A local classification of a class of (α,β) metrics with constant flag curvature

Differential Geometry and its Applications 28 (2010) 170–193

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Differential Geometry and its Applications www.elsevier.com/locate/difgeo

A local classification of a class of (α , β) metrics with constant flag curvature Linfeng Zhou Mathematics Department, Peking University, Beijing, 100871, PR China

a r t i c l e

i n f o

a b s t r a c t

Article history: Received 30 April 2009 Available online 10 June 2009 Communicated by J. Slovak

We first compute Riemannian curvature and Ricci curvature of (α , β) metrics. Then we β apply these formulae to discuss a special class (α , β) metrics F = α (1 + α ) p (| p |  1) which have constant flag curvature. We obtain the sufficient and necessary conditions that (α +β)2

have constant flag curvature. Then we prove that such metrics must be locally F= α projectively flat and complete their local classification. Using the same method we find a

MSC: 53B40 53C60 58B20

2

necessary condition that flag curvature of F = αα+β is constant and proved that there are no non-trivial Matsumoto metrics. Furthermore, we give a negative answer whether there β are non-trivial metrics F = α (1 + α ) p (| p |  1) of constant flag curvature when β is closed. © 2009 Elsevier B.V. All rights reserved.

Keywords: (α , β ) metrics Locally projectively flat

1. Introduction In Finsler geometry one important open problem is to classify those metrics of constant flag curvature, which is the generalization of sectional curvature in Riemannian geometry. For the indicatrix of various Finsler metrics are quite different from each other, this problem is quite difficult. Obviously the simplest metrics are that the indicatrix are symmetric ellipses. This means that Finsler metrics are Riemannian. In the late 1920s a completely rigorous result as well-known is available: for every real number k there exists an uniqueness simple-connected complete Riemannian manifold of constant sectional curvature equal to k up to an isometry [4]. The next simpler case is Randers metrics. Its indicatrix can be viewed as a parallel translation of the indicatrix of a Riemannian metric along a vector. So they are non-symmetric metrics and quite close to Riemannian metrics. Yasuda and Shimada [15] and Matsumoto [7] first gave a characterization of Randers metrics with constant flag curvature in 1980s. Under the guidance of Yasuda–Shimada’s theorem, Bao and Shen constructed a family of Randers metrics on Lie group S 3 to prove there really exists the examples stated in there theorem [3]. Later some other examples were constructed by Shen [13], Yasuda–Shimada’s result was found to be some wrong. Soon Bao and Robles gave a corrected characterization by three conditions [1]. At the same time this was also done by Matsumoto and Shimada independently [9]. Another viewpoint of Randers metrics is Zermelo navigation problem. Roughly speaking, a Randers metric can be considered as a disturbance of a Riemannian metric by a vector field. By this navigation point, Bao, Robles and Shen simplified the three characterized conditions and gave a local classification [2]. Are there any other class Finsler metrics of constant flag curvature and how to classify them? In 1990s R. Bryant constructed a 2-parameter family of locally projectively flat Finsler metrics on S 2 with flag curvature K = 1 by using moving frame and complex geometry [6]. Bryant’s metrics are not (α , β) metrics. However, its indicatrix are the quartic curves.

E-mail address: [email protected]. 0926-2245/$ – see front matter doi:10.1016/j.difgeo.2009.05.008

© 2009 Elsevier B.V.

All rights reserved.

L. Zhou / Differential Geometry and its Applications 28 (2010) 170–193

171

In fact, in early 1920s L. Berwald gave a Finsler metric on unit ball B n of zero flag curvature which belongs to the class β F = α (1 + α )2 and is projectively flat [5]. Recently Mo, Shen and Yang found more such kinds of metrics by deformation of β

Randers metrics [12]. Soon Shen and Yildirim proved that these are all the non-trivial metrics F = α (1 + α )2 of constant flag curvature which are locally projectively flat [14]. β Hence one question is whether there are any other metrics F = α (1 + α )2 of constant flag curvature. In this paper, we β

proved that the answer is negative. In other words, we conclude that if F = α (1 + α )2 have constant flag curvature, then F must be locally projectively flat. Thus we complete the local classification of such metrics. The main idea of the proof is a classical method of separating irrational part and rational part of Riemannian curvature and Ricci curvature in Finsler geometry. We improve this method in Lemma 4.1 and it can simplify the proof of Bao and Robles in [1]. On the other hand We use Ricci curvature instead of Riemannian curvature to get some necessary conditions β because Ricci curvature is simpler. By these thoughts we obtained the necessary and sufficient conditions that F = α (1 + α )2 have constant flag curvature and found that these conditions can conclude that F are locally projectively flat. β Furthermore, we use the same methods to analyze a more general class (α , β) metrics F = α (1 + α ) p (| p |  1) and get some results. Especially we proved that there are no non-trivial Matsumoto metrics with constant flag curvature which were called a slop of mountain in [10]. 2. Notation and definitions The  (α , β) metrics were first introduced by Matsumoto [8]. They are Finsler metrics built from a Riemannian metric

α = ai j (x) y i y j , 1-form β = bi (x) y i and a C ∞ function φ(s) on a manifold M. A Finsler function of (α , β) metrics is given by the form

F := α φ(s),

β

s=

α

.

To satisfy that F is positive and strongly convex on T M \0, it is known that if and only if

  φ(s) − sφ  (s) + B − s2 φ  (s) > 0,

φ(s) > 0,

|s|2  B < b0

where B := b i b j ai j = β2α . There are many examples of (α , β) metrics. The most familiar such kind of metrics are Randers metrics F = α + β . As we know that Randers metrics satisfy to be positive and strongly convex if and only if B = β2α < 1. Till now mathematician have mastered their curvatures quite well. β Another examples, we will discuss in this paper, are a more general class F = α (1 + α ) p in (α , β) metrics which include Randers metrics and Matsumoto metrics [10]. It is clear that there exist a positive real number  ( p ) > 0 depending on index p s.t. when B <  and | p |  1, F satisfy the positive and strongly convex conditions. Let x denote points on the manifold M, y ∈ T x M denote tangent vectors at point x. The fundamental tensor of a Finsler metric F is formally analogous to the metric tensor in Riemannian geometry. In local coordinates it is defined by

g i j :=

1 2

F2

 yi y j

.

Spray coefficients G i are defined by

G i :=

1 4



g il 2

 ∂ g jl ∂ g jk − y j yk . ∂ xk ∂ xl

A Finsler metric is called locally projectively flat if there exists a local coordinate such that the geodesics can be parametrized as a straight line. This is equivalent to that under this local coordinate spray coefficients must satisfy

G i = P yi where P is a smooth positively homogeneous function on T M \{0} and is called a projective factor. As done in Randers metrics, there are two key quantities:

r i j :=

1 2

(b i | j + b j |i ),

si j :=

1 2

(b i | j − b j |i )

where b i | j means the coefficients of the covariant derivative of β with respect to

r

i

j

ik

:= a rkj , i

s

i

j

s0 := si y i ,

s00 := si j y i y j = 0, r := r i j b i b j ,

:= a skj ,

r i0 := r i j y j ,

r00 := r i j y y , si := b j s j i ,

j

ik

si0 := si j y j ,

sb := si j b i b j = 0.

α . Furthermore, we denote

172

L. Zhou / Differential Geometry and its Applications 28 (2010) 170–193

3. The flag and Ricci curvature of (α , β ) metrics Now we recall the definition of Riemannian curvature. For a vector y = y i ∂ i |x ∈ T x M, define R y = R i j dx j ⊗ ∂ i |x : ∂x ∂x T x M → T x M by

 

R i j := 2 G i

xj

        − yk G i xk y j + 2G k G i yk y j − G i yk G k y j .

For any tangent plane P = span{ y , u } ⊂ T x M define

K ( P , y ) :=

g y ( R y (u ), u ) g y ( y , y ) g y (u , u ) − g y ( y , u ) g y ( y , u )

and K is called flag curvature. Usually, K ( P , y ) depends on the direction y ∈ P . In Riemannian case, K ( P , y ) is independent of y ∈ P . So flag curvature generalizes sectional curvature in Riemannian geometry. Later we will use a basic fact time and again that a Finsler metric F has constant flag curvature if and only if [1]



R i j = K F 2 δ ij −

yi F



Fyj .

As we know that the spray coefficients G i of an (α , β ) metric F := α φ(s) and the spray coefficients α G i of the Riemannian metric α are related by [14]

G i = α G i + Θ(−2α Q s0 + r00 )li + Ψ (−2α Q s0 + r00 )b i + α Q si 0 where

φ , φ − sφ  φφ  − s(φφ  + φ  φ  ) , Θ := 2φ((φ − sφ  ) + ( B − s2 )φ  ) φ  Ψ := 2((φ − sφ  ) + ( B − s2 )φ  ) Q :=

yi

β

li := α and s := α . Denote ζ i := Θ(−2α Q s0 + r00 )li + Ψ (−2α Q s0 + r00 )b i + α Q si 0 . By Berwald’s formula for Riemannian curvature:





R i j = α R i j + 2ζ i | j − yk ζ i |k



yj

 

    − ζ i y k ζ k y j + 2ζ k ζ i y j y k ,

we can compute Riemannian curvature of (α , β) metrics using Maple and have the following proposition: Proposition 3.1. For an (α , β) metric F := α φ(s), its Riemannian curvature tensor R i j is locally given by

Ri j = α Ri j + T i j where



T i j = δ i j d + li l j ll + li b j lb + li s j ls + li r j lr + li s0 j ls1 + li r j0 lr1 − li s j |0 (u α ) + li s0| j (2u α ) − li sk sk j u α 2 Q

+ li r00| j (2 f ) − li r j0|0 (2 f ) − li rk0 sk j (2Q α f ) + li r jk sk 0 (4Q α f ) + b i l j bl + b i b j bb + b i s j bs + b i r j br + b i s0 j bs1 + b i r j0 br1 − b i s j |0 ( v α ) + b i s0| j (2v α )   − b i sk sk j v Q α 2 + b i r00| j (2g ) − b i r j0|0 (2g ) − b i rk0 sk j (2Q α g ) + b i r jk sk 0 (4Q α g ) + r i j cr + si j cs + si 0l j sl + si 0 b j sb1 + si 0 s j ss + si 0 s0 j ss1 + si 0 r j0 sr   + si l j sl1 + si b j sb2 − si s j Q v α 2 + si r j0 (2g Q α ) + r i 0l j rl + r i 0 b j rb + r i 0 s j rs + r i 0 r j0 rr   + si k sk 0l j ssl + si k sk 0 b j ssb − si k sk j Q 2 α 2 + si 0| j (2Q α ) − si j |0 ( Q α ) + si 0|0l j sl2 − si 0|0 b j Q s and

d=

2  r00

α2 +





2 f s g B − s2 − f s + f 2 − 2sf g

r00 s0 

α





 



2v f s B − s2 + 2 f u + 2gu s B − s2 − 2v f s − u s + 2Q f s − 2 f B

    + s20 −2u B + u 2 + 2u s v B − s2 + 2Q u s + r0 s0 (−2u B + 4v f )





L. Zhou / Differential Geometry and its Applications 28 (2010) 170–193

r00 r0

+

α

2  r00

ll =

α2

(−2 f B + 4 f g ) −

α

f + 2α sk sk 0 Q u + rk0 sk 0 (4Q f ) − s0|0 u ,

   (sf s g s − 4 f s g − 2sf ss g ) B − s2 + 6sf g + f 2 + 6s2 f s g + sf f s + sf ss + 2 f s − s2 f g s

r00 s0 

+

r00|0

173

α

  (−2sgu ss + sg s u s + sf s v s − 2sf ss v − 5 f s v ) B − s2 + 2sf Bs

+ f u + sf s u + 4s2 gu s − 5Q f s + su ss + sf u s + s Q s f s − 2Q f ss s − s2 f v s       + 6s2 f s v − 2gu s B − s2 + 7sf v + u s + 2 f B + s20 (su s v s − 3u s v − 2su ss v ) B − s2  + 4s2 u s v − 2Q u ss s + suu s − 3Q u s + 2u Bs s + s Q s u s + r0 s0 (−4sf s v + 2sf v s + 2su Bs − 6 f v ) r00 r0

+

α

(2 f B + 2sf Bs + 2sf g s − 4 f g − 4sf s g ) + α sk sk 0 (s Q s u − Q u − 2s Q u s ) +

r00|0

α

( f + sf s )

+ s0|0 (su s ) + rk0 sk 0 (2s Q s f − 4s Q f s − 6Q f ), lb =

2  r00

   (2 f ss g − f s g s ) B − s2 − 2 f g − f f s + sf g s − 6sf s g − f ss α     + s20 (2u ss v − u s v s ) B − s2 − uu s − Q s u s − 2u Bs − 4su s v + 2Q u ss   r00 s0  (2 f ss v + 2gu ss − g s u s − f s v s ) B − s2 − Q s f s + α  − f s u + 2Q f ss − 2 f Bs − 4sgu s + sf v s − u ss − f u s − 2 f v − 6sf s v − r0 s0 (2u Bs + 2 f v s − 4 f s v ) 2

r00|0

r00 r0

(−2 f g s + 4 f s g − 2 f Bs ) + rk0 sk 0 (4Q f s − 2Q s f ) + α sk sk 0 (2Q u s − Q s u ) − s0|0 u s − f s, α     ls = r00 (2gu s − f s v ) B − s2 − u f + 4 f B + sf v − u s − Q f s    + α s0 2u B − u 2 + Q u s + u s v B − s2 − α r0 (2 f v + 2u B ), +

α

lr = r00 (4 f B + 4 f g ) + α s0 (4u B + 4 f v ), r00 ls1 = (3Q f + 3 f s + 3s Q f s ) + s0 (3s Q u s + 3u s ),

α

   r00  lr1 = 2 f s g B − s2 − 2sf g − f s − 2 f 2

α

    + s0 (4 f s v − 2gu s ) B − s2 + 4Q f s + u s − 2 f u − 4 f B − 4sf v − r0 (4 f B + 4 f g ), 2  r00

bl =

α2 +

sg s2 − 2sgg ss − 2gg s

r00 s0 

α





B − s2 + sg ss + g s + 4s2 gg s



  (2sg s v s − 3g s v − 2sg ss v − 2sg v ss ) B − s2 + sv ss

 r00 r0 + 2sg Bs − 2Q sg ss + 5s2 g s v + Q s sg s + 2s2 g v s − 2sg v − 3Q g s + (−2sgg s + 2sg Bs ) α     + s20 sv 2s − 2sv v ss − v v s B − s2 + 3s2 v v s − 2s Q v ss − 3sv 2 − Q v s − 2v B + s Q s v s + 2sv Bs + s0 r0 (2sv Bs − 2g v + 2sg v s − 4sg s v − 2v B ) +

r00|0

α

(sg s ) + rk0 sk 0 (−2Q g + 2sg Q s − 4s Q g s )

+ s0|0 (sv s − v ) + α sk sk 0 (s Q s v + Q v − 2s Q v s ), bb =

2  r00

α2 + +

2gg ss − g s2

r00 s0 

α s20







B − s2 − g ss − 4sgg s



   (2g ss v + 2g v ss − 2g s v s ) B − s2 − 2g Bs − v ss + 2v g − 2sg v s − 5sg s v + 2Q g ss − Q s g s

2v v ss − v 2s







B − s2 − Q s v s − 2v Bs + 2v 2 + 2Q v ss − 3sv v s + r00|0

r00 r0

α

(−2g Bs + 2gg s )

+ s0 r0 (−2g v s + 4g s v − 2v Bs ) − g s − s0|0 v s + α sk sk 0 (2Q v s − Q s v ) + rk0 sk 0 (4Q g s − 2Q s g ), α     bs = r00 (− g s v + 2g v s ) B − s2 + 4g B − Q g s − v s + 2sg v     + α s0 v v s B − s2 + 2v B + Q v s + sv 2 − α r0 (2g v + 2v B ),   br = r00 4g 2 + 4g B + α s0 (4v B + 4g v ),

174

L. Zhou / Differential Geometry and its Applications 28 (2010) 170–193

bs1 = br1 =

r00

α

  (3g s + 3Q sg s ) + s0 3v s − 3( v − sv s ) Q ,

r00 

α



















2gg s B − s2 − g s + s0 (4g s v − 2g v s ) B − s2 − 2v s − 2sg v + 4Q g s − 4g B + r0 −4g 2 − 4g B ,

cr = r00 (2g ) + α s0 (2v ),

















cs = α s0 2Q s v B − s2 + 2Q Q s + 2Q sv + 2v + r00 2g − Q s + 2g Q s B − s2 + 2Q sg ,

   ( Q s sg s − 2sg Q ss ) B − s2 + 2Q s2 g − 2Q sg + Q s2 g s + s Q ss + sg s α    + s0 (s Q s v s − Q s v − 2Q ss sv ) B − s2 + Q s2 v s − Q s Q + s Q s2 − 3Q sv  − v + sv s − 2Q Q ss s + 2Q s2 v ,    r00  (2Q ss g − Q s g s ) B − s2 + 2g Q − 2sg Q s − Q sg s − Q ss − g s sb1 = α     + s0 (2Q ss v − Q s v s ) B − s2 − v s − Q s2 + 2Q v − 2Q s sv + 2Q Q ss − Q sv s ,     ss = α − Q s v B − s2 − Q Q s − v − Q sv , sl =

r00 

ss1 = −3Q 2 + 3Q Q s s + 3Q s ,









sr = −2 Q s B − s2 + s Q + 1 g + Q s , sl1 = r00 (−2g Q + 2Q s sg − Q sg s ) + α s0 ( Q v + Q sv s − 2Q s sv ), sb2 = r00 (−2Q s g + Q g s ) + α s0 (2Q s v − Q v s ), ss2 = −α 2 Q v , sr1 = 2α Q g , r00 rl = sg s + s0 (sv s − v ),

α

rb = −

r00

α rs = − v α ,

g s − s0 v s ,

rr = −2g , ssl = α ( Q − s Q s ) Q , ssb = α Q Q s , sl2 = −( Q − s Q s ). Here we denote

f (s, B ) := Θ,

g (s, B ) := Ψ,

u (s, B ) := −2Θ Q ,

v (s, B ) := 2Ψ Q .

Proof. As the calculation is tiresomely long and can be done by Maple programme, we write the procedure in Appendix A. 2 Remark. For a Randers metric F = α + β , its spray coefficients are given by

Gi = α Gi +

1 2(1 + s)

(−2α s0 + r00 )li + α si 0 .

So

f (s, B ) =

1 2(1 + s)

u (s, B ) = −

1 1+s

,

,

g ( s , B ) = 0, v ( s , B ) = 0,

Q = 1.

According to Proposition 3.1 we can compute Riemannian curvature R i j :

L. Zhou / Differential Geometry and its Applications 28 (2010) 170–193

 R i j = α R i j + δ ij d + li l j ll + li b j lb + li s0 j

3s0







1



2

175







1

+ αli s j |0 − α l i s 0| j + α 2li sk sk j 1+s 1+s 1+s         1 1 1 2 − li r j0|0 − αli rk0 sk j + αli r jk sk 0 − 3si 0 s0 j + li r00| j 1+s 1+s 1+s 1+s 1+s

where 2 d = r00

3 4α 2 (1 + s)2 3 2

− r00 s0

3

α (1 + s)2

+ s20

3

(1 + s)2

− r00|0

1 2α (1 + s)

+ rk0 sk 0

2 1+s

− sk sk 0

2α 1+s

+ s0|0

1 1+s

,

3 3 + r00 s0 − s20 4α 2 (1 + s)3 α (1 + s)3 (1 + s)3 1−s s+3 1 s + α sk sk 0 − rk0 sk 0 + r00|0 + s0|0 , 2 2 2 (1 + s) (1 + s) 2α (1 + s) (1 + s)2 3 3 3 2 lb = −r00 + r00 s0 − s20 4α 2 (1 + s)3 α (1 + s)3 (1 + s)3 2 2 1 1 − rk0 sk 0 + α sk sk 0 + r00|0 − s0|0 . (1 + s)2 (1 + s)2 2α (1 + s)2 (1 + s)2 ll = −r00

The above equation coincides with the formula of Riemannian curvature in [1]. Ricci curvature is the trace of Riemannian curvature. When we research flag curvature, Ricci curvature is often more convenient and simpler. In Proposition 3.1 we get a local expression of Riemannian curvature of an (α , β) metric. It is natural to compute Ricci curvature by Maple and we have a similar formula. Proposition 3.2. Under the same notation in Proposition 3.1 Ricci curvature R m m of an (α , β) metric is given by

Rmm = α Rmm + T mm where

T m m := (n − 1)

2 r00

2 r00

α

α

c + 2 1

c 2 2

+ (n − 1)s20 c 3 + s20 c 4 + (n − 1)

r00 s0

α

c5 +

r00 s0

α

c 6 + (n − 1)

  r00|0 + (n − 1) r0 s0 (4 f v − 2u B ) + sk sk 0 α (2Q u ) + rk0 sk 0 (4Q f ) − f − us0|0 k

+ rr00 c 10 + r k r00 c 11 + s0 r0 c 12 + α s0 rc 15 + α s0 r + α sk 0 rk c 23 + s0|0 c 24 +

k

2 k c 16 + r 0 c 17

r00 r0

α

c7 +

r00 r0

α

c8

α + α sk sk 0 c 20 + sk 0 s0k c 21 + sk 0 rk0 c 22

r00|0

c 25 + α rk0 sk c 26 + bk r00|k c 27 + α bk sk|0 c 28 + α bk s0|k c 29 + α r k 0 sk c 30 α + r k 0 rk0 c 31 + bk rk0|0 c 32 − α 2 si k sk i Q 2 + 2α sk 0|k Q and





c 1 = f 2 + 2g f s B − s2 − f s − 2sf g ,



c 2 = 2gg ss − g s2





B − s2

2



  + (− g ss − 6sgg s ) B − s2 + 2sg s ,

c 3 = u 2 + 2u s v B − s2 + 2Q u s − 2u B ,



c 4 = 2v v ss − v 2s



B − s2

2

   + 2Q v ss − 2v Bs − 4sv v s + 2Q ss v − 2Q s v s + 2v 2 B − s2

+ 6Q v + 2Q ss Q − 4s Q v s − 4s Q s v − 4v s − s2 v 2 − Q s2 − 2sv B ,     c 5 = 2Q f s − 2 f B − u s + 2 f u + 2 f s v B − s2 + 2gu s B − s2 − 2sf v , 2  c 6 = (2g ss v + 2g v ss − 2g s v s ) B − s2

  + (2Q ss g − 2sg v s − 2Q s g s − 8sg s v − 2g Bs + 2Q g ss − v ss + 2g v ) B − s2 − 4g s − 8Q sg s − v + sv s − Q ss ,

c7 = 4 f g − 2 f B ,





c 8 = (4gg s − 2g Bs ) B − s2 − 2g s ,





c 9 = 4g B + (4g v s − 2g s v ) B − s2 + 4sg v − v s − 2Q s g , c 10 = 4g 2 + 4g B , c 11 = 2g ,

176

L. Zhou / Differential Geometry and its Applications 28 (2010) 170–193





c 12 = (8g s v − 2v Bs − 4g v s ) B − s2 − 2sv B − 4sg v − 4g B + 4Q g s − 3v s ,









c 13 = 2 v v s B − s2 + sv 2 + v B + Q s v , c 15 = 4g v + 4v B ,

c 16 = 2v ,

2

c 17 = −4g B − 4g ,

c 18 = −2v B − 4g v ,

2





c 19 = − v ,

c 20 = (−2Q s v + 2Q v s ) B − s2 − v ,

c 21 = 2Q s + 2s Q Q s − 2Q 2 ,

c 22 = (4Q g s − 4Q s g ) B − s2 − 4Q sg + 2Q s − 2g ,

c 23 = 4g Q ,

c 24 = − v s B − s2 − sv − Q s ,











c 25 = − g s B − s2 ,

c 26 = 4g Q ,

c 27 = 2g ,

c 28 = − v ,

c 29 = 2v ,

c 30 = − v ,

c 31 = −2g ,

c 32 = −2g .



Proof. In Proposition 3.1 contract the upper index and down index:

T m m = n(d) + ll + s(lb) +

s0

α

ls +

r0

α

lr +

r00

α

lr1 − s0|0 u + s0|0 (2u ) − sk sk 0 (u α Q ) +

r00|0

α

(2 f ) −

r00|0

α

(2 f )

− rk0 sk 0 (2Q f ) + rk0 sk 0 (4Q f ) + s(bl) + B (bb) + r (br) − s0 (bs1 ) + r0 br1   − bk sk|0 ( v α ) + bk s0|k (2v α ) + sk sk v Q α 2 + bk r00|k (2g ) − bk rk0|0 (2g ) + rk0 sk (2Q α g ) + rk sk 0 (4Q α g )   s0 r00 rl + r k k cr + sk k cs + s0 sb1 + sk 0 sk ss + sk 0 s0k ss1 + sk 0 rk0 sr + sl1 − sk sk Q v α 2 + sk rk0 (2g Q α ) +

α

+ r0 rb + rk0 sk rs + r k 0 rk0 rr + s0k sk 0

ssl

α

α

+ sk sk 0 ssb − si k sk i Q 2 α 2 + sk 0|k (2Q α ) − sk k|0 Q α − s0|0 Q s

+ sk 0 rk0 Q s − sk 0 s0k Q s . Substituting the coefficients in Proposition 3.1 into above equation and simplifying it, then we can get the result.

2

One important case of (α , β) metrics is that 1-form β is closed i.e. si j = 0. Under this condition Riemannian curvature and Ricci curvature are much easier. Corollary 3.3. If 1-form β is closed, Riemannian curvature tensor R i j and Ricci curvature R m m of an (α , β) metric can be locally expressed by:

Ri j = α Ri j + T i j, Rmm = α Rmm + T mm where

T i j = δ i j d + li l j ll + li b j lb + li r j lr + li r j0 lr1 + li r00| j (2 f ) − li r j0|0 (2 f ) + b i l j bl + b i b j bb + b i r j br + b i r j0 br1

+ b i r00| j (2g ) − b i r j0|0 (2g ) + r i j cr + r i 0l j rl + r i 0 b j rb + r i 0 r j0 rr, T m m :=

2 r00

α2 +

   r0 r00  r00|0 (n − 1) f 2 + 2 f s g B − s2 − 2sf g − f s + (n − 1)(4 f g − 2 f B ) − (n − 1) f

α

2  r00

α2

 2

2gg ss − g s

α

   − (6gg s s + g ss ) B − s2 + 2sg s

  r00|0    (4gg s − 2g sB ) B − s2 − 2g s − g s B − s2 α α     − r02 4g 2 + 4g B + rr00 4g 2 + 4g B + 2gr00|i b i − 2gri0 b i + 2gr00 r ii − 2gri0 r i 0 , +

and

r0 r00 

B−s

 2 2

L. Zhou / Differential Geometry and its Applications 28 (2010) 170–193

2  r00

d=

α

2







2 f s g B − s2 − f s + f 2 − 2sf g +

r00 r0

α

(−2 f B + 4 f g ) −

r00|0

α

177

f,

2  r00

ll =

   (sf s g s − 4 f s g − 2sf ss g ) B − s2 + 6sf g + f 2 + 6s2 f s g + sf f s + sf ss + 2 f s − s2 f g s

α2

r00 r0

+ lb =

α

(2 f B + 2sf Bs + 2sf g s − 4 f g − 4sf s g ) +

r00|0

α

( f + sf s ),

2  r00

α

  r00 r0  r00|0 (2 f ss g − f s g s ) B − s2 − 2 f g − f f s + sf g s − 6sf s g − f ss + (−2 f g s + 4 f s g − 2 f Bs ) − f s,

2

α

α

lr = r00 (4 f B + 4 f g ),    r00  lr1 = 2 f s g B − s2 − 2sf g − f s − 2 f 2 − r0 (4 f B + 4 f g ),

α

bl =

2  r00

α2

bb =

sg s2 − 2sgg s − 2gg s

2  r00

α2



2gg ss − g s2









B − s2 + sg ss + g s +





B − s2 − g ss − 4sgg s +

r00 r0

α



r00 r0

α

(−2sgg s + 2sg Bs ) +

(−2g Bs + 2gg s ) −

r00|0

α

r00|0

α

(sg s ),

gs ,

br = r00 4g 2 + 4g B ,      r00  br1 = 2gg s B − s2 − g s + r0 −4g 2 − 4g B ,

α

cr = r00 (2g ), r00 rl = sg s ,

α

rb = −

r00

α

gs ,

rr = −2g . Proof. Substituting si j = 0 into Propositions 3.1 and 3.2 will get the result. 4. A local classification of Finsler metrics F :=

(α +β)2

α

2

with constant flag curvature

L. Berwald constructed a projectively flat (α , β) metric F = example is given by [5]

(α +β)2

α

with zero flag curvature on B n ⊂ Rn . The detailed

 ( | y |2 − (|x|2 | y |2 − x, y 2 ) + x, y )2  F := . (1 − |x|2 )2 | y |2 − (|x|2 | y |2 − x, y 2 )

A slight generalization of Berwald’s example was worked out by Mo, Shen and Yang [12]. It can expressed as following form:

 [(1 + a, x )( | y |2 − (|x|2 | y |2 − x, y 2 ) + x, y ) + (1 − |x|2 ) a, y ]2  F := (1 − |x|2 )2 | y |2 − (|x|2 | y |2 − x, y 2 )

where a ∈ Rn is an arbitrary constant vector with |a| < 1. This metric is also projectively flat with zero flag curvature. Later (α +β)2

which are Shen and Yildirim concluded that except the above examples there are no other non-trivial metrics F = α both locally projectively flat and have constant flag curvature [14] up to a local isometry. How about this kind of metrics if we get rid of the condition of projective flatness? This was asked by Shen in [11]. Now let us discuss this question. β

Lemma 4.1. Suppose r00 and s0 of (α , β) metrics F = α (1 + α ) p (| p |  1) on a manifold M satisfy

(r00 + c α s0 )2 ≡ 0 mod(s + a), then

r00 − β

c a



α ss0 = σ α 2 s2 − a2



where s = α and σ is a smooth function on a manifold M.

178

L. Zhou / Differential Geometry and its Applications 28 (2010) 170–193

Proof.

r00 + c α s0 = r00 −

c a

c

α ss0 + α s0 (s + a). a

(1)

Since

(r00 + c α s0 )2 ≡ 0 mod(s + a), this means

r00 + c α s0 ≡ 0 mod(s + a). From (1) we know that

r00 −

c a

α ss0 ≡ 0 mod(s + a).

So we may assume

 r00 −

c a



α ss0 = α (s + a) D

where D is a polynomial of s. We separate the irrational part and rational part, then obtain

 r00 −

c a



α ss0 + α s Rat( D ) + αa Irrat( D ) = 0,

α s Irrat( D ) + αa Rat( D ) = 0.

(2) (3)

From (3) we can solve

s Rat( D ) = − Irrat( D ). a Substituting to (2) we obtain

 r00 −

c a



α ss0 −

α a



s2 − a2 Irrat( D ) = 0.

This means that

r00 −

c a





α ss0 ≡ 0 mod s2 − a2 .

Thus

r00 −

c a





α ss0 = σ α 2 s2 − a2 .

2

Remark. The above lemma can simplify the proof of Bao and Robles in [1] and can immediately conclude one condition: r00 − 2β s0 = σ α 2 (1 − s2 ). Lemma 4.2. Suppose F := satisfy

(α +β)2

α

is a Finsler metric on an n-dimensional manifold M with constant flag curvature K , then F must

(1) r00 = σ (1 + 2B − 3s2 )α 2 , (2) s0 = 0, (3) sk 0 s0k = 0, where σ is a smooth function on M. Proof. The spray coefficients of F are given by

Gi = α Gi − i.e.

2s − 1 1 + 2B − 3s2



   1 2 −4 −4 α s0 + r00 li + α s + r bi + α si 0 0 00 2 1−s 1−s 1−s 1 + 2B − 3s

L. Zhou / Differential Geometry and its Applications 28 (2010) 170–193

2s − 1

f (s, B ) = − u=

1 + 2B − 3s2 4(2s − 1)

(1 + 2B − 3s2 )(1 − s)

Q =

2 1−s

g (s, B ) =

,

v=

,

1 1 + 2B − 3s2 −4

179

,

(1 + 2B − 3s2 )(1 − s)

,

.

Since F have constant flag curvature, F also have constant Ricci curvature. This means that α Rm

m

+ T m m − K (n − 1) F 2 = 0.

(4)

By Proposition 3.2 we can compute T m m by Maple

T m m = (n − 1)

+

2 r00 c¯ 1

+

α 2 A 31 c¯ 6

r00 s0

α A 41 A 2

2 r00 c¯ 2

α 2 A 41

+ (n − 1)s20

c¯ 3 3 2 A1 A2

r00 r0 c¯ 7

r00 r0 c¯ 8

+ (n − 1)

α A 21

+

+ s20

c¯ 4 4 3 A1 A2

+ (n − 1)

r00 s0

c¯ 5

α A 31 A 2

α A 31

  r00|0 + (n − 1) r0 s0 (4 f v − 2u B ) + sk sk 0 α (2Q u ) + rk0 sk 0 (4Q f ) − f − us0|0

α

+ rr00

c¯ 10 A 21

+ sk 0 s0k

+ r k k r00

c¯ 21 A 32

c¯ 11

+ sk 0 rk0

c¯ 22

A 31 A 22

+ α s0 r

+ α sk 0 rk

A 21 A 2

c¯ 15

c¯ 23 A1 A2

A 21 A 2

+ s0|0

+ α s0 r k k c¯ 24 A 21 A 2

+

c¯ 16 A1 A2

+ r02

r00|0 c¯ 25

α A 21

c¯ 17 A 21

+ α sk sk 0

+ α rk0 sk

c¯ 20 A 21 A 22

c¯ 26 A1 A2

c¯ 28 c¯ 29 c¯ 30 c¯ 31 + α bk s0|k + α r k 0 sk + r k 0 rk0 A1 A2 A1 A2 A1 A2 A1 4 4 + bk rk0|0 − α 2 si k sk i 2 + α sk 0|k A1 A2 A2

+ bk r00|k

c¯ 27

+ s0 r 0

A1

c¯ 12

A1 c¯ 32

+ α bk sk|0

where

A 1 = 1 + 2B − 3s2 ,

A2 = 1 − s

and c¯ i (i = 1, 2, . . . , 33) are polynomials of variations s and B. Substitute T m m to Eq. (4) and multiple it by A 41 : α R m 1 + 2B − 3s2 4 + T m 1 + 2B − 3s2 4 − K (n − 1) F 2 1 + 2B − 3s2 4 = 0. m m

Obviously α R m 1 + 2B − 3s2 4 − K (n − 1) F 2 1 + 2B − 3s2 4 ≡ 0 m

So



T m m 1 + 2B − 3s2

4

  ≡ 0 mod 1 + 2B − 3s2 .

From (5) we can find out that 2 r00

α2

c¯ 2 + s20

c¯ 4 A 32

+

For

r00 s0 c¯ 6

α A2 



1 + 2B − 3s2 = −3 s − we have 2 r00

α2

c¯ 2 +

s20

c¯ 4 A 32

+

r00 s0 c¯ 6

α A2

and 2 r00

c¯ 4

α

A 32

c¯ + s20 2 2

+

  ≡ 0 mod 1 + 2B − 3s2 .

r00 s0 c¯ 6

α A2

1 + 2B 3



 s+

 ≡ 0 mod s +

1 + 2B 3



1 + 2B

 , 

3

  1 + 2B . ≡ 0 mod s − 3





mod 1 + 2B − 3s2 .

(5)

180

L. Zhou / Differential Geometry and its Applications 28 (2010) 170–193

Simplify above two equations by Maple and get



r00

α and



r00

α

12( B + 2 +

√

2

3 + 6B )

 ≡ 0 mod s +



1 + 2B



−

s0 √ (5 + B ) 3 + 6B + 9 + 9B

+

√ 2   3 + 6B ) 1 + 2B . s0 ≡ 0 mod s − √ 3 (5 + B ) 3 + 6B − 9 − 9B

3

12( B + 2 −

(6)

(7)

From (6) and Lemma 4.1 we know that

√

r00 +

√

  3 + 6B ) α ss0 = σ1 α 2 1 + 2B − 3s2 . √ √ ((5 + B ) 3 + 6B + 9 + 9B ) 1 + 2B 12 3( B + 2 +

Similarly from (7) and Lemma 4.1 we can conclude that

√

r00 +

√   3 + 6B ) α ss0 = σ2 α 2 1 + 2B − 3s2 . √ √ ((5 + B ) 3 + 6B − 9 − 9B ) 1 + 2B 12 3( B + 2 −

So





mod 1 + 2B − 3s2 .

ss0 ≡ 0

This holds if and only if that

s 0 = 0. Thus





r00 = σ α 2 1 + 2B − 3s2 . Substituting T m m and s0 = 0 to Eq. (4) and multiplying it by A 32 , we have

sk 0 s0k c¯ 21 ≡ 0

mod(1 − s),

where c¯ 21 = 6(3s − 1). Hence

sk 0 s0k = 0.

2 (α +β)2

Theorem 4.3. Suppose F := is a Finsler metric on an n-dimensional manifold M. Then F has constant flag curvature K if and α only if the following three conditions hold: (1) β is closed and K = 0, (2) r00 = σ (1 + 2B − 3s2 )α 2 and σ0 + 2σ 2 β = 0, (3) α R ki jl = ai j (6σ 2 bk bl − (4B + 5)σ 2 akl ) + (aik a jl + ail a jk )σ 2 (4B + 5) − 6(b j aik + b i a jk )bl σ 2 − 6(b j ail + b i a jl )bk σ 2

+ 6b i b j akl σ 2 . Proof. “⇐”: It can be proved by a direct calculation. “⇒”: From Lemma 4.2 we know that r00 = σ (1 + 2B − 3s2 )α 2 , s0 = 0 and sk 0 s0k = 0. Thus we have





r i j = σ (1 + 2B )ai j − 3b i b j ,

s i = 0,

si k sk j = 0.

So

r0 = σ β(1 − B ),







r = σ B − B2 ,





r00|0 = σ0 α 2 1 + 2B − 3s2 + σ 4r0 α 2 − 6r00 β

 2

r00| j b j = σb α 1 + 2B − 3s

 2



+ σ 4r α 2 − 6r0 β

r j0|0 b j = σ0 β(1 − B ) + σ (r0 β − 3Br00 ),





r ii = σ (1 + 2B )n − 3B ,





r i0 r i 0 = σ 2 (1 + 2B )2 − (6 + 3B )s2 ,





r j0 = σ α (1 + 2B )l j − 3sb j , r j = σ (1 − B )b j ,

 

where σ0 = σxi y i , where σb = σxi b i ,

L. Zhou / Differential Geometry and its Applications 28 (2010) 170–193







r j0|0 = σ0 α (1 + 2B )l j − 3sb j + σ α 4r0l j − 3

r00

 2

α

b j − 3sr j0 + 3ss0 j ,

r00| j = σ α (4α r j − 6sr j0 − 6ss0 j ) + σ j α (1 + 2B ) − 3s2 i



i

i

181

 

where σ j = σx j ,



= σ α (1 + 2B )l − 3sb ,   r i j = σ (1 + 2B )δ ij − 3b i b j . r

0

Substitute the above equations to the expression of T i j in Proposition 3.1 and simplify it by Maple:

T i j = −α δ ij











σ 2 α 2s2 + 2s − 4B − 5 + σ0 (1 − 2s) − αli l j σ 2 α (2s + 4B + 5) + σ0



      + αli b j σ 2 α (2s + 4) + 2σ0 − α 2li σ j (4s − 2) − α b i l j 2σ0 − 2σ 2 α s − α 2 b i b j 2σ 2 + 2α 2 b i σ j + l i s0 j σ α

6( s − 2) A2

2

− si j |0 α

A2

− b i s0 j σ α

+ si 0|0l j

2(2s − 1) A 22

12 A2

− si 0l j σ α

− si 0|0 b j

2(2s − 9) A 22

− si 0 b j σ α

18s A 22

+ s i 0 s0 j

6(3s − 1) A 32

+ s i 0| j α

4 A2

2 A 22

where

A 1 = 1 + 2B − 3s2 ,

A 2 = 1 − s.

For F have constant flag curvature, this is equivalent to





i

α R i + T i = K F 2 δi − y F j . j j j F y

(8)

For the same reason in Lemma 4.2 we know that



si 0 s0 j 6(3s − 1) ≡ 0 mod(1 − s). This means that si 0 = 0 or s0 j = 0, i.e. β is closed. On the other hand, under above conditions Ricci curvature is given by





T m m = −2β 2 σ 2 (n − 2) − β 4σ0 − 2σ0n + 2σ 2 α (n − 1)

  − α 2 −2σb + 3σ 2 (1 + 2B ) + 2σ 2 − 3σ 2n − 2σ 2 (1 + 2B )n − ασ0 (n − 1).

Then we can separate the rational part and the irrational part from (8)

    + Rat T m m = K (n − 1)α 2 1 + 6s2 + s4 ,     Irrat T m m = α 2 K (n − 1) 4s + 4s3 . α Rm

m

From (9) we have:





Irrat T m m = −2β σ 2 α (n − 1) − ασ0 (n − 1). Therefore

K =−



n−1 4β(1 +

s2 )



2σ 2 β + σ0 .

Notice K is a constant number, the above equation holds if and only if

2σ 2 β + σ0 = 0

and

K = 0.

So we can express Riemannian curvature of α R i = −T i = δi σ j j j

 2

α







6β 2 − (5 + 4B )α 2 + li l j (5 + 4B )σ 2 α 2 − 6σ 2 α β li b j + b i l j + 6b i b j σ 2 α 2 .

Thus we have αR

ki jl

  = ai j 6σ 2 bk bl − (4B + 5)σ 2 akl + (aik a jl + ail a jk )σ 2 (4B + 5) − 6(b j aik + b i a jk )bl σ 2 − 6(b j ail + b i a jl )bk σ 2 + 6b i b j akl σ 2 .

2

Furthermore, we find the metrics in Theorem 4.3 are locally projectively flat. To prove this we need several lemmas.

(9)

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L. Zhou / Differential Geometry and its Applications 28 (2010) 170–193

˜ are two conformal Riemannian metrics on M, that is, there exists a smooth function Lemma 4.4. Suppose α and α a˜ i j = e 2ρ ai j . Their curvature tensors are related by

ρ such that

i i R˜ kjl = R kjl + ρkj δli − ρkl δ ij + akj aip ρ pl − akl aip ρ p j

where

ρi j =

∂ 2ρ ∂ρ ∂ρ ∂ρ 1 ∂ρ ∂ρ − Γikj k − i j + ai j akl k l . i j 2 ∂x ∂x ∂x ∂x ∂x ∂x ∂x

Proof. It can be proved by a direct computation.

2

Using the notation in Theorem 4.3 we have the following lemma: Lemma 4.5. Suppose F :=

˜ , then K˜ = −1. of α

(α +β)2

α

˜ 2 = σ 2 α 2 and K˜ be sectional curvature satisfy the three conditions in Theorem 4.3 and σ = 0. Let α

Proof. From Lemma 4.1 we have the relationship of Riemannian curvature between

α and α˜ :

α˜ R ˜ i j = α R i j + ρ j0 y i − ρ00 δ i + αl j aip ρ p0 − α 2 aip ρ p j j

where ρ j0 := ρ jk yk , ρ00 := ρi j y i y j . ˜ 2 = σ 2 α 2 we have For α

ρ = ln |σ |. Hence

∂ ln σ ∂ρ 1 = = σ xi , i i σ ∂x ∂x σ xi σ x j σ xi x j ∂ 2ρ =− + . σ σ2 ∂ xi ∂ x j Because σ satisfies

σ x i + 2σ 2 b i = 0, we can obtain

∂ 2ρ ∂ρ ∂ρ ∂ρ 1 ∂ρ ∂ρ − Γikj k − i j + ai j akl k l 2 ∂ xi ∂ x j ∂x ∂x ∂x ∂x ∂x k 2σ i σ j σ i j Γi j σxk ai j akl σxk σxl =− x2 x + xx − + 2

ρi j =

σ

σ

=−

σ

2σ

2 

  (2σ 2 b i )| j ai j akl (−2σ 2 bk )(−2σ 2 bl ) −2σ 2 b i −2σ 2 b j + + 2

σ σ2 = −8σ 2 b i b j − 4σx j b i − 2σ b i | j + 2σ 2 Bai j

2σ

  = −8σ 2 b i b j + 8σ 2 b i b j − 2σ 2 (1 + 2B )ai j − 3b i b j + 2σ 2 Bai j = −2σ 2 (1 + B )ai j + 6σ 2 b i b j . So

ρi0 = −2σ 2 αli (1 + B ) + 6σ 2 β bi , aip ρ p0 = −2σ 2 α (1 + B )li + 6σ 2 β b i , aip ρ p j = −2σ 2 (1 + B )δ ij + 6σ 2 b i b j . Then

  α˜ R ˜ i = α R i j − 2σ 2 α 2 (1 + B )li l j + 6σ 2 α β li b j − −2σ 2 α 2 (1 + B ) + 6σ 2 β 2 δ i j j − 2σ 2 α 2 (1 + B )li l j + 6σ 2 α β b i l j + 2σ 2 α 2 (1 + 2B )δ ij − 6σ 2 α 2 b i b j   = α R i j + δ ij 4σ 2 α 2 (1 + B ) − 6σ 2 β 2 − 4(1 + 2B )σ 2 α 2li l j + 6σ 2 α β li b j + 6σ 2 α β b i l j − 6σ 2 α 2 b i b j .

L. Zhou / Differential Geometry and its Applications 28 (2010) 170–193

183

Furthermore α R i j satisfies: α R i = δ i σ 2 6β 2 − (5 + 4B )α 2  + l i l (5 + 4B )σ 2 α 2 − 6σ 2 α β l i b + b i l  + 6b i b σ 2 α 2 . j j j j j j

Thus we have

  α˜ R ˜ i j = σ 2 δ i −α 2 + li l j σ 2 α 2 j   = −α˜ 2 δ ij − ˜li ˜l j .

˜ have constant sectional curvature −1. This means that α (α +β)2

Theorem 4.6. Suppose F := α must be locally projectively flat.

2

is a Finsler metric on an n-dimensional manifold M. If F has constant flag curvature K , then F

Proof. From Theorem 4.3 we know that

s i j = 0,



r00 = σ 1 + 2B − 3s2



α2

and

σ x i + 2σ 2 b i = 0. Thus

G i = α G i − σ α (2s − 1) y i + σ α 2 b i . If σ = 0, then r00 = 0 and sectional curvature of α is zero. This means β is parallel with respect to α and α is flat. Thus F is locally Minkowskian [14]. Hence it is locally projectively flat. ˜ is projectively flat. Then there is a local coordinate system {xi } in which, the spray If σ = 0, from Lemma 4.5 we know α ˜ can be expressed in the following form coefficients of α α˜ G i = P˜ y i .

Notice that

σ0 y i σxk g ki α 2 + σ 2σ i σ y 0 = α˜ G i − − σ α 2bi . σ

α G i = α˜ G i −

So

G i = α˜ G i −

σ0 y i − σ α (2s − 1) y i . σ

In the same local coordinate system, the spray coefficients G i of F can be expressed in the form

G i = P yi . Thus F is locally projectively flat.

2 β

Remark. Since the locally projectively flat Finsler metrics F = α (1 + α )2 with constant flag curvature have been classified [14], thus we complete the local classification of such metrics with constant flag curvature. 5. Some results on more general case Now we discuss a more general class (α , β) metrics F = Matsumoto metrics. The spray coefficients of F = α (1 +

α p

α (1 + β ) β p ) are given by α

(| p |  1) which include Randers metrics and

G i = α G i + f (s, B )r00 li + g (s, B )r00 b i + u (s, B )α s0 li + v (s, B )α s0 b i + Q α si 0 where

184

L. Zhou / Differential Geometry and its Applications 28 (2010) 170–193

f (s, B ) = g (s, B ) =

2( p − 1) ps − p 2(( p 2 − 1)s2 + ( p − 2)s − Bp 2 + Bp − 1) −( p − 1) p 2(( p 2 − 1)s2 + ( p − 2)s − Bp 2 + Bp − 1)

, ,

(2( p − 1)s − 1) p 2 , (( p 2 − 1)s2 + ( p − 2)s − Bp 2 + Bp − 1)(( p − 1)s − 1) ( p − 1) p 2 v (s, B ) = . (( p 2 − 1)s2 + ( p − 2)s − Bp 2 + Bp − 1)(( p − 1)s − 1)

u (s, B ) =

Here we called an (α , β) metric F is trivial if

r i j = s i j = 0. So if an (α , β) metric F is trivial, we have

Gi = α Gi. In Section 4 we have discussed the case of p = 2. When p 2 = 1 i.e. p = 1 or −1, we can see that the denominator of f (s, B ) and g (s, B ) is 1 degree polynomial of s. These are quite different from other case. 5.1. The case of p = 1 When p = 1, F is a Randers metric and we have already knew the answer [1,2]. 5.2. The case of p = −1 2

When p = −1, F = (αα+β) including the famous Matsumoto metrics [10]. By a similarly analysis, we obtain a necessary condition that such kind of metrics have constant flag curvature. 2

Theorem 5.1. Suppose a Finsler metric F := (αα+β) has constant flag curvature K , then F must satisfy



18

r00 +

(4B − 1)(2B + 1)



α ss0 = σ α 2 (1 + 2B )2 − 9s2 .

Proof. The spray coefficients of F are given by









G i = α G i + f (s, B )r00 + u (s, B )α s0 li + g (s, B )r00 + v (s, B )α s0 b i + Q α si 0 where

f (s, B ) = − g (s, B ) =

,

,

1 + 2B + 3s 1 + 4s

u (s, B ) = − v (s, B ) =

1 + 4s 2(1 + 2B + 3s) 1

(1 + 2B + 3s)(1 + 2s) 2

(1 + 2B + 3s)(1 + 2s)

,

.

Because F has constant flag curvature, it has constant Ricci curvature. Hence α Rm

m

+ T m m = (n − 1) K F 2 .

By Proposition 3.2 we can compute Ricci curvature R m m and find out that



r00

α

−

2

6 4B − 1

s0

≡ 0 mod(1 + 2B + 3s).

According Lemma 4.1 we have

r00 +

18

(4B − 1)(2B + 1)





α ss0 = σ α 2 (1 + 2B )2 − 9s2 .

2

L. Zhou / Differential Geometry and its Applications 28 (2010) 170–193

185

2

Corollary 5.2. If 1-form β of F := (αα+β) is closed, then there is no non-trivial metric which has constant flag curvature. Proof. Assume that F has constant flag curvature. By Theorem 5.1 we know that

r00 +



18

(4B − 1)(2B + 1)



α ss0 = σ α 2 (1 + 2B )2 − 9s2 .

Since β is closed i.e. si j = 0, then





r00 = σ α 2 (1 + 2B )2 − 9s2 . Hence

r0 =

1 2





σ α s 2(1 + 2B )2 − 18B , 



r = σ B (1 + 2B )2 − 9B ,













 2

r00|0 = 2σ α 4(1 + 2B )α r0 − 9r00 s + σ0 (1 + 2B )2 − 9s2



r00|i b i = 2σ α 4r α (1 + 2B ) − 9r0 s + σb (1 + 2B )2 − 9s



α2,

α2, 

r i0|0 b i = σ (18B − 1)sα r0 − 9σ Br00 + σ0 α s (1 + 2B )2 − 9B ,





r ii = σ n(1 + 2B )2 − 9B ,



r i0 r i 0 = σ 2 (1 + 2B )4 − 18(1 + 2B )2 s2 + 81Bs2



α2.

Substitute these equations to T m m and obtain



T m m = 9(8n − 11)σ 2 α 2 s4 + 18 (5 − 4B )n + 6B − 7



1

2





σ 2 α 2 s3 7





+3 16B − 50B + n − 32B + 82B + σ α − (2n − 3)σ0 α s2 4 4       3  2 1 5 × 8B − 36B − 11 n − 32B 2 + 60B + 11 σ 2 α − 6σb α + σ0 + 4B n − 8B − αs 2 2 2    11 11 + 16B 3 + 33B 2 + 18B + n + 32B 3 − 102B 2 − 24B − σ 2 α 2 + 2(1 + 2B )σb α 2 4 4    1 1 + B+ n + 2B − σ0 α . 2

2

2

So its rational part and irrational part are given by







Rat T m m = 9(8n − 11)σ 2 α 2 s4 + 3



2



1

16B 2 − 50B + 5



1 4

 n − 32B 2 + 82B +

7 4



σ 2 α 2 s2

+ σ0 + 4B n − 8B − αs 2 2    11 11 + 16B 3 + 33B 2 + 18B + n + 32B 3 − 102B 2 − 24B − σ 2 α 2 + 2(1 + 2B )σb α 2 , 4 4     Irrat T m m = 18 (5 − 4B )n + 6B − 7 σ 2 α 2 s3 − 3(2n − 3)σ0 α s2        3  2 1 1 + 8B − 36B − 11 n − 32B 2 + 60B + 11 σ 2 − 6σb α 2 s + B+ n + 2B − σ0 α . 2

2

Because F has constant flag curvature, it also have constant Ricci curvature. This means that α Rm

i.e.



m

+ T m m = (n − 1) K F 2 = (n − 1) K α 2

1

(1 + s)2

 1 + 2s + s2 α R m m + T m m = (n − 1) K α 2 .

Again we separate its rational part and irrational part and obtain:



       1 + s2 α R m m + 1 + s2 Rat T m m + 2s Irrat T m m = K α 2 (n − 1), 











2sα R m m + 2s Rat T m m + 1 + s2 Irrat T m m = 0.

2

186

L. Zhou / Differential Geometry and its Applications 28 (2010) 170–193

We can solve K and α R m m

  −(1 − s2 )2 Irrat R m m , 2(n − 1)sα 2 2 α R m = − Rat R m  − 1 + s Irrat R m . m m m K=

2s

Hence

K=

   −(1 − s2 )2 2 σ 18 (5 − 4B )n + 6B − 7 σ 2 α 2 s3 − 3(2n − 3)σ0 α s2 2 2(n − 1)sα         3  2 1 1 + 8B − 36B − 11 n − 32B 2 + 60B + 11 σ 2 − 6σb α 2 s + B+ n + 2B − σ0 α . 2

2

2

(10)

Since K is constant, (10) holds if and only if

σ = 0. This means that

r i j = s i j = 0.

2

Thus complete the proof.

Remark. From this corollary we know that there are no non-trivial Matsumoto metrics with constant flag curvature. 5.3. 1-form β is closed β

Next we discuss the other case i.e. p = −1, 1, 2. We mainly concerns the metrics F = α (1 + α ) p when 1-form β is closed. β

Theorem 5.3. There are no non-trivial Finsler metrics F = α (1 + α ) p (| p |  1) which have constant flag curvature when β is closed expect p = 1, 2. Proof. When p = −1, we have proved in Corollary 5.2. When p = −1, 1, 2 and β is closed, we can compute T m m

T mm =

2 r00

c¯ 1

α

A 31

(n − 1) 2

+

r0 r00

α

(n − 1)

c¯ 2 A 21

+

r00|0

α

(n − 1)

c¯ 3 A1

+

2 r00 c¯ 4

α

2

A 41

+

r0 r00 c¯ 5

α

A 31

+

r00|0 c¯ 6

α A 21

 c¯ 7   c¯ 8  + r02 − rr00 2 + r00|i b i − r i0 b i + r00 r ii − r i0 r i 0

A1

A1

where A 1 := ( p 2 − 1)s2 + ( p − 2)s − Bp 2 + Bp − 1 and c¯ i (i = 1, 2, . . . , 8) are polynomials of s and B. From the equation of constant Ricci curvature

R m m + T m m = K (n − 1) F 2 , we know that







2 r00 ≡ 0 mod p 2 − 1 s2 + ( p − 2)s − Bp 2 + Bp − 1 .

Because p = −1, 0, 1, A 1 = ( p 2 − 1)s2 + ( p − 2)s − Bp 2 + Bp − 1 is a 2 degree polynomial of s. Thus

r00 = σ α 2







p 2 − 1 s2 + ( p − 2)s − Bp 2 + Bp − 1 .

(11)

Notice that p = 2, the right side includes an irrational part. Hence (11) holds if and only if

σ = 0. This means that there are no non-trivial such metrics which have constant flag curvature.

2

Remark. Because the calculation is too tedious and too long, we do not continue to discuss this kind of metrics without the condition β is closed. But we conjecture that above theorem also holds if we get rid of the restriction on 1-form β .

L. Zhou / Differential Geometry and its Applications 28 (2010) 170–193

187

6. Explanation & Acknowledgement The author thanks the supervisor Professor Xiaohuan Mo for his support while this paper was written. The author also thanks Yu Changtao and Sun Fang for some benefic discussion with them. After the first version of the paper appeared, Mastooreh Farahmand, Zhongmin Shen and other unknown reviewers patiently went through the tedious computation in it and pointed out some mistakes recently: especially the errors in the formula of Riemannian curvature and Ricci curvature of (α , β) metrics. They are deserved the author’s appreciation. Finally the author would like to thank for communicating editor in the discussion preceding the revision. Furthermore, the author gave many corrections in this second revised version. To distinguish them, the paper’s title is also slightly changed. But the main results are still correct and remained to be unchanged. Appendix A. How to compute the flag and Ricci curvature of (α , β ) metric As we know that the spray coefficients G i of an (α , β) metric F := α φ(s) and the spray coefficients α G i of a Riemannian metric α are related by:

G i = α G i + Θ(−2α Q s0 + r00 )li + Ψ (−2α Q s0 + r00 )b i + α Q si 0 where

Q =

−2φ  , φ − sφ 

Θ=

φφ  − s(φφ  + φ  φ  ) , 2φ((φ − sφ  ) + ( B − s2 )φ  )

Ψ=

φ  2((φ − sφ  ) + ( B − s2 )φ  )

yi

β

and li := α , s = α , si 0 = si j y j , s0 = si y i , r00 = r i j y i y j , B = ai j b i b j . To simplify our computation, we separate the spray coefficients into three parts. Let

f (s, B ) := Θ,

g (s, B ) := Ψ,

u (s, B ) := −2Θ Q ,

v (s, B ) := −2Ψ Q

and

P := f (s, B )r00 ,

L := g (s, B )r00 ,

M := u (s, B )α s0 ,

W := Q α ,

N := v (s, B )α s0 .

Then

G i = α G i + ζ1i + ζ2i

(12)

if we denote

ζ1i := Pli + Lb i , ζ2i := Mli + W si 0 + Nb i . By Berwald’s formula for Riemannian curvature and (12) we have

R i j = α R i j + T 1i j + T 2i j + T 3i

j

where



      − ζ pi yk ζ pk y j + 2ζ pk ζ pi y j yk ( p = 1, 2),         − ζ2i yk ζ1k y j + 2ζ1k ζ2i y j yk + 2ζ2k ζ1i y j yk .

T pi j := 2ζ pi | j − yk ζ pi |k

   k ζ yk 2 y j

T 3i j := − ζ1i



yj

Before calculation we need some basic preparation:

188

L. Zhou / Differential Geometry and its Applications 28 (2010) 170–193

 i l

 yj

=

yk

1

=

yj

i

α

 δ ij − li l j ,

   1  i 1  = l y k l y j − lk l i y k y j = l y j = 2 δ ij − li l j , α α     1 i 1  i i i δj − l lj = = 2 −δ j lk − δk l j + 3li l j lk − li a jk ,

l

 i l



α

 i  k l

yi

y j yk

yj

 k

α

α

yk



k

1

(li ) y j = l aki y j = (ai j − li l j ), α   β 1 sy j = = (b j − sl j ),

α

s yi y j =

α

yj

1

(−b i l j − b j li + 3sli l j − sai j ),  B |i = b j b j |i = 2b j |i b j = 2(r i + si ),   β 1 1 s |i = = b j |i y j = (r i0 + s0i ), 

α2

α

α

|i

α

i

l | j = 0, b i | j = r i j + si j . Let us first compute T 1i j : We can express (ζ1i ) y j and (ζ1i ) y j yk by

 i   ζ1 y j = P y j l i + P l i y j + L y j b i ,        i ζ1 y j yk = P y j yk li + P y j li yk + P yk li y j + P li y j yk + L y j yk b i . Hence

 i   k        ζ1 y k ζ 1 y j = P y k l i + P l i y k + L y k b i P y j l k + P l k y j + L y j b k =

=

2P

P y j li +

α

yj

2P

2L

k

   k

yj

i

li + P 2 li

 i

b + P yk b l L y j + P L y j l P y j b i + P P yk

α

 

2ζ1k ζ1i

yk y j

k

α

α

+

δkj



1

α

yk

l

yj

yk





k

b + L y k bk b i L y j

  1  δkj − lk l j li + P 2 2 δ ij − li l j

α

   − l l j b + P yk b l L y j + P L y j δki − li lk bk + L yk bk b i L y j α α   P2 i 3P 3P 2 i Ps i 2L i k = 2 δj + P y j l − 2 l l j + P yk b − l Lyj + P y j bi + P L yk

1



i

+ P L yk l

α

P y j b i + P P y k lk

α

 k

P y j li +

 

2L

2P

α



i

k

i

1

α

α

2P L

L y j bi −

α2

α

 k



b i l j + L yk b b i L y j ,

         = 2 Plk + Lbk P y j yk li + P y j li yk + P yk li y j + P li y j yk + L y j yk b i      P 2  i P P 2  i P =2 P y j li + 2 l yj − l y j + L y j b i + L P y j y k bk l i + L P y j l i y k bk α α α α         + L P y k bk l i y j + L P l i y j y k bk + L L y j y k bk b i  =2

P

P y j li +

α 

+ L P yk b  =2

P

2

α2

P2 

α

2

  k 1

+

α L

α

 P    1 δ ij − li l j + L y j b i + L P y j yk bk li + L P y j δki − li lk bk

α



δ ij − li l j + L P 

P y k bk −

P Ls

α2



1 

α2

α

   −δ ij lk − δki l j + 3li l j lk − li a jk bk + L L y j yk bk b i

   3P Ls i P2 L P y k bk + l lj δ ij + − 2 − 2

α

α

α



L. Zhou / Differential Geometry and its Applications 28 (2010) 170–193

189

       Ls P PL i PL i P L i k i i i k i + − + P y j l + L P y j yk b l − 2 l b j − 2 b l j + L y j b + P y j b + L L y j yk b b .

α

α

α

α

α

α

Similarly we can write ζ1i | j and yk (ζ1i |k ) y j by

  ζ1i | j = P | j li + L | j b i + L r i j + si j ,     yk ζ1i |k y j = ζ1i |k yk y j − ζ1i | j          = P |k yk li + L |k yk b i + L r i 0 + si 0 y j − P | j li + L | j b i + L r i j + si j  = P

|k y

k





yj

li + P |k yk

1

α

   δ ij − li l j + L |k yk y j b i + L y j r i 0 + L y j si 0

+ Lr i j + Lsi j − P | j li − L | j b i − Lr i j − Lsi j      1 1 = δ ij P |k yk − li l j P |k yk + li P |k yk y j − li P

α

α

|j

  + b i L |k yk y j − L | j b i + L y j r i 0 + L y j si 0 .

Therefore

 

      − ζ1i yk ζ1k y j + 2ζ1i | j − yk ζ1i |k y j  2  2    2P Ls  6P Ls P |k yk i P |k yk i P 2L  P 2L  k k = + P kb − − − P kb + + l lj δj + α y α α y α α2 α2 α2 α2           2Ls P i Ps 2P L − + l Pyj + − P yk bk li L y j + 2L P y j yk bk li − 2 li b j + 2L L y j yk bk b i − b i L y j L yk bk

T 1i j = 2ζ1k ζ1i

yk y j

α

α

+ 3l P

i

i

|j



−l P

|k y

k

α



i

yj

i



+ 3L | j b − b L |k y

k

α



i

yj

i

+ 2Lr j + 2Ls j − L y j r i 0 − L y j si 0 .

Then we compute T 2i j :

 i   ζ2 y j = M y j l i + M l i y j + W y j s i 0 + W s i j + N y j b i ,  i       ζ2 y j yk = M y j yk li + M y j li yk + M yk li y j + M li y j yk + W y j yk si 0 + W y j si k + W yk si j + N y j yk b i . So

  i   k       ζ2 yk ζ2 y j = M yk li + M li yk + W yk si 0 + W si k + N yk b i M y j lk + M lk y j + W y j sk 0 + W sk j + N y j bk =

3M

li M y j +

α + −

2W

α

si 0 M y j +

2N

α

bi M y j −

3M 2

α2

li l j +

α

δ ij

α

α

α2

α2

   2M W i 2M W i 2M k + W yk sk 0 si 0 W y j − s l + s + + N b bi N y j k 0 j j y 2

2M

2M N

M2



i

k

b l j + M yk s

0

i



i

k

l W y j + W s ks

 0

α



k

W y j + N yk s

 0

bi W y j

MW

+ W li M yk sk j − 2 li s0 j + W si 0 W yk sk j + W 2 si k sk j + W b i sk j N yk α     Ms + bk M y k − l i N y j + W y k bk s i 0 N y j + W s i N y j ,

α

2ζ2k

 i         ζ2 y j yk = 2 Mlk + W sk 0 + Nbk M y j yk li + M y j li yk + M yk li y j + M li y j yk  + W y j yk si 0 + W y j si k + W yk si j + N y j yk b i    2 W ( M yk sk 0 ) + N ( M yk bk ) i M − Ns i M − M Ns δj l Myj + + =2 2  +  + +

α

− M + 3M Ns

α2

MW

α W

α

α

α

2

−

W ( M yk sk 0 ) + N ( M yk bk )

α



li l j +

M

α

W y j si 0

   M + W W yk sk 0 + N W yk bk si j + N y j b i + W li M y j yk sk 0

Myjs

α

i

0

−

MW

α2

i

s 0l j −

MW

α2

i

l s j0 + W W y j yk sk 0 si 0 + W W y j sk 0 si k

190

L. Zhou / Differential Geometry and its Applications 28 (2010) 170–193

N

+ W N y j yk sk 0 b i + N M y j yk bk li +

α

M y j bi −

MN

α2

bi l j −

MN

α2

   + N W y j y k bk s i 0 + N W y j s i + N N y j y k bk b i ,

li b j

  ζ2i | j = M | j li + N | j b i + N r i j + si j + W | j si 0 + W si 0| j ,       yk ζ2i |k y j = yk M |k li + N |k b i + N si k + r i k + W |k si 0 + W si 0|k y j 1

= yk ( M |k ) y j li + yk M |k

α

   δ ij − li l j + ( N |k ) y j yk b i + N y j si 0 + r i 0

+ ( W |k ) y j yk si 0 + yk W |k si j + W y j si 0|0 + W si j |0  k   k  y M |k y M |k − li l j + li yk ( M |k ) y j + b i ( N |k ) y j yk = δ ij +s

i

 0

α

α

N y j + ( W |k ) y j y

k



+ si j W |0 + si 0|0 W y j + si j |0 W + r i 0 N y j .

Thus

 

T 2i j = 2ζ2k ζ2i



= δ ij

M2

α2



i

+l lj

      − ζ2i yk ζ2k y j + 2ζ2i | j − yk ζ2i |k y j

yk y j

+ M

2W 

α

2

2W 

−

α2



2N 

M yk sk 0 +

α

k

M yk s



M y k bk −

α 

0

−

2N 

α

M yk b

k

2M Ns

α2 

yk M |k

−

α

6M Ns

+

α2



yk M |k

+



α

      3M W i M + 2Ns Ms i i k + li W y j − M yk sk 0 + 2W li M y j yk sk 0 + + l N y j −b M yk + l s0 j − l Myj 2

α

α

 i

 k

α

 i

2M N





 k

+ 2Nl M y j yk b − W li ( M yk sk j ) − li b j + l 2M | j − yk ( M |k ) y j − b i N y j N yk b α2       − b i W y j N yk sk 0 + 2W b i N y j yk sk 0 + 2Nb i N y j yk bk − W b i sk j N yk + b i 2N | j − ( N |k ) y j yk     + 2si j W W yk sk 0 + N W yk bk − si 0 W y j W yk sk 0 + si k sk 0 W y j W − si 0 W yk sk j W − W 2 si k sk j     − si 0 N y j W yk bk − si N y j W + 2Nsi W y j + 2W sk 0 si 0 W y j yk + 2Nsi 0 W y j yk bk   + si 0 2W | j − N y j − ( W |k ) y j yk + si j (2N − W |0 ) + 2W si 0| j − si 0|0 W y j − si j |0 W + 2Nr i j − r i 0 N y j . Finally we compute T 3i j :

 i   k        ζ1 yk ζ2 y j = P yk li + P li yk + L yk b i M y j lk + M lk y j + W y j sk 0 + W sk j + N y j bk =

2P

α +

li M y j + M

α

2L

α

bi L y j −

bi M y j +

2LM

α2

+ W P yk sk j li + +

P

α

i

b Nyj −

α

α

P y j li − li l j

2P M

α2





b i l j + P yk sk 0 li W y j +

WP

Ps

M

α

si j −



i

WP

α

2



+ P

α

MP 

α2

δ ij − li l j

 



si 0 W y j + L yk sk 0 b i W y j







li s0 j + W L yk sk j b i + P yk bk li N y j



l N y j + L y k bk b i N y j ,

        i   k ζ2 yk ζ1 y j = M yk li + M li yk + W yk si 0 + W si k + N yk b i P y j lk + P lk y j + L y j bk =

2M

α +

li P y j +

MP 

α



2

W

α

si 0 P y j +

W

α

si 0 P y j +

2N

α

bi P y j +

P

α

li M y j −

2P M

α2

li l j

 P P P 2P N WP i WP δ ij − li l j + si 0 W y j − 2 W si 0l j + b i N y j − 2 b i l j + s j − 2 si 0l j k

i

+ M yk b l L y j +

α

M

α

α

i

b Lyj −

Ms

α

i



α

k



α

i



i

k



α



α



l L y j + W y k b s 0 L y j + W s k b L y j + N y k bk b i L y j ,

L. Zhou / Differential Geometry and its Applications 28 (2010) 170–193

191

           ζ1k ζ2i y j yk = Plk + Lbk M y j yk li + M y j li yk + M yk li y j + M li y j yk + W y j yk si 0 + W y j si k + W yk si j + N y j yk b i P

=

PM

li M y j +

α

α2

 P   L PW i P s j + b i N y j + Lli M y j yk bk + b i M y j δ ij − li l j + si 0 W y j +

α

α

α

  LMs LM 3LMs i LM − li M y j + M yk bk δ ij − li l j − 2 δ ij − 2 b i l j + l l j − 2 li b j α α α α α2 α       i k i k i i k + Ls 0 W y j yk b − Ls W y j + L W yk b s j + Lb N y j yk b ,          k = Ml + W sk 0 + Nbk P y j yk li + P y j li yk + P yk li y j + P li y j yk + L y j yk b i Ls

  ζ2k ζ1i y j yk

M

=

α +

α

L

MP 

P y j li + W

α

Pyjs

α

i

0

2

 M   δ ij − li l j + b i L y j + W li P y j yk sk 0

α



k

+ W P yk s

1 0

α

 WP      δ ij − li l j + 2 −l j si 0 − li s j0 + W b i L y j yk sk 0 + Nli P y j yk bk

α

  1 i  NP     N i + δ j − li l j + 2 −sδ ij − b i l j + 3sli l j − li b j + Nb i L y j yk bk . b − sli P y j + N P yk bk

α

α

α

Hence

       k     ζ − ζ2i yk ζ1k y j + 2ζ1k ζ2i y j yk + 2ζ2k ζ1i y j yk yk 2 y j    2N   2N P s L 2LMs 2W  P i 2P M k k k = δj − + P yk s 0 + P yk b − + 2 M yk b − l i M y j 2 2 2

T 3i j = − ζ1i

α

−

M

α

α

i



i

l Pyj +l lj

α

2P M

6LMs

+

α2

i



α

α2

 i

2W 

−



α 3W P

− P yk sk 0 l W y j − W l P yk sk j +

α

α

α2

k

P yk s

 0

−

2N 

α

P yk b

k

l i s 0 j − P y k bk l i N y j +



α

6N P s

+

Ps

α

α2

− 2 M yk b

li N y j

   2Ls i  Ms i 2LM i − M y k bk l i L y j + l L y j + 2Lli M y j yk bk − l Myj − l bj 2 i



k



α

i



k

α



2Ns

α

2N P

i

i

+ 2W l P y j yk s 0 + 2Nl P y j yk b − l Pyj − l bj α α2         − L yk sk 0 b i W y j − W b i L yk sk j − L yk bk b i N y j − N yk bk b i L y j       + 2Lb i N y j yk bk + 2W b i L y j yk sk 0 + 2Nb i L y j yk bk       + si j 2LW yk bk + si 0 2L W y j yk bk − ( W yk bk ) L y j + si (−2LW y j + W L y j ). Thus we need to compute

P y j = ( f r00 ) y j = f s s y j r00 + 2 f r j0 = f s P y k bk = f s k

P yk s

0

=

P yk sk j =

1

α



α α

f s + 2 f rk0 sk 0 ,



P y j y k bk = P y k bk

=lj

r00

fss j −



(b j − sl j )r00 + 2 f r j0 ,

B − s2 r00 + 2 f r0 ,

r00 s0 r00

1

α

 yj

r00 

α2 

+bj  P y j yk sk 0 = l j −

α2

sf s s0 j + 2 f rk0 sk j ,



=

fs

1

α



 2



 2

−sf ss B − s

r00 

α

2

r00 s0

α2 k

+ 2r jk s

0

f ss B − s



− fs B − s

s0 j r00

α2

 2



− 2sf s + 2

( f s + sf ss ) − 2

f +





B − s2 r00 + 2 f r0

rk0 sk 0

sf s + 2

α

 sf s + 2s2 f s − 2 r0

fs

α

r0

f s,



α





+ 2 f s B − s2



sf s

r j0 s0

α

yj

 +bj

r00 s0

α2

 r j0

α

f ss + 2

+ 2 f r j,

rk0 sk 0

α

 fs

L

α

 k

192

L. Zhou / Differential Geometry and its Applications 28 (2010) 170–193

P | j = f | j r00 + f r00| j = fs

P |k yk =



P |k yk

f ss

=

yj

(r j0 + s0 j )r00 + 2 f B (r j + s j )r00 + f r00| j ,

2 r00 + 2 f B (r0 + s0 )r00 + f r00|0 ,

α



fs

α

2 (b j − sl j )r00 −

α2

fs

α2

2 r00 lj +

4 fs

α

r00 r j0 +

2 f Bs

α

(b j − sl j )(r0 + s0 )r00

fs

+ 2 f B (r j + s j )r00 + 4 f B (r0 + s0 )r j0 + (b j − sl j )r00|0 + 2 f r j0|0 + f r00| j α    2 r r00 2 f Bs s sf s 2 f Bs = l j − 2 ( f ss s + f s ) − (r0 + s0 )r00 − r00|0 + b j 002 f ss + (r0 + s0 )r00 α α α α α  fs 4 fs + r00|0 + r00 r j0 + 2 f B r00 (r j + s j ) + 4 f B (r0 + s0 )r j0 + 2 f r j0|0 + f r00| j .

α

α

By a similar calculation we can get

L y j = gs

1

(b j − sl j )r00 + 2gr j0 ,

α

L y j b j = gs

α

r00



B − s2 r00 + 2gr0 ,

r00 s0

L yk sk 0 = L yk sk j =

1

α

g s + 2grk0 sk 0 ,

r00 

0

gs

L |j =

α

L |k yk =



L |k y

k

sg s s0 j + 2grk0 sk j ,

α

α

 L y j yk s

α2

      sg s 2 2 2 − − + − sg B − s g B − s 2s g 2 r ss s s 0 α α2       r j0 g r00   s g ss B − s2 − 2sg s + 2 r0 + 2g s B − s2 + 2gr j , +bj 2

L y j y k bk = l j

k

r00

gs s j −

α

=lj −

r00 s0

α2

( g s + sg ss ) − 2

k

rk0 s

0

α



sg s

 +bj

α

r00 s0

α2

g ss + 2

rk0 sk 0

α

 gs

+ 2r jk sk 0 g +

s0 j r00

α2

sg s + 2

gs

α

 yj

2 r00 + 2g B (r0 + s0 )r00 + gr00|0 ,

   2  r00 r00 2g Bs s sg s 2g Bs gs = l j − 2 ( g ss s + g s ) − (r0 + s0 )r00 − r00|0 + b j g + (r0 + s0 )r00 + r00|0 2 ss

α

α

4g s

α

α

α

α

α

r00 r j0 + 2g B r00 (r j + s j ) + 4g B (r0 + s0 )r j0 + 2gr j0|0 + gr00| j ,

M y j = (u α s0 ) y j = u s s0 b j + s0 l j (u − su s ) + u α s j ,





M yk bk = s0 Bu s + (u − su s )s , M yk sk 0 = u s s20 + u α sk sk 0 , s0 M yk sk j = u s s0 s j + (u − su s )s0 j + u α sk sk j ,



M y j y k bk = l j

s0 

α



M y j yk sk 0 = l j −

α

            s0   −u ss s B − s2 + u s s2 − us + b j u ss B − s2 + u − su s + s j u s B − s2 + us ,

α

s20

α

  2  s s0 u ss s + sk sk 0 (u − su s ) + b j 0 u ss + sk sk 0 u s + s j (s0 u s ) + s j0 (u − su s ),

α

α

M | j = u s s0 (s0 j + r j0 ) + 2u B α s0 (s j + r j ) + u α s0| j , M |k yk = 2u B s20 α + u α s0|0 + 2r0 u B α s0 + u s s0 r00 , M |k yk

α

(r j0 + s0 j )r00 + 2g B (r j + s j )r00 + gr00| j ,

+



r j0 s0

 yj

  s0 r00 = l j s20 (2u B − 2u Bs s) + s0|0 (u − su s ) + r0 s0 (2u B − 2u Bs s) − su ss α   r s 00 0 + b j 2u Bs s20 + u s s0|0 + 2u Bs r0 s0 + u ss + s j (4u B s0 α + 2r0 u B α + u s r00 )

α

gs ,

L. Zhou / Differential Geometry and its Applications 28 (2010) 170–193

193

+ u α s j |0 + u α s0| j + r j (2u B α s0 ) + r j0 (2u s s0 ), N y j = v s s0 b j + s0l j ( v − sv s ) + v α s j ,





N yk bk = s0 B v s + ( v − sv s )s , N yk sk 0 = v s s20 + v α sk sk 0 , s0 N yk sk j = v s s0 s j + ( v − sv s )s0 j + v α sk sk j ,



k

N y j yk b = l j

α

            s0   − v ss s B − s2 + v s s2 − vs + b j v ss B − s2 + v − sv s + s j v s B − s2 + vs ,

s0 

α



N y j yk sk 0 = l j −

α

s20

α

 2   s s0 v ss s + sk sk 0 ( v − sv s ) + b j 0 v ss + sk sk 0 v s + s j (s0 v s ) + s j0 ( v − sv s ),

α

α

N | j = v s s0 (s0 j + r j0 ) + 2v B α s0 (s j + r j ) + v α s0| j , N |k yk = 2v B s20 α + v α s0|0 + 2r0 v B α s0 + v s s0 r00 ,



N |k yk

  s0 r00 = l j s20 (2v B − 2v Bs s) + s0|0 ( v − sv s ) + r0 s0 (2v B − 2v Bs s) − sv ss α   r00 s0 + b j 2v Bs s20 + v s s0|0 + 2v Bs r0 s0 + v ss + s j (4v B s0 α + 2r0 v B α + v s r00 )

 yj

α

+ v α s j |0 + v α s0| j + r j (2v B α s0 ) + r j0 (2v s s0 ), W y j = ( Q α ) y j = b j Q s + l j ( Q − s Q s ), W yk bk = B Q s + s( Q − s Q s ), W yk sk 0 = Q s s0 , W yk sk j = s j Q s + W y j y k bk = k

W y j yk s

0

s0 j

α

( Q − s Q s ),

  b j     lj  s s Q − Q − Q ss B − s2 + Q ss B − s2 + Q − s Q s ,

α

=

lj

α

α

(−s Q ss s0 ) +

bj

α

( Q ss s0 ) +

W | j = ( Q α )|k = Q s (r j0 + s0 j ),

s j0

α

( Q − s Q s ),

W |k yk = Q s r00 ,



W |k yk



yj

= −l j

r00

α

s Q ss + b j

r00

α

Q ss + 2Q s r j0 .

Substituting these equations into the formula of Riemannian curvature R i j and computing by Maple we obtain Proposition 3.1. References [1] [2] [3] [4] [5] [6] [7] [8] [9] [10] [11] [12] [13] [14] [15]

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