The S-curvature of homogeneous (α, β)-metrics
Shaoqiang Deng and Xiaoyang Wang
Abstract. In this paper we give a formula of the S-curvature of homogeneous (α, β)-metrics. Then we use this formula to deduce a formula of
the mean Berwald curvature Eij of Randers metrics.
M.S.C. 2000: 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 the S-curvature of homogeneous (α, β)-metrics in Finsler geometry, for fundamental theory of Finsler geometry
we refer to [1] and [4]. Let M be a connected smooth manifold and α be a Riemannian 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 metric α. 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
β
form F = αφ(s) are called (α, β)-metric, here s = α
, and φ(s) is a function of s. It is
clear that (α, β)-metrics are the generalization of Randers metrics, in fact, if we set
β
φ(s) = 1 + s, then F = α(1 + α
) = 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 vanishing S-curvature.
It is a very interesting fact that the S-curvature and the flag curvature are subtly
related with each other. We now recall the definition of S-curvature. Let V be an
n-dimensional real vector space and F be a Minkowski norm on V . For a basis {ei }
of V , let
Vol(B n )
,
σF =
Vol{(y i ) ∈ Rn |F (y i ei ) < 1}
∗
Balkan
Journal of Geometry and Its Applications, Vol.15, No.2, 2010, pp. 39-48.
c Balkan Society of Geometers, Geometry Balkan Press 2010.
°
40
Shaoqiang Deng and Xiaoyang Wang
where Vol means the volume of a subset in the standard Euclidean space Rn and B n
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
p
det(gij (y))
τ (y) = ln
, y ∈ V \{0}
σF
is independent of the choice of the basis. τ = τ (y) is called the distortion of (V , F ).
Now let (M, F ) be a Finsler space. Let τ (x, y) be the distortion of the Minkowski
norm Fx on Tx M . For y ∈ Tx M − {0}, let τ (t) be the geodesic with τ (0) = x and
τ̇ (0) = y. Then the quantity
S(x, y) =
d
[τ (σ(t), σ̇(t))]|t=0
dt
is called the S-curvature of the Finsler space (M , F ). A Finsler space (M , F ) is said
to have almost isotropic S-curvature if there exists a smooth function c(x) on M and
a closed 1-form η such that:
S(x, y) = (n + 1)(c(x)F (y) + η(y)),
x ∈ M, y ∈ Tx M.
If in the above equation η = 0, then (M , F ) is said to have isotropic S-curvature. If
η = 0 and c(x) is a constant, then (M, F ) is said to have constant S-curvature.
In this paper we will obtain an explicit formula of the S-curvature of homogeneous
(α, β)-metrics, without using local coordinate systems. As an application, we prove
that a homogeneous (α, β)-metric has isotropic S-curvature if and only if it has vanishing S-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 the S-curvature of a G-invariant homogeneous (α, β)
β
. Since
metric F = αφ(s) on the coset space G/H of a Lie group G, where s = α
(G/H, F ) is homogeneous, we only need to compute this at the origin o = H. By [2],
in a local coordinate system, the S-curvature of the (α, β) metric F = αφ(s) with the
underlying Riemann metric α can be expressed as
S = (2Ψ −
where
Q=
φ0
φ−sφ0 ,
f 0 (b)
Φ
)(r0 + s0 ) − α−1
(r00 − 2αQs0 ),
bf (b)
2∆2
∆ = 1 + sQ + (b2 + s2 )Q0 ,
Ψ=
Q0
2∆ ,
Φ = −(Q − sQ0 ){n∆ + 1 + sQ} − (b2 − s2 )(1 + sQ)Q00 ,
rij = 12 (bi;j + bj;i ),
sij = 12 (bi;j − bj;i ),
bi = bj aji ,
rj = bi rij , sj = bi sij , r00 = rij y i y j , s0 = si y i , r0 = ri y i ,
and the function f (b) in the formula is defined as follows. The Busemann-Hausdorff
volume form dVBH = σBH (x)dx is defined by
ωn
,
σBH (x) =
∂
i
V ol{(y ) ∈ R : F (x, y i ∂x
i ) < 1}
S-curvature
41
and the Holmes-Thompson volume form, dVHT = σHT (x)dx is defined by
Z
1
σHT (x) =
det(gij )dy,
ωn {(yi )∈Rn |F (x,yi ∂ i )<1}
∂x
where Vol denotes the Euclidean volume, gij =
ωn = V ol(B n (1)) =
∂2
2
∂y i ∂y j [F ],
1
1
V ol(S n−1 ) = V ol(S n−2 )
n
n
and
Z
π
sinn−2 (t)dt.
0
p
When F = gij (x)y i y j is a Riemannian metric, both volume forms reduce to the
p
same Riemannian volume form dVHT = dVBH = det(gij (x))dx. Now for the (α, β)
β
metric F = αφ(s), s = α
, b = ||βx ||α , let dV = dVBH or dVHT . Then
f (b) =
 R π n−2
sin
tdt


,
R π0 sinn−2 t



 0 φ(b cos t)n dt
if dV = dVBH ,
Rπ



(sinn−2 t)T (b cos t)dt


R π n−2
,
 0
sin
tdt
0
if dV = dVHT ,
where T (s) = φ(φ − sφ0 )n−2 {(φ − sφ0 ) p
+ (b2 − s2 )φ00 }. Then the volume form dV is
given by dV = f (b)dVα , where dVα = det(αij )dx, denote the Riemannian volume
form of α.
In [2] the authors showed that if b = ||βx ||α is a constant, then r0 + s0 = 0. So in
this case we have
Φ
S = −α−1
(r00 − 2αQs0 ).
2∆2
Now we will deduce a formula of the S-curvature of homogeneous (α, β)-metrics.
Recall that the group I(M, F ) of isometries of a Finsler space (M, F ) is a Lie transformation group of M ([7]). If I(M, F ) acts transitively on M , then (M, F ) is called
homogeneous. Let (G/H, F ) be a homogeneous (α, β) metric of the form F = αφ(s),
β
where s = α
with α a G-invariant Riemannian metric on G/H and β 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 on To (G/H)
and o = H is the origin of G/H. It is clear that b = ||βx ||α is a constant. Therefore
the S-curvature can be expressed as
S = −α−1
Φ
(r00 − 2αQs0 ).
2∆2
(∗)
Also, G/H is a reductive homogeneous manifold in the sense of Nomizu ([10], see also
[9]), i.e, the Lie algebra of G has a decomposition:
(2.1)
g = h + m,
(direct sum of subspaces)
such that Ad(m) ⊂ m, ∀h ∈ H. Then we can identify m with the tangent space of
(G/H) at the origin o = H and in this way β corresponds a vector in m which is
42
Shaoqiang Deng and Xiaoyang Wang
invariant under the adjoint action of H on m. In the following we still denote this
vector by u.
Let h, i be 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 group ϕt ,t ∈ R of G/H by
φt (gH) = (exp(tv)g)H,
g ∈ G.
Then ϕt generates a vector field on G/H which is a Killing vector field (this is called
the fundamental vector field generated by v in [9]). We denote this vector field by v̄.
The following formula is a direct consequence of the formula in [9, Vol.2, page 201]
(see also [13]):
(2.2)
h(∇v̄1 v̄2 )|o , wi =
1
(−h[v1 , v2 ]m , wi + h[w, v1 ]m , v2 i + h[w, v2 ]m , v1 i),
2
where v1 , v2 , w ∈ m, and [v1 , v2 ]m denote the protection of [v1 , v2 ] to m 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. Let u1 · · · um be an orthonormal basis of m
with respect to h · , · i. Then by [8] there exists a neighborhood U of o in G/H such
that the mapping
(exp x1 u1 exp x2 u2 · · · exp xm um )H 7→ (x1 , x2 · · · xm )
defines a local coordinate system on U .
In the following we adopt the computation of r00 and s0 on (G/H, F ) in [5]. Let
u
(u1 , u2 , · · · , um ) be an orthonormal basis of (m, h · , · i) such that um = |u|
. Recall
1
2
that u is the vector in m corresponding to the 1-from β. Let (U, (x , x · · · xn )) be
the local coordinate system defined as above. Then in [5] it was shown that
1 2 l
c y h[un , ul ]m , un i
2
1
1
= h[cun , y l ul ]m , cun i = h[u, y]m , ui,
2
2
s0 (y)|o = y l sl (o) = cy l snl (o) =
and
c
rij (o) = − (h[un , ui ]m , uj i + h[un , uj ]m , ui i).
2
Moreover,
c
r00 |o = rij (o)y i y j = − (h[un , ui ]m , uj i + h[un , uj ]m , ui i)y i y j
2
c
= − (h[un , y i ui ]m , y j uj i + h[un , y j uj ]m , y i ui i)
2
c
= − (h[un , y]m , yi + h[un , y]m , yi) = −ch[un , y]m , yi.
2
S-curvature
43
Substituting the above into the formula (∗) we obtain the formula of S-curvature:
1 Φ
(rij (o)y i y j − 2α(y)Qs0 (y))
α(y) 2∆2
1 Φ
=−
(−ch[u, y]m , yi − α(y)Qh[u, y]m , ui).
α(y) 2∆2
S(o, y) = −
Summarizing, we get
Theorem 2.1 Let F = αφ(s) be a G-invariant (α, β)-metric on the reductive homogeneous manifold G/H with a decomposition of the Lie algebra
g = h + m.
Then the S-curvature of F has the form
S(o, y) = −
1 Φ
(−ch[u, y]m , yi − α(y)Qh[u, y]m , ui),
α(y) 2∆2
y ∈ m,
where u is the vector in m corresponding to the 1-form β, and we have identified m
with the tangent space of G/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 Scurvature if and only if F has vanishing S-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 = u and using the formula in Theorem 2.1, we get c(o) = 0.
Hence S(o, y) = 0, ∀y ∈ To (G/H). Since F 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])
µ m¶
∂G
1 ∂2
,
Eij =
2 ∂y i ∂y j ∂y m
where Gm = Gm (x, y) are the spray coefficients. We know that
S=
∂Gm
− (ln σ(x))xk y k
∂y m
44
Shaoqiang Deng and Xiaoyang Wang
here (ln σ(x))xk is the function of x because ln σ(x) is the function of x. Hence
0=
∂2
[(ln σ(x))xk y k ]
∂y i ∂y j
This means that
∂2S
∂2
=
i
j
∂y ∂y
∂y i ∂y j
·
¸
µ m¶
∂Gm
∂2
∂G
k
−
(ln
σ(x))
y
=
= 2Eij
k
x
∂y m
∂y i ∂y j ∂y m
Now we compute
∂2S
∂ 2 S(o, y)
(o,
y)
=
= 2Eij (o, y).
∂y i ∂y j
∂y i ∂y j
By Theorem 2.1 we have
∂ 2 S(o, y)
∂2
=
∂y i ∂y j
∂y i ∂y j
µ
¶
µ
¶
cΦ
∂2
ΦQ
h[u,
y]
,
yi
+
h[u,
y]
,
ui
.
m
m
2∆2 α(y)
∂y i ∂y j 2∆2
Before the computation, we recall that
∂s
1³
ym ´
∂α
ym
=
bm − s
,
=
,
m
∂y
α
α
∂y
α
where ym = amj y j . Since u1 , u2 , · · · , um in Section 2 is an orthonormal basis, we
have amj |o = δjm . Therefore at the origin we have ym = y m . Now we consider the
special case of the Randers metric φ = 1 + s. Let ψ = φ − sφ0 . Then we have
Q=
φ0
= 1,
ψ
Q0 = 0,
Ψ=
Q0
= 0,
2∆
∆ = 1 + sQ + (b2 − s2 )Q0 = 1 + s,
Θ=
Q − sQ0
1
1
=
=
,
2∆
2∆
2(1 + s)
Φ = −(Q − sQ0 ){n∆ + 1 + sQ} − (b2 − s2 )(1 + sQ)Q00 = −(n + 1)(1 + s).
Letting c = 1 we get
1 Φ
(−ch[u, y]m , yi − α(y)Qh[u, y]m , ui)
α(y) 2∆2
n+1
n+1
h[u, y]m , yi −
h[u, y]m , ui.
=−
2(1 + s)α(y)
2(1 + s)
S(o, y) =
Therefore
µ
¶
∂2
n+1
n+1
∂ 2 S(o, y)
=
−
h[u,
y]
,
yi
−
h[u,
y]
,
ui
m
m
∂y i ∂y j
∂y i ∂y j
2(1 + s)α(y)
2(1 + s)
µ
¶
µ
¶
n+1
∂2
n+1
∂2
−
h[u,
y]
,
yi
−
h[u,
y]
,
ui
.
=
m
m
∂y i ∂y j
2(1 + s)α(y)
∂y i ∂y j 2(1 + s)
2Eij (o, y) =
S-curvature
45
Now
µ
¶
n+1
h[u, y]m , yi
2(1 + s)α(y)
½
µ
¶
∂ 2 h[u, y]m , yi
n+1
∂2
1
1
=−
h[u, y]m , yi i j
+
2
∂y ∂y
(1 + s)α(y)
(1 + s)α(y)
∂y i ∂y j
µ
¶
µ
¶¾
∂h[u, y]m , yi ∂
∂h[u, y]m , yi ∂
Φ
1
+
+
,
∂y j
∂y i (1 + s)α(y)
∂y i
∂y j (1 + s)α(y)
∂2
∂y i ∂y j
−
where
∂h[u, y]m , yi
= h[u, ui ]m , yi + h[u, y]m , ui i,
∂y i
∂ 2 h[u, y]m , yi
= h[u, ui ]m , uj i + h[u, uj ]m , ui i,
∂y i ∂y j
and
∂
∂y j
µ
1
(1 + s)α(y)
¶
=−
=−
=−
∂s
∂y j
∂α
α(y) + (1 + s) ∂y
j
(1 + s)2 α2 (y)
1
(b
α(y) j
j
j
y
y
− s α(y)
)α(y) + (1 + s) α(y)
(1 + s)α(y)
(bj −
yj
)
s α(y)
j
y
+ (1 + s) α(y)
=−
(1 + s)2 α2 (y)
bj +
yj
α(y)
2 2
(1 + s) α (y)
.
Hence
∂2
∂y i ∂y j
=−
=−
µ
∂
(bj
∂y i
1
(1 + s)α(y)
+
yj
)(1
α(y)
¶


yj
bj + α(y)
∂ 

=
−
∂y i
(1 + s)2 α2 (y)
+ s)2 α2 (y) − (bj +
yj
) ∂ [(1
α(y) ∂y i
+ s)2 α2 (y)]
(1 + s)4 α4 (y)
j
∂
( y )(1
∂y i α(y)
+ s)2 α2 (y) − (bj +
yj
)2(1
α(y)
4 4
∂α(y)
∂s 2
+ s)α(y)[ ∂y
i α (y) + s ∂y i ]
(1 + s) α (y)
∂y j
∂y i
y j ∂α(y)
∂y i
α(y) −
1
2 2
α2 (y)
(1 + s) α (y)
µ
¶·
¸
2
yj
1
yi
yi
+
b
+
(b
−
s
)α(y)
+
s
j
j
(1 + s)α3 (y)
α(y)
α(y)
α(y)
α(y)
·
¸
µ
¶
i
1
2
yj
j
j y
=−
δ
α(y)
−
y
+
b
+
bi
j
i
α(y)
(1 + s)α3 (y)
α(y)
(1 + s)2 α4 (y)
=−
j
y
2bi (bj + α(y)
)
δij
yi yj
+
+
=−
.
2 3
2 5
3
(1 + s)α (y)
(1 + s) α (y)
(1 + s) α (y)
46
Shaoqiang Deng and Xiaoyang Wang
Therefore we have
µ
¶
∂2
n+1
−
h[u, y]m , yi
∂y i ∂y j
2(1 + s)α(y)


yj

2bi (bj + α(y)
)
δij
yi yj
h[u, y]m , yi
= −
+
+
 (1 + s)2 α3 (y)
(1 + s)α3 (y) 
(1 + s)2 α5 (y)
1
h[u, ui ]m , uj i + h[u, uj ]m , ui i
(1 + s)α(y)
+
bj +
−
yj
α(y)
2 2
(1 + s) α (y)
Note that
h[u, ui ]m , yi + h[u, y]m , ui i −
bi +
yi
α(y)
2 2
(1 + s) α (y)
h[u, uj ]m , yi + h[u, y]m , uj i.
¶
µ
¶
h[u, y]m , yi
n+1
n + 1 ∂2
−
h[u, y]m , ui = −
2(1 + s)
2 ∂y i ∂y j
(1 + s)
(
2 1
2
∂
n+1
1 ∂ h[u, y]m , ui
=−
+ h[u, y]m , ui i1+sj
2
1+s
∂y i ∂y j
∂y ∂y
)
1
∂ 1 ∂h[u, y]m , ui
∂ 1+s
∂h[u, y]m , ui
+ 1+s
+
,
∂y i
∂y j
∂y j
∂y i
∂2
∂y i ∂y j
µ
where
∂h[u, y]m , ui
= h[u, uj ]m , ui,
∂y j
1
∂ 1+s
∂y j
Hence
=−
∂ 2 h[u, y]m , ui
∂
=
h[u, uj ]m , ui = 0,
i
j
∂y ∂y
∂y i
j
y
− s α(y)
)
1
α(y) (bj
2
(1 + s)
.


yj
1
∂  α(y) (bj − s α(y) ) 
=
−
∂y i ∂y j
∂y i
(1 + s)2
1
∂ 2 1+s
=−
∂
[ 1 (b
∂y i α(y) j
j
y
− s α(y)
)](1 + s)2 −
1
(b
α(y) j
j
y
− s α(y)
) ∂(1+s)
∂y i
(1 + s)4
µ
¶
2(1 + s)
yj
∂s
bj − s
4
i
2
α(y)
∂y
(1 + s) α (y)
(1 + s) α(y)




j
µ
¶ i
i
∂ y
yj
1
y 
 1 − ∂s y − s α(y)  − 1
b
−
s
=−
j
2
α(y)
∂y i α(y)
∂y i
α2 (y)
α(y) α(y)
(1 + s)
¶
µ
¶
µ
2
1
yi
yj
+
bi − s
bj − s
α(y) α(y)
α(y)
(1 + s)3 α(y)


µ
¶
∂y j
∂α

α(y) − y j ∂y
i
1
1 
1
yi
yj
∂y i
=−
−
b
−
s
−
s
i
2 
2
α(y)
α(y)
α(y) α(y)
α (y)
(1 + s)
µ
¶ i ¸
µ
¶µ
¶¾
yj
y
2
yj
yi
1
bj − s
+
b
−
s
b
−
s
− 2
j
i
α (y)
α(y) α(y)
(1 + s)α2 (y)
α(y)
α(y)
j
=−
j
∂α(y)
y
y
∂
α(y) ∂y
i (bj − s α(y) ) − (bj − s α(y) ) ∂y i
2
+
S-curvature
47
·
µ
¶
µ
¶
i
1
1
yi
s
j
i
j y
−
b
−
s
y
−
α(y)δ
−
y
i
j
α3 (y)
α(y)
α3 (y)
α(y)
(1 + s)2
µ
¶
µ
¶µ
¶¸
i
j
j
y
y
2
y
yi
− 3
bj − s
+
b
−
s
b
−
s
.
j
i
α (y)
α(y)
(1 + s)α2 (y)
α(y)
α(y)
=−
So,
¶
n+1
h[u, y]m , ui
2(1 + s)
·
µ
¶
µ
¶
i
h[u, y]m , ui
1
yi
s
j y
j
i
=−
−
−
y
b
−
s
y
−
α(y)δ
i
j
α3 (y)
α(y)
α3 (y)
α(y)
(1 + s)2
µ
¶
µ
¶
µ
¶¸
i
j
j
y
y
2
y
yi
− 3
bj − s
+
b
−
s
b
−
s
j
i
α (y)
α(y)
(1 + s)α2 (y)
α(y)
α(y)
∂2
∂y i ∂y j
−
µ
−
j
y
− s α(y)
)
1
(b
α(y) j
(1 + s)2
h[u, ui ]m , ui −
1
(b
α(y) i
i
y
− s α(y)
)
(1 + s)2
h[u, uj ]m , ui.
Finally we have
µ
¶
µ
¶
∂ 2 S(o, y)
∂2
n+1
∂2
n+1
=
−
h[u,
y]
,
yi
−
h[u,
y]
,
ui
m
m
∂y i ∂y j
∂y i ∂y j
2(1 + s)α(y)
∂y i ∂y j 2(1 + s)
·
µ
¶
µ
¶
i
i
h[u, y]m , ui
1
y
s
j
i
j y
− 3
bi − s
y − 3
α(y)δj − y
=−
α (y)
α(y)
α (y)
α(y)
(1 + s)2
µ
¶
µ
¶µ
¶¸
i
j
j
y
y
2
y
yi
− 3
bj − s
+
b
−
s
b
−
s
j
i
α (y)
α(y)
(1 + s)α2 (y)
α(y)
α(y)
2Eij (o, y) =
−
1
(b
α(y) j
j
y
− s α(y)
)
1
(b
α(y) i
h[u, ui ]m , ui −
(1 + s)2
(1 + s)2
1
+
(h[u, ui ]m , uj i + h[u, uj ]m , ui i)
(1 + s)α(y)
−
−
bj +
yj
α(y)
2 2
(1 + s) α (y)
bi +
i
y
− s α(y)
)
h[u, uj ]m , ui
(h[u, ui ]m , yi + h[u, y]m , ui i)
yi
α(y)
2 2
(h[u, uj ]m , yi + h[u, y]m , uj i)
(1 + s) α (y)


yj
j
i j
2bi (bj + α(y)
)
δ
y
y
i
 h[u, y]m , yi.
+ −
+
+
(1 + s)α3 (y)
(1 + s)2 α3 (y)
(1 + s)2 α5 (y)
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.
48
Shaoqiang Deng and Xiaoyang Wang
[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 Publishers, 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: dengsq@nankai.edu.cn