2011), 91-107. [12] K. Grove, W. Ziller, Cohomogeneity one spaces with positive Ricci curvature, Invent. Math., 149 (2002), 619-646. [13] K. Grove, W. Ziller, Curvature and symmetry of Milnor spheres, Ann. of Math. 152 (2000), 331-367. [14] K. Grove, W. Ziller, Cohomogeneity one manifolds with positive Ricci curvature, Invent. math. 149 (2002), 619-646. [15] K. Grove, L. Verdiani, B. Wilking and W. Ziller, Non-negative curvature obstruction in cohomogeneity one and the Kervaire spheres, Ann. del. Scuola Norm. Sup. 5 (2006), 159170. [16] K. Grove, B. Wilking and W. Ziller, Positively curved cohomogeneity one manifolds and 3-Sasakian geometry, J. Diff. Geom. 78 (2008), 33-111. [17] Z. Hu, S. Deng, Homogeneous Randers spaces with isotropic S-curvature and positive flag curvature, Math. Z., 270 (2012), 989–1009 [18] J. Li, Z. Hu and S. Deng, cohomogeneity one Randers metrics, Can. Math. Bull., to appear. [19] P. Mostert, On a compact Lie group acting on a manifold, Ann. of Math, 65 (1955), 447-455; Errata, Ann. of Math. 66 (1957) 589.

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[20] G. Randers, On an assymmetric metric in the four-space of general relativity, Phys. Rev., 59 (1941), 195-199. [21] Z. Shen, Volume comparison and its applications in Riemannian-Finsler Geometry, Adv. Math., 128 (1997), 306-328. [22] L. Verdiani, Cohomogeneity one Riemannian manifolds of even dimension with strictly positive sectional curvature, I, Math. Z. 241 (2002), 329-339. [23] L. Verdiani, Cohomogeneity one manifolds of even dimension strictly positive sectional curvature, J. Diff. Geom. 68 (2004), 31-72. [24] W. Ziller, On the geometry of cohomogeneity one manifolds with positive curvature, in: Riemannian Topology and Geometric Sturctures on Manifolds, in honor of Charles P. Boyer’s 65th birthday, Progress in Mathematics 271 (2009), 233-262.

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