aa r X i v : . [ m a t h . DG ] N ov ON MATSUMOTO METRICS OF SCALE FLAG CURVATURE
XIAOLING ZHANG
College of Mathematics and Systems Science, Xinjiang University, Urumqi, Xinjiang Province, 830046,P.R.China, [email protected]
Abstract.
This paper contributes to the study of the Matsumoto metric F = α α − β , where the α is aRiemannian metric and the β is a one form. It is shown that such a Matsumoto metric F is of scalarflag curvature if and only if F is projectively flat. Keywords:
Finsler metric; Matsumoto metric; scalar flag curvature; projectively flat Introduction
In Finsler geometry, there are several important geometric quantities. In this paper, our main focusis on the flag curvature.For a Finsler manifold (
M, F ), the flag curvature K at a point x is a function of tangent planes P ⊆ T x M and nonzero vectors y ∈ P . This quantity tells us how curved the space is. When F isRiemannian, K depends only on the tangent plane P ⊆ T x M and is just the sectional curvature inRiemannian geometry. A Finsler metric F is said to be of scalar flag curvature if the flag curvature K at a point x is independent of the tangent plane P ⊆ T x M , that is, the flag curvature K is a scalarfunction on the slit tangent bundle T M \{ } . It is known that every locally projectively flat Finslermetric is of scalar flag curvature. However, the converse is not true.( α, β )-metrics form a special and important class of Finsler metrics which can be expressed in theform F = αφ ( s ), s = βα , where α = p a ij ( x ) y i y j is a Riemannian metric, β = b i ( x ) y i is a 1-formon M and φ ( s ) is a C ∞ positive function satisfying (2.3) on some open interval. In particular, when φ ( s ) = − s , the Finsler metric F = α α − β is called a Matsumoto metric, which was first introduced byM.Matsumoto to study the time it takes to negotiate any given path on a hillside(cf. [1]). Recently,some researchers have studied Matsumoto metrics([8, 9, 14, 17]).Randers metrics are the simplest ( α, β )-metrics. Bao, etc., finally classify Randers metrics of con-stant flag curvature by using the navigation method (see [2]). Further, Shen, etc., classify Randersmetrics of weakly isotropic flag curvature (see [12]). There are Randers metrics of scalar flag curvaturewhich are not of weakly isotropic flag curvature or not locally projectively flat(see [3, 11]). Besides,some relevant researches are refereed to [5, 10, 15], under additional conditions. So far, Randers met-rics of scalar flag curvature are still unknown. Xia and Yang obtain the results for Kropina metricsand m-Kropina metrics, respectively(see [13, 16]).Our main result concerns Matsumoto metrics of scalar flag curvature. Theorem 1.1.
Let F = α α − β be a non-Riemannian Matsumoto metric on an n -dimensional manifold M , n ≥ . Then F is of scalar flag curvature if and only if F is projectively flat, i.e., α is locallyprojectively flat and β is parallel with respect to α . Li obtains that an n ( ≥ α is locally projectively flat and β is parallel with respect to α (see [7]). It is known that α islocally projectively flat is equivalent to that α is of constant curvature. Hence, a Matsumoto metric,which is projectively flat (i.e., of scalar flag curvature), must be locally Minkowskian.The content of this paper is arranged as follows. In § § β , with respect to α , is a constant Killing vector field. By using it, Theorem 1.1 isproved in § Preliminaries
In this section, we give a brief description of several geometric quantities in Finsler geometry. See[4] in detail.Let F be a Finsler metric on an n -dimensional smooth manifold M and ( x, y ) = ( x i , y i ) the localcoordinates on the tangent bundle T M . The geodesics of F are locally characterized by a system ofsecond order ordinary differential equations d x i dt + 2 G i (cid:18) x ( t ) , dx ( t ) dt (cid:19) = 0 , where G i = 14 g ij { [ F ] x k y j y k − [ F ] x j } .G i are called spray coefficients of F.The Riemann curvature R (in local coordinates R ik ∂∂x i ⊗ dx k ) of F is defined by R ik = 2 ∂G i ∂x k − ∂ G i ∂x j ∂y k y j + 2 G j ∂ G i ∂y j ∂y k − ∂G i ∂y j ∂G j ∂y k . It is known that F is of scalar flag curvature if and only if, in a local coordinate system, R ik = K ( x, y ) { F δ i k − F F y k y i } . In particular, F is of weakly isotropic(resp.isotropic or constant) flag curvature if K ( x, y ) = θF + κ ( x )(resp. K ( x, y ) = κ ( x ) or constant), where κ = κ ( x ) is a scalar function and θ is a 1-form on M .In dimension n ≥
3, a Finsler metric F is of isotropic flag curvature if and only if F is of constant flagcurvature by Schur’s Lemma. In general, a Finsler metric of weakly isotropic flag curvature and thatof isotropic flag curvature are not equivalent.The Ricci curvature(or Ricci scalar) of F is defined by Ric := R mm . For a vector y ∈ T x M \ { } , define W y ( u ) = W ij ( y ) u j ∂∂x i | x by W ij ( y ) := A i j − n + 1 ∂A kj ∂y k y i , (2.1)where A i j := R i j − n − R mm δ ij . (2.2)We call W := { W y } y ∈ T x M \{ } the Weyl curvature. It is easy to check that W is a projective invariantwhich means that if G i = e G i + P y i , then W ij = f W ij . As is known that a Finsler metric is of scalarflag curvature if and only if its Weyl curvature W = 0. For a Riemannian space ( M, g ) of dimension n ≥
3, the following conditions are equivalent: (a) W = 0, (b) g is of scalar flag curvature, (c) g is ofconstant curvature, (d) g is locally projectively flat.( α , β )-metrics are an important class of Finsler metrics. An ( α , β )-metric is expressed by thefollowing form, F = αφ ( s ) , s := βα , where α = p a ij ( x ) y i y j is a Riemannian metric and β = b i ( x ) y i is a 1-form. φ = φ ( s ) is a C ∞ positivefunction on an open interval ( − b , b ) satisfying φ ( s ) − sφ ′ ( s ) + ( b − s ) φ ′′ ( s ) > , | s | ≤ b < b , (2.3)where b := k β ( x ) k α = p a ij b i b j . It is known that F = αφ ( s ) is a Finsler metric if and only if k β ( x ) k α < b for any x ∈ M . In particular, if φ ( s ) = 1 + s , then it is a Randers metric. If φ ( s ) = − s ,then it is a Matsumoto metric F = α α − β , etc. It is known that a Matsumoto metric F is a Finslermetric if and only if b < b = . N MATSUMOTO METRICS OF SCALE FLAG CURVATURE 3
Let G i ( x, y ) and α G i ( x, y ) denote spray coefficients of F and α , respectively. To express formulae forspray coefficients G i of F in terms of α and β , let’s introduce some notations. Let b i | j be a horizontalcovariant derivative of b i with respect to α . Denote r ij := 12 ( b i | j + b j | i ) , s ij := 12 ( b i | j − b j | i ) ,s ij := a ik s kj , s j := b i s ij = b i s ij , r j := b i r ij , r := b i r i ,r := r i y i , s := s i y i , r := r ij y i y j . Throughout this paper, we use a ij to raise and lower the indices of b i , r ij , s ij , r i , s i and y i , etc. Withthese, we can express the spray coefficients G i as follows(see [6]) G i = α G i + αQs i + Θ( r − αQs ) y i α + ψ ( r − αQs ) b i , (2.4)where Q : = φ ′ φ − sφ ′ , Θ : = φ ′ ( φ − sφ ′ )2 φ [ φ − sφ ′′ + ( B − s ) φ ′′ ] − sψ,ψ : = φ ′′ φ − sφ ′ + ( B − s ) φ ′′ ] . Here B := b . In particular, for a Matsumoto metric F = α α − β , it follows from (2.4) that G i = α G i + αQs i + Θ( r − αQs ) y i α + ψ ( r − αQs ) b i , (2.5)where Q = 1 A , Θ = 1 − s A , ψ = 1 A , (2.6)and A = 1 − s , A = 1+ 2 B − s are functions of s, B respectively. Note that B < for a Matsumotometric F = α α − β . The variations s or B is also a linear combination of two functions A and A , whichmeans A , A can be regarded as new variations in place of the variations s, B .3. Basic Formulas for ( α, β ) -metrics From (2.5), we have that G i = α G i + P y i + Q i , (3.1)where P = α ( r − αQs )Θ , Q i = αQs i + ψ ( r − αQs ) b i . Let G i = α G i + Q i . (3.2)By (3.1), (3.2) and (2.1), we have W ij = W i j . (3.3)Assume that F is of scalar flag curvature. Now we have(3.4) 0 = W ij = W i j = R i j − R mm n − δ i j − n + 1 ( R mj − R ss n − δ mj ) . m y i , Contracting (3.4) with b j b i yields0 = R i j b j b i − B R mm n − − sαn + 1 ( R i j b j − R mm n − b i ) . i . (3.5) XIAOLING ZHANG
Using formulas and notations in [6], we have(3.6) R i j b j = α R i j b j + b i ( sC + BC + rC − C s + C r − vαb j s j | + 2 vαb j s | j + vQα s k s k + 2 ψb j r | j − ψb j r j | + 2 Qψαs k r k + 4 Qψαr k s k )+ s i ( sC + BC + 2 Qψαr ) + s i ( sC + BC − C s )+ r i C + r i ( sC + BC − ψr ) + s i k s k ( sC + BC ) + s i k s k Q α + 2 b j s i | j Qα − b j s i j | Qα + s i | ( sC − BQ s ) , where v ( s, B ) := − ψQ,C = r α (( sψ s − sψψ ss − ψψ s )( B − s ) + sψ ss + ψ s + 4 s ψψ s )+ r s α ((2 sψ s v s − vψ s − svψ ss − sψv ss )( B − s ) + sv ss + 2 sψ Bs − sQψ ss + 5 s vψ s + sQ s ψ s + 2 s v s ψ − svψ − Qψ s )+ 2 r r α ( − sψψ s + sψ Bs ) + s (( sv s − svv ss − vv s )( B − s ) + 3 s vv s − sQv ss − sv − Qv s − v B + sQ s v s + 2 sv Bs )+2 s r ( sv Bs − ψv + sψv s − svψ s − v B ) + r | α sψ s +2 r k s k ( − Qψ + sψQ s − sQψ s ) + s | ( sv s − v )+ αs k s k ( sQ s v + Qv − sQv s ) ,C = r α ((2 ψψ ss − ψ s )( B − s ) − ψ ss − sψψ s )+ r s α (2( vψ ss + ψv ss − ψ s v s )( B − s ) − ψ Bs − v ss + 2 vψ − sψv s − svψ s + 2 Qψ ss − Q s ψ s )+ 2 r r α ( − ψ Bs + ψψ s ) + 2 s r ( − ψv s + 2 ψ s v − vB s )+ s ((2 vv ss − v s )( B − s ) − Q s v s − v Bs + 2 v + 2 Qv ss − svv s ) − r | α ψ s − s | v s + as k s k (2 Qv s − Q s v ) + 2 r k s k (2 Qψ s − Q s ψ ) ,C = r (( − vψ s + 2 ψv s )( B − s ) + 4 ψ B − Qψ s − v s + 2 svψ )+ αs ( vv s ( B − s ) + 2 v B + Qv s + sv ) − αr ( ψv + v B ) ,C = 4 r ( ψ + ψ B ) + 4 αs ( v B + ψv ) ,C = 3 r α (1 + sQ ) ψ s + 3 s ( v s − ( v − sv s ) Q ) ,C = r α (2 ψψ s ( B − s ) − ψ s ) − r ( ψ + ψ B )+ s ((4 vψ s − ψv s )( B − s ) + v s − svψ + 4 Qψ s − ψ B ) ,C = r ( − Qψ + 2 sQ s ψ − sQψ s ) − αs ( vQ + sQv s − sQ s v ) ,C = r ( − Q s ψ + Qψ s ) − αs (2 Q s v − Qv s ) ,C = r α (( sQ s ψ s − sψQ ss )( B − s )+2 s Q s ψ − sQψ + s Qψ s + sQ ss + sψ s )+ s (( sQ s v s − Q s v − sQ ss v )( B − s ) + s Qv s − QQ s + sQ s − sQv − v + sv s − sQQ ss + 2 s Q s v ) ,C = r α ((2 Q ss ψ − Q s ψ s )( B − s ) + 2 Qψ − sψQ s − sQψ s − Q ss − ψ s )+ s ((2 Q ss v − Q s v s )( B − s ) − v s − Q s + 2 Qv − sQ s v + 2 QQ ss − sQv s ) , N MATSUMOTO METRICS OF SCALE FLAG CURVATURE 5 C = − Q + 3 sQQ s + 3 Q s ,C = 2 ψr + 2 vαs ,C = r α sψ s + s ( sv s − v ) ,C = − r α ψ s − s v s ,C = α ( Q − sQ s ) Q,C = αQQ s ,C = sQ s − Q, (3.7)and R mm = α R mm + r α c + 1 α ( r s c + r r c + r | c )+ s c + ( rr − r ) c + r s c + ( r r mm − r m r m + r | m b m − r m | b m ) c + r m s m c + s | c + s m s m c + α { rs c + s m s m c + (3 s m r m − s r mm + 2 r m s m − s | m b m + s m | b m ) c + s m | m c } + α ( s m s m c + s i m s mi c ) , (3.8)where c = (2 ψψ ss − ψ s )( B − s ) − (6 sψψ s + ψ ss )( B − s ) + 2 sψ s ,c = ( − ψ (2 Qψ ss + Q s ψ s + Q ss ψ ) + 4 Qψ s )( B − s ) +( − ψ ( Q − sQ s ) + 2(2 Q ss ψ + Q s ψ s + 2 Qψ ss ) − ψ sB + 20 sQψψ s )( B − s )+2 ψ ( Q − sQ s ) − ψ s − Q ss − sQψ s ,c = 2(2 ψψ s − ψ sB )( B − s ) − ψ s ,c = − ψ s ( B − s ) ,c = (4 ψ (2 QQ ss − Q s ) + 8 Qψ ( Qψ ss + Q s ψ s ) − Q ψ s )( B − s ) +( − sQψ ( Qψ s + Q s ψ ) − ψ (2 QQ ss − Q s ) − Q ( Qψ ss + Q s ψ s )+4( Qψ sB + Q s ψ B ) + 8 Q ψ )( B − s ) − s Q ψ + 4(2 + 3 sQ )( Qψ s + Q s ψ ) − Q ψ + 2 QQ ss − Q s + 4 sQψ B ,c = 4 ψ + 4 ψ B ,c = (8 ψ ( Q s ψ − Qψ s ) + 4( Qψ sB + Q s ψ B ))( B − s ) + 8 sQψ + 4 Qψ s − − sQ ) ψ B ,c = 2 ψ,c = − Q s ψ − Qψ s )( B − s ) + 2 Q s − sQ ) ψ,c = 2( Q s ψ + Qψ s )( B − s ) − Q s + 2 sQψ,c = − Q + 2(1 + sQ ) Q s ,c = − Q ( ψ + ψ B ) ,c = − Q ψ s ( B − s ) + 2 Qψ,c = 2 Qψ,c = 2 Q,c = − Q ψ,c = − Q . (3.9) 4. A Special Case for Matsumoto metrics
Lemma 4.1. ( [14] ) (1) − s and (1 + 2 B ) − s are relatively prime polynomials in s if and only if B = 1 ; (2) B − s and (1 + 2 B ) − s are relatively prime polynomials in s if and only if B = 1 ; (3) B − s (or (1 + 2 B ) − s ) and − s are relatively prime polynomials in s if and only if B = . For an ( α, β )-metrics, the form β is said to be Killing (resp. closed) form if r ij = 0 (resp. s ij = 0). β is said to be a constant Killing form if its dual is a Killing vector field and has constant length withrespect to α , equivalently r ij = 0 , s i = 0. XIAOLING ZHANG
Lemma 4.2.
Let F = α α − β be a non-Riemannian Matsumoto metric on an n -dimensional manifold M , n ≥ . Suppose β is a constant Killing form. Then F is of scalar flag curvature if and only if F is projectively flat.Proof. Suppose β is a constant Killing form, i.e., r ij = 0 , s i = 0. We need to rewrite the equation(3.5). By (3.6), we have(4.1) R i j b j = α R i j b j + α [( B − s ) QQ s + sQ + Q ] s i k s k + 2 αQb j s i | j − [( B − s ) Q s + sQ ] s i | , (1) The calculations for R i k b k b i .By (4.1), we have(4.2) R i j b j b i = α R i j b j b i − [( B − s ) Q s + sQ ] s k s k . (2) The calculations for R mm .By (3.8), R mm can be rewritten as(4.3) R mm = α R mm + 2( − Q + sQQ s + Q s ) s k s k + 2 αQs k | k − α Q s i k s ki . (3) The calculations for ( R i j b j ) . i .By (4.1), ( R i j b j ) . i can be rewritten as(4.4) ( R i j b j ) . i =( α R i j ) . i + 1 α ( B − s )( QQ s − sQ s − sQQ ss − Q ss ) s k s k − [( B − s ) Q s + sQ ] s k | k + α [( B − s ) QQ s + sQ + Q ] s i k s ki . (4) The calculations for ( R mm b i ) . i .By (4.3) and a direct computation, we obtain(4.5) ( R mm ) . i b i =( α R mm ) . i b i + 2 α ( B − s )( − QQ s + sQ s + sQQ ss + Q ss ) s k s k + 2[( B − s ) Q s + sQ ] s k | k − α [( B − s ) QQ s + sQ + Q ] s i k s ki . Notice that A − = (1 − s ) − only exist in the term ( R i j b j ) . i and ( R mm b i ) . i . Let V := 1 α ( B − s )( sQ s + sQQ ss ) s k s k = 12 s ( B − s ) αA s k s k Plugging (4.2)-(4.5) into (3.5) yields(4.6) 0 = − sαn + 1 ( − V − n − V ) + ( · · · )= 12 s ( B − s )( n − A s k s k + ( · · · ) , = 12 s ( B − s )(1 + 4 s ) ( n − − s ) s k s k + ( · · · ) , where ( · · · ) does not contain the factor A − .For the Matsumoto metric F , we have B < /
4, which implies B − s can not be divided by 1 − s from Lemma 4.1. Obviously, s can not be divided by 1 − s either. Thus s k s k must be divided by1 − s = α − β α . This implies there exists some function ρ = ρ ( x ), such that s k s k = ρ ( α − β ).Differentiating both sides of it with y i y j yields s kj s ki = ρ ( a ij − b i b j ). Then contracting it with b j gives 0 = ρ (1 − B ) b j , which means ρ = 0 and 0 = s kj s ki . Since | s ij | α = − s i k s ki = − a ij s jk s ki = 0, s ij = 0. Hence b i | j = 0, which means that β is parallel with respect to α . So by (4.6), we have W ij = W i j = α W i j = 0. Then α is projectively flat, so is F .The converse is obvious. This completes the proof of Lemma 4.2. (cid:3) N MATSUMOTO METRICS OF SCALE FLAG CURVATURE 7 Proof of Theorem 1.1
Lemma 5.1.
Let F = α α − β be a non-Riemannian Matsumoto metric on an n ( ≥ -dimensionalmanifold M . If F is of scalar flag curvature, then β is a conformal 1-form with respect to α . i.e.there is a function σ = σ ( x ) on M such that r = σα .Proof. Rewriting the equation (3.5).(1) The calculations for R i j b j b i .By (3.6), we have(5.1) R i j b j b i = α R i j b j b i + B ( sC + BC + rC − C s + C r − αvb j s j | + 2 αvb j s | j + α vQs k s k + 2 ψb j r | j − ψb j r j | + 2 αQψs k r k + 4 αQψr k s k )+ s ( sC + BC − C s ) + rC + r ( sC + BC − ψr ) + s k s k ( sC + BC )+ α Q s k s k + 2 αQb i b j s i | j − αQb i b j s i j | + ( sC − BQ s ) b i s i | , which can be rewritten as(5.2) R i j b j b i = α R i j b j b i + f ( s, B ) α A r + X i ,j ,k α i A j A k ( · · · ) , where the sum indices i , j and k satisfy − ≤ i ≤ , ≤ j ≤ ≤ k ≤ f ( s, B ) is a nonzero polynomial of s, B given by f ( s, B ) /A = B ( B − s ) (2 ψψ ss − ψ s ) − B ( B − s )(6 sψψ s + ψ ss ) + Bsψ s ,f ( s, B ) = − B { s + 3(1 − B ) s + (8 B − B − s + 3 B ( B + 2) } , ( · · · ), independent of r , is a polynomial of A , A (or s, B ), and the degree of ( · · · ) in s is no morethan deg( A j A k ) = j + k .(2) The calculations for R mm .By (3.8), BR mm can be rewritten as(5.3) BR mm = B α R mm + f ( s, B ) α A r + X i ,j ,k α i A j A k ( · · · ) , where the sum indices i , j and k satisfy − ≤ i ≤ , ≤ j ≤ ≤ k ≤ f ( s, B ) is a nonzero polynomial of s, B given by f ( s, B ) /A = Bc = B ( B − s ) (2 ψψ ss − ψ s ) − B ( B − s )(6 sψψ s + ψ ss ) + 2 Bsψ s ,f ( s, B ) = − B { s − B ) s + 6(1 + 2 B ) s + 2(2 B − B − s + 3 B ( B + 2) } , ( · · · ), independent of r , is a polynomial of A , A (or s, B ), and the degree of ( · · · ) in s is no morethan deg( A j A k ) = j + k .(3) The calculations for ( R i j b j ) . i .By (3.6), ( R i j b j ) . i can be rewritten as(5.4) ( R i j b j ) . i = ( α R i j b j ) . i + f ( s, B ) α A r + X i ,j ,k α i A j A k ( · · · ) , where the sum indices i , j and k satisfy − ≤ i ≤ , ≤ j ≤ ≤ k ≤ f ( s, B ) is a nonzero polynomial of s, B given by f ( s, B ) /A = 2( B − s ) ψψ sss − ( B − s ) (18 sψψ ss + 6 ψψ s + ψ sss )+ ( B − s )(5 sψ ss + ψ s + 24 s ψψ s ) − s ψ s ,f ( s, B ) = − s + 27(16 B + 5) s − B + 2 B − s − B + 31 B + 1) s + 9(40 B − B − B − s + 9 B ( − B + 58 B + 7) s − B (16 B + 12 B + 54 B − , XIAOLING ZHANG ( · · · ), independent of r , is a polynomial of A , A (or s, B ), and the degree of ( · · · ) in s is no morethan deg( A j A k ) + 1 = j + k + 1.(4) The calculations for ( R mm b i ) . i .By (3.8) and a direct computation, we obtain(5.5)( R mm ) . i b i =( α R mm ) . i b i + r α { ( B − s ) c s − sc } + 1 α { r s (( B − s ) c s − sc )+ r r (( B − s ) c s − sc + 2 c ) + r | (( B − s ) c s − sc ) } + 1 α { s ( B − s ) c s + rr (( B − s ) c s + c ) + r ( − ( B − s ) c s + 2 c )+ r s (( B − s ) c s + 2 c ) + r r mm ( B − s ) c s + r m s m ( − ( B − s ) c s + c s )+ r | m b m (( B − s ) c s + c ) + r m | b m ( − ( B − s ) c s + 2 c )+ s | c s + s m s m c s } + rs (( B − s ) c s + sc + c ) + s m s m (( B − s ) c s + sc + 2 c )+ s m r m (3( B − s ) c s + 3 sc − c ) − s r mm (2( B − s ) c s + 2 sc )+ r m s m (2( B − s ) c s + 2 sc + c ) − s | m b m (2( B − s ) c s + 2 sc − c )+ s m | b m (( B − s ) c s + sc + c ) + s m | m (( B − s ) c s + sc )+ (2 r r mm − r m r m + b k b m ( r k | m − r mk | )) c + α { s m s m (( B − s ) c s + 2 sc − c ) + s i m s mi (( B − s ) c s + 2 sc ) + b k s mk | m c } . Thus, we have(5.6) ( R mm b i ) . i = ( α R mm b i ) . i + f ( s, B ) α A r + X i ,j ,k α i A j A k ( · · · ) , where the sum indices i , j and k satisfy − ≤ i ≤ , ≤ j ≤ ≤ k ≤ f ( s, B ) is a nonzero polynomial of s, B given by f ( s, B ) /A = ( B − s ) c s − sc = 2( B − s ) ψψ sss − ( B − s ) (18 sψψ ss + 6 ψψ s + ψ sss )+ ( B − s )(6 sψ ss + 2 ψ s + 24 s ψψ s ) − s ψ s ,f ( s, B ) = − s + 216(1 + 2 B ) s − B + 4 B + 1) s + 54(4 B + B + 1) s + 18(16 B − B − B − s + 54 B ( − B + 9 B + 1) s − B (4 B + 24 B − , ( · · · ), independent of r , is a polynomial of A , A (or s, B ), and the degree of ( · · · ) in s is no morethan deg( A j A k ) + 1 = j + k + 1.Plugging (5.2), (5.3), (5.4) and (5.6) into (3.5) yields(5.7) 0 = α R i j b j b i + f ( s, B ) α A r + X i ,j ,k α i A j A k ( · · · ) − n − { B α R mm + f ( s, B ) α A r + X i ,j ,k α i A j A k ( · · · ) }− n + 1 { sα ( α R i j b j ) . i + sf ( s, B ) α A r + s X i ,j ,k α i − A j A k ( · · · ) } + 1 n − { sα ( α R mm b i ) . i + sf ( s, B ) α A r + s X i ,j ,k α i − A j A k ( · · · ) } . N MATSUMOTO METRICS OF SCALE FLAG CURVATURE 9
Multiplying both sides of (5.7) with α yields(5.8) 0 = α { α R i j b j b i − n − B α R mm − n + 1 sα ( α R i j b j ) . i + 1 n − sα ( α R mm b i ) . i } + α X i,j,k α i A j A k ( · · · ) + r A { A f ( s, B ) − A f ( s, B ) n − − sf ( s, B ) n + 1 + sf ( s, B ) n − } . Consequently, the term r A { A f ( s, B ) − A f ( s, B ) n − − sf ( s, B ) n + 1 + sf ( s, B ) n − } (5.9)must be divided by α . Observe that the coefficient of r in (5.9) can not be divided by α from thedefinition of A and f i ( s, b ) (i=1,...,4). Thus r must be divided by α , which means r = σα forsome function σ = σ ( x ). This completes the proof of Lemma 5.1. (cid:3) Proof of Theorem 1.1.
Assume that F is of scalar flag curvature. Note that B < for a Matsumoto metric F .By (5.1), we conclude that there does not exist A − in the term R i j b j b i . Also, there does not exist A − in the term R mm by (3.8). So we focus on the term ( R i j b j ) . i and ( R mm b i ) . i .By (3.6), ( R i j b j ) . i can be rewritten as(5.10) ( R i j b j ) . i = 2 ψψ sss ( B − s ) ( r − αQs ) α + [ · · · ]= 324( B − s ) ( r − αQs ) α A + [ · · · ] , where [ · · · ] does not contain the factor A − .For the same reason, by (5.5), ( R mm b i ) . i can be rewritten as(5.11) ( R mm b i ) . i = 2 ψψ sss ( B − s ) ( r − αQs ) α + [ · · · ]= 324( B − s ) ( r − αQs ) α A + [ · · · ] , where [ · · · ] does not contain the factor A − .Thus, (3.5) can also be rewritten as(5.12) 0 = − n + 1 ( 324( B − s ) ( r − αQs ) α A − B − s ) ( r − αQs ) ( n − α A ) sα + [ · · · ]= − n − s ( B − s ) ( r − αQs ) ( n − α A + [ · · · ] , = − n − s (1 + 2 B + 3 s ) (1 + 2 s )( B − s ) ( A r − αs ) ( n − α (1 − s )((1 + 2 B ) − s ) + [ · · · ] , where [ · · · ] does not contain the factor A − .By Lemma 5.1, we have(5.13) A r − αs = α ( σα − σβ − s ) . Note(5.14) s (1 + 2 B + 3 s ) (1 + 2 s ) = 16(1 + 2 B )(5 + 4 B )9 (1 + 2 B + 3 s ) mod { (1 + 2 B ) α − β } . For the same reason in discussing (4.6), we get that [(1 + 2 B ) α + 3 β ]( σα − σβ − s ) must be dividedby (1 + 2 B ) α − β from (5.12), (5.13) and (5.14), i.e.,(5.15) [(1 + 2 B ) α + 3 β ]( σα − σβ − s ) = 0 mod { (1 + 2 B ) α − β } . Case one : σ = 0. By (5.15), we have s = ρ { (1 + 2 B ) α − β } for some function ρ = ρ ( x ).Differentiating both sides yields (1 + 2 B ) ρa ij = 9 ρb i b j + s i s j . Since n ≥
3, we get ρ = 0 and s = 0. Hence, β is a constant Killing form. Case two : σ = 0. Let q := [(1+2 B ) α +3 β ]( σα − σβ − s ) . q can be rewritten as q = α q even + q odd ,where(5.16) q even := (1 + 2 B ) σ α − − B ) σ β − σ (1 − B ) βs + 4(1 + 2 B ) s and(5.17) q odd := − (1 + 8 B ) σ α β − B ) σα s + 12 σ β + 24 σβ s + 12 βs . Hence, q even can be divided by (1 + 2 B ) α − β .Meanwhile, we have(5.18) ( σα − σβ − s )( σα + 2 σβ + 2 s ) = σ α − σ β − σβs − s can be divided by (1 + 2 B ) α − β from (5.12). σ { (5.16) + (1 + 2 B ) × (5.18) } yields σ (1 + 2 B ) α − σβ − βs = 0 mod { (1 + 2 B ) α − β } . That is(5.19) σ (1 + 2 B ) α − σβ − βs = ̺ { (1 + 2 B ) α − β } holds for some function ̺ = ̺ ( x ). (5.19) can be rewritten as(1 + 2 B )[ σ − (1 + 2 B ) ̺ ] α = 3 β ( − ̺β + 2 σβ − s ) , which is equivalent to(5.20) σ = (1 + 2 B ) ̺ and(5.21) 3 ̺β = 2 σβ − s . Differentiating both sides of (5.21) and contracting it with b i yield 3 ̺ = 2 σ , which is a contractionwith (5.20).Above all, we have r = s = 0, i.e., β is a constant Killing form. By Lemma 4.2, F is projectivelyflat.The converse is obvious. This completes the proof of Theorem 1.1. (cid:3) From Theorem 1.1, one obtains the following
Corollary 5.1. ( [17] ) Let F = α α − β be a non-Riemannian Matsumoto metric on an n -dimensionalmanifold M , n ≥ . Then the following conditions are equivalent:(a) F is of weakly isotropic flag curvature K ; (b) F is of constant flag curvature K ; (c) α is a flatmetric and β is a constant one form.In this case, F is locally Minkowskian. Acknowledgements
Author would like to express her sincere thanks to Prof. Yibing Shen for hisconstant encouragement and many helpful discussions on this paper.
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