We classify some special Finsler metrics of constant flag curvature on a manifold of dimension n>2.
1. Introduction
One of the important problems in Finsler geometry is to study and characterize Finsler metrics of constant flag curvature, which is the generalization of sectional curvature in Riemannian geometry. The local structure of Finsler metrics of constant flag curvature has been historically mysterious and their classification seems to be far from being solved.
The (α,β)-metrics are an important class of Finsler metrics including Randers metrics as the simplest class. By making use of navigation problem, Bao et al. gave a local classification of Randers metrics with constant flag curvature [1]. Recently, Zhou has classified that the (α,β)-metrics with constant flag curvature in the following form,
(1)F=(α+β)2α,
are locally projectively flat [2].
Lately, Shen and Zhao have studied projectively flat (α,β)-metrics
(2)F=α(1+εs+2ks2-k23s4),s=βα,
where ε is a constant and k is a nonzero constant, and they proved such projectively flat Finsler metrics with constant flag curvature must be locally Minkowskian [3]. Hence, one natural problem is to consider the classification of such metrics with constant flag curvature. In this paper, we prove the following rigidity result.
Theorem 1.
Let F=α+εβ+(2kβ2/α)-(k2β4/3α3), where the 1-form β is nonzero, ε is a constant, and k is a nonzero constant, be an (α,β)-metric on a manifold of dimension n>2. Suppose that F is of constant flag curvature; then it must be locally Minkowskian.
2. Preliminaries
Let F be a Finsler metric on an n-dimensional manifold M and Gi the geodesic coefficients of F, which are defined by
(3)Gi=14gil{[F2]xkylyk-[F2]xl}.
For any x∈M and y∈TxM∖{0}, the Riemann curvature Ry=Rji(∂/∂xi)⊗dxj is defined by
(4)Rji=2∂Gi∂xj-∂2Gi∂xk∂yjyk+2Gk∂2Gi∂yk∂yj-∂Gi∂yk∂Gk∂yj.
(α,β)-metrics were first introduced by Matsumoto [4]. They are expressed in the following form:
(5)F=αϕ(s),s=βα,
where α=aij(x)yiyj is a Riemannian metric and β=bi(x)yi is a 1-form. ϕ=ϕ(s) is a smooth positive function satisfying
(6)ϕ(s)>0,ϕ(s)-sϕ′(s)+(b2-s2)ϕ′′(s)>0,kkkkkkkkkkkkkkkkkkkkkkkkik∀|s|≤b<b0.
For any flag (P,y), where P=span{y,u}⊂TxM, the flag curvature is defined by
(7)K(P,y):=gy(Ry(u),u)gy(y,y)gy(u,u)-[gy(y,u)]2.
When F is Riemannian, K(P,y)=K(P) is independent of y∈P. It is just the sectional curvature of P in Riemannian geometry. F is said to be of scalar curvature if, for any y∈TpM, the flag curvature K(P,y)=K(y) is independent of P containing y∈TpM that is equivalent to the following system of equations in a local coordinate system (xi,yi) in TM,
(8)Rji=KF2(δji-F-1Fyjyi).
If K is a constant, then F is said to be of constant flag curvature.
Let
(9)rij:=12(bi|j+bj|i),sij:=12(bi|j-bj|i),
where “|” denotes the covariant derivative with respect to the Levi-Civita connection of α. Clearly β is closed if and only if sij=0. Moreover, we denote
(10)rji:=aikrkj,sji:=aikskj,r00:=rijyiyj,ri0:=rijyj,si:=bjsij,s0:=siyi,r:=rijbibj,si0:=sijyj,ri:=bjrji,ri:=aijrj,si:=aijsj,
where (aij):=(aij)-1 and bi:=aijbj. Let Gi and αGi be the geodesic coefficients of F and α, respectively. Then we have the following.
Lemma 2 (see [5]).
For an (α,β)-metric F=αϕ(s), s=(β/α), the geodesic coefficients Gi are given by
(11)Gi=Gαi+αQs0i+Θ{r00-2Qαs0}li+Ψ{r00-2Qαs0}bi,
where
(12)Q=ϕ′ϕ-sϕ′,Θ=ϕϕ′-s(ϕϕ′′+ϕ′ϕ′)2ϕ[(ϕ-sϕ′)+(B-s2)ϕ′′],Ψ=ϕ′′2[(ϕ-sϕ′)+(B-s2)ϕ′′].
Here li:=(yi/α) and B:=b2.
Let u(s,B):=-2ΘQ, υ(s,B):=-2ΨQ; one has the following.
Proposition 3 (see [6]).
For any (α,β)-metric F=αϕ(s), s=(β/α), the Riemannian curvature is given by
(13)Rji=Rαji+Tji,
where
(14)Tji=Δδji+C11lilj+C12libj+C13lisj+C14lirj+C130lis0j+C140lirj0-(uα)lisj|0+(2uα)lis0|j-(uQα2)lisksjk+(2Θ)lir00|j-(2Θ)lirj0|0-(2QαΘ)lirk0sjk+(4QαΘ)lirjks0k+C21bilj+C22bibj+C23bisj+C24birj+C230bis0j+C240birj0-(υα)bisj|0+(2υα)bis0|j-(υQα2)bisksjk+(2Ψ)bir00|j-(2Ψ)birj0|0-(2QαΨ)birk0sjk+(4QαΨ)birjks0k+C3sji+[C31silj+C32sibj+(υQα2)sisj+(2QαΨ)sirj0]+[C310s0ilj+C320s0ibj+C330s0isjmi+C331s0is0j+C340s0irj0]+C4rji+[C410r0ilj+C420r0ibjkkkkkkkk-(υα)r0isj-(2Ψ)r0irj0]+[C332skis0klj+C333skis0kbj-(Q2α2)skisjk]+(2Qα)s0|ji-(Qα)sj|0i+[C311s0|0ilj-Qss0|0ibj],Δ=r002α2[2ΘsΨ(B-s2)-Θs+Θ2-2sΘΨ]+r00s0α[2(υΘs+usΨ)(B-s2)+2uΘmmmm-2υsΘ-us+2QΘs-2ΘB2(υΘs+usΨ)(B-s2)]+2r00r0α(2ΘΨ-ΘB)-r00|0αΘ+s02[-2uB+u2+2usυ(B-s2)+2Qus]+2r0s0(-uB+2υΘ)+2uQα(sks0k)+4QΘ(rk0s0k)-us0|0,C11=r002α2[s2ΘΨs(sΘsΨs-4ΘsΨ-2sΘssΨ)(B-s2)+6sΘΨmm+Θ2+6s2ΘsΨ+sΘΘs+sΘss+2Θsmm-s2ΘΨs]+r00s0α[(B-s2)(-2sΨuss+susΨs+sυsΘs-2sυΘssm1111mi-5υΘs-2Ψus)(B-s2)+2sΘBs+Θummmm+suΘs+4s2Ψus-5QΘs+sussmmmm+sΘus+sQsΘs-2sQsΘss-s2Θυsmmmm+6s2υΘs+7sυΘ+us+2ΘB]+2r00r0α(ΘB+sΘBs+sΘΨs-2ΘΨ-2sΘsΨ)+r00|0α(Θ+sΘs)+s02[(B-s2)(susυs-3usυ-2sussυ)miiiii×(B-s2)+4s2usυ-2Qusssmiimi+suus-3Qus+2suBs+susQs(B-s2)]+2r0s0(-2sυΘs+sΘυs+suBs-3υΘ)+αsks0k(sQsu-Qu-2sQus)+s0|0(sus)+2rk0s0k(sQsΘ-2sQΘs-3QΘ),C12=r002α2[(2ΘssΨ-ΘsΨs)(B-s2)mm-2ΘΨ-ΘΘs+sΘΨs-6sΘsΨ-Θss(B-s2)]+r00s0α[(2υΘss+2Ψuss-usΨs-υsΘs)(B-s2)mmmim-(Qs+u+6sυ)Θs+2QΘss-2ΘBsmmmim-4susΨ+sυsΘ-uss-Θus-2υΘ(B-s2)]+2r00r0α(-ΘΨs+2ΘsΨ-ΘBs)+s02[(2ussυ-usυs)(B-s2)-uusmmii-Qsus-2uBs-4susυ+2Quss(2ussυ-usυs)(B-s2)(B-s2)]-2r0s0(uBs+Θυs-2υΘs)+2rk0s0k(2QΘs-QsΘ)+αsks0k(2Qus-Qsu)-s0|0us-r00|0αΘs,C13=r00[(2usΨ-Θsυ)(B-s2)-uΘmm+4ΘB+sυΘ-us-QΘs(B-s2)]+αs0[2uB-u2+Qus+usυ(B-s2)]-αr0(2υΘ+2uB),C14=4r00(ΘB+ΘΨ)+4αs0(uB+υΘ),C130=3r00α[QΘ+(1+sQ)Θs]+3s0(sQ+1)us,C140=r00α[2ΘsΨ(B-s2)-2sΘΨ-Θs-2Θ2]+s0[(4υΘs-2Ψus)(B-s2)miim+4QΘs+us-2uΘ-4ΘB-4sυΘ(4υΘs-2Ψus)(B-s2)(B-s2)]-4r0(ΘB+ΘΨ),C21=r002α2[(sΨs2-2sΨΨss-2ΨΨs)(B-s2)m1i+sΨss+Ψs+4s2ΨΨs(B-s2)]+r00s0α[(2sΨsυs-3υΨs-2sυΨss-2sΨυss)(B-s2)mmmim+sυss+2sΨBs-2QsΨss+5s2υΨsmmmim+sQsΨs+2s2υsΨ-2sυΨ-3QΨs(2sΨsυs-3υΨs-2sυΨss-2sΨυss)(B-s2)]+2r00r0α(-sΨΨs+sΨBs)+s02[(sυs2-2sυυss-υυs)(B-s2)+3s2υυsmim-2sQυss-3sυ2-Qυs-2υB+sQsυs+2sυBs(sυs2-2sυυss-υυs)(B-s2)]+2s0r0(sυBs-Ψυ+sΨυs-2sυΨs-υB)+r00|0α(sΨs)+2rk0s0k(-QΨ+sΨQs-2sQΨs)+s0|0(sυs-υ)+αsks0k(sQsυ+Qυ-2sQυs),C22=r002α2[(2ΨΨss-Ψs2)(B-s2)-Ψss-4sΨΨs]+r00s0α[2(υΨss+Ψυss-Ψsυs)(B-s2)hhhhhhh-2ΨBs-υss+2υΨhhhhhhh-2sΨυs-5sυΨs+2QΨss-QsΨs(υΨss+Ψυss-Ψsυs)(B-s2)(B-s2)]+2r00r0α(-ΨBs+ΨΨs)+2s0r0(-Ψυs+2Ψsυ-υBs)+s02[(2υυss-υs2)(B-s2)-Qsυshhhih-2υBs+2υ2+2Qυss-3sυυs(2υυss-υs2)(B-s2)]-r00|0αΨs-s0|0υs+αsks0k(2Qυs-Qsυ)+2rk0s0k(2QΨs-QsΨ),C23=r00[(-υΨs+2Ψυs)(B-s2)hhh+4ΨB-QΨs-υs+2sυΨ(-υΨs+2Ψυs)(B-s2)(B-s2)]+αs0[υυs(B-s2)+2υB+Qυs+sυ2]-2αr0(Ψυ+υB),C24=4r00(Ψ2+ΨB)+4αs0(υB+Ψυ),C230=3r00α(1+sQ)Ψs+3s0[υs-(υ-sυs)Q],C240=r00α[2ΨΨs(B-s2)-Ψs]+s0[(4υΨs-2Ψυs)(B-s2)+υshhhh-2sυΨ+4QΨs-4ΨB(B-s2)]kkkk-4r0(Ψ2+ΨB),C3=2αs0[Qsυ(B-s2)+QQs+sQυ+υ]+r00[2QsΨ(B-s2)+2Ψ-Qs+2sQΨ],C31=r00(-2QΨ+2sQsΨ-sQΨs)-αs0(υQ+sQυs-2sQsυ),C32=r00(-2QsΨ+QΨs)-αs0(2Qsυ-Qυs),C310=r00α[(QssΨs-2sΨQss)(B-s2)hhih+2Qss2Ψ-2sQΨ+Qs2Ψs+sQss+sΨs]+s0[(sQsυs-Qsυ-2sQssυ)(B-s2)hhhh+Qs2υs-QQs+sQs2-3sQυhhhh-υ+sυs-2sQQss+2Qss2υ],C320=r00α[(2QssΨ-QsΨs)(B-s2)hhih+2QΨ-2sΨQs-sQΨs-Qss-Ψs(B-s2)]+s0[(2Qssυ-Qsυs)(B-s2)-υs-Qs2hhhh+2υQ-2sQsυ+2QQss-sQυs(B-s2)],C330=-α[υQs(B-s2)+QQs+υ+sQυ],C331=-3Q2+3sQQs+3Qs,C340=-2[Qs(B-s2)+sQ+1]Ψ+Qs,C4=(2Ψ)r00+(2υ)αs0,C410=r00αsΨs+s0(sυs-υ),C420=-r00αΨs-s0υs,C332=α(Q-sQs)Q,C333=αQQs,C311=sQs-1.
Proposition 4 (see [6]).
For an (α,β)-metric F=αϕ(s),s=(β/α), the Ricci curvature of F is related to the Ricci curvature Ricα of α by
(15)Ric=Ricα+Tmm,
where
(16)Tmm=r002α2[(n-1)c1+c2]+1α{+r00|0[(n-1)c7+c8]r00s0[(n-1)c3+c4]hhhh+r00r0[(n-1)c5+c6]hhhh+r00|0[(n-1)c7+c8]}+{s02[(n-1)c9+c10]+(rr00-r02)c11hhhh+r0s0[(n-1)c12+c13]hhhh+(r00rmm-r0mr0m+r00|mbm-r0m|0bm)c14hhhh+r0ms0m[(n-1)c15+c16]hhhh+s0|0[(n-1)c17+c18]+s0ms0mc19(rr00-r02)}+α{mm-2s0|mbm+sm|0bm)c23+s0|mmc24rs0c20+sms0m[(n-1)c21+c22]hhhh+(-2s0|mbm+sm|0bm3smr0m-2s0rmm+2rms0mhhhhhhh-2s0|mbm+sm|0bm)c23+s0|mmc24}+α2(smsmc25+smisimc26),c1=2ΨΘs(B-s2)-2sΨΘ+Θ2-Θs,c2=(2ΨΨss-Ψs2)(B-s2)2-(6sΨΨs+Ψss)(B-s2)+2sΨs,c3=-4(2QΨΘs+QsΨΘ)(B-s2)+2(QsΘ+2QΘs)+4QΘ(sΨ-Θ)-2ΘB,c4=[(B-s2)2-4Ψ(2QΨss+QsΨs+QssΨ)+4QΨs2(B-s2)2](B-s2)2+[(B-s2)2-4Ψ2(Q-sQs)+2(2QssΨ+QsΨs+2QΨss)m-2ΨsB+20sQΨΨs(B-s2)2](B-s2)+2Ψ(Q-sQs)-4Ψs-Qss-10sQΨs,c5=4ΨΘ-2ΘB,c6=2(2ΨΨs-ΨsB)(B-s2)-2Ψs,c7=-Θ,c8=-Ψs(B-s2),c9=8QΨ(QΘs+QsΘ)(B-s2)+4Q2(Θ2-Θs)+4Q(ΘB-ΘQs),c10=[4Ψ2(2QQss-Qs2)+8QΨ(QΨss+QsΨs)ii-4Q2Ψs2](B-s2)2ii+[-16sQΨ(QΨs+QsΨ)-4Ψ(2QQss-Qs2)hhh-4Q(QΨss+QsΨs)hhh+4(QΨsB+QsΨB)(2QQss-Qs2)](B-s2)-12s2Q2Ψ2+4(2+3sQ)(QΨs+QsΨ)-8Q2Ψ+2QQss-Qs2+4sQΨB+8BΨ2Q2,c11=4Ψ2+4ΨB,c12=4Q(-2ΨΘ+ΘB),c13=[8Ψ(QsΨ-QΨs)+4(QΨsB+QsΨB)](B-s2)+8sQΨ2+4QΨs-4(1-sQ)ΨB,c14=2Ψ,c15=4QΘ,c16=-4(QsΨ-QΨs)(B-s2)+2Qs-2(1+2sQ)Ψ,c17=2QΘ,c18=2(QsΨ+QΨs)(B-s2)-Qs+2sQΨ,c19=-2Q2+2(1+sQ)Qs,c20=-8Q(Ψ2+ΨB),c21=-4Q2Θ,c22=-4Q2Ψs(B-s2)+2QΨ,c23=2QΨ,c24=2Q,c25=-4Q2Ψ,c26=-Q2.
Remark 5.
In Proposition 4, we have corrected some terms in the formulas for coefficients c9 and c10, which are not printed.
Definition 6 (see [7]).
Let
(17)Djkli=∂3∂yj∂yk∂yl(Gi-1n+1∂Gm∂ymyi),
where Gi are the spray coefficients of F. The tensor D:=Djkli∂i⊗dxj⊗dxk⊗dxl is called Douglas tensor. A Finsler metric is called Douglas metric if the Douglas tensor vanishes.
Note that an (α,β)-metric F=α+εβ+(2kβ2/α)-(k2β4/3α3) is a Douglas metric if and only if bi|j=σ[(1+4kB)aij-5kbibj] holds for some scalar function σ=σ(x) [8].
Definition 7 (see [7]).
Put
(18)Wki:=Aki-1n+1∂Akm∂ymyi,
where R=(Rkk/(n-1)),Aji=Rji-Rδji. Then W=Wki(∂/∂xi)⊗dxk is a tensor on TM∖{0}. it is called the Weyl curvature tensor.
It is Hilbert’s fourth problem in the regular case to study and characterize Finsler metrics on an open domain u⊂Rn whose geodesics are straight lines. Finsler metrics with this property are called projectively flat metrics. A famous theorem of Douglas is in the following.
Theorem 8 (see [9]).
A Finsler metric F on a manifold M (dim M>2) is locally projectively flat if and only if D=0 and W=0.
3. Proof of Theorem 1
In this section, we will prove Theorem 1. First, we will study the following lemma, because Lemma 4.1 [3] is found to have some wrong. For example, when k=-(1/4b2), F does not satisfy the definition of (α,β)-metrics at s=0.
Lemma 9.
F=α+εβ+(2kβ2/α)-(k2β4/3α3) is a Finsler metric if and only if one of the following holds.
If k>0, then b<1/k and ε∈[-k,k].
If k<0, then b<(1/2-k) and ε∈[-(23/24)-k,(23/24)-k],
where b:=∥βx∥α, for any x∈M.
Proof.
Let
(19)ϕ=1+εs+2ks2-k2s43,s=βα.
Direct computations yield
(20)ϕ-sϕ′=(ks2-1)2,ϕ-sϕ′+(b2-s2)ϕ′′=(ks2-1)(5ks2-1-4kb2).
From the definition of (α,β), we know that, for all |s|≤b<b0, ϕ must satisfy
(21)ϕ>0,ϕ-sϕ′+(b2-s2)ϕ′′>0.
That is,
(22)1+εs+2ks2-k2s43>0,(ks2-1)(5ks2-1-4kb2)>0.
Particularly, when s=0 and s=b, it is easy to see that
(23)1+4kb2>0,kb2-1≠0.
Now we first discuss the second inequality of (22). It is equivalent to the following two cases.
Let y1=ks2-1,y2=5ks2-1-4kb2; it is easy to see that y1 and y2 are quadratic functions with respect to s.
Case 1. When
(24)ks2-1>0,5ks2-1-4kb2>0,
k>0.
By the graph of quadratic function and (23), we find that there is no b such that ks2-1>0 and 5ks2-1-4kb2>0 always hold for all |s|≤b.
Consider (II) k<0.
By the graph of quadratic function and (23), we conclude the same result with (I).
In any case, Case 1 cannot exist for all |s|≤b.
Case 2. When
(25)ks2-1<0,5ks2-1-4kb2<0,
k>0.
By the graph of quadratic function and (23), the above system of inequality always holds if and only if b<(1/k).
Consider (IV) k<0.
Similarly, the above system of inequality always holds if and only if b<(1/2-k).
In the following, we will discuss the condition satisfying the first inequality of (22). Since
(26)1+εs+2ks2-k2s43>0⟺(ks2-3)2-3εs-12<0,
let y=(ks2-3)2-3εs-12.
Consider (1) k>0.
In this case, by (III), we know that |s|≤b<(1/k). By the graph and monotonicity of quadratic functions, we find
(27)-8-3εs<y≤-3-3εs.
Hence, for all |s|≤b<(1/k), (26) always holds if and only if 1+εs>0. Moreover, if ε>0, we have ε≤k. If ε=0, 1+εs>0 holds forever for all |s|≤b<(1/k). If ε<0, we get ε≥-k.
In a word, ε∈[-k,k].
Consider (2) k<0.
In this case, by (IV), we know that |s|≤b<(1/2-k). By the same way as above, we have
(28)-3-3εs≤y<-2316-3εs.
Hence, for all |s|≤b<(1/2-k), (26) always holds if and only if (23/16)+3εs≥0. Furthermore,
if ε>0, we have ε≤(23/24)-k. If ε=0, (23/16)+3εs≥0 holds forever for all |s|≤b<(1/k). If ε<0, we get ε≤-(23/24)-k.
In any case, ε∈[-(23/24)-k,(23/24)-k]. We complete the proof of the lemma.
Lemma 10.
Let F=α+εβ+(2kβ2/α)-(k2β4/3α3) be a Finsler metric on a manifold Mn, where ε is a constant and k is a nonzero constant. If F is of constant flag curvature, then it must satisfy
r00=σ(1+4kB-5ks2)α2,
s0=0,
s0ks0k=0,
where σ=σ(x) is a smooth function on M.
Proof.
By a direct computation, we have
(29)Q=3ε+12ks-4k2s33(-1+ks2)2,Ψ=2k1+4kB-5ks2,Θ=3ε-40k2s3-15εks2+8k3s52(1+4kB-5ks2)(3+3εs+6ks2-k2s4).
Since F is of constant flag curvature, F is also of constant Ricci curvature; that is, the following system of equations holds
(30)αRii+Tii-K(n-1)F2=0.
By Proposition 4, we can calculate Tii by Maple
(31)Tii=r002α2[(n-1)c-1A13A32+c-2A14]+r00s0α[(n-1)c-3A13A22A32+c-4A14A22]+1α{r00r0[(n-1)c-5A12A3+c-6A13]mmii+r00|0[(n-1)c-7A1A3+c-8A12]}+s02[(n-1)c-9A13A24A32+c-10A14A25]-16k2A12(rr00-r02)+r0s0[(n-1)c-12A12A22A3+c-13A13A22]+4kA1(r00rii-r0ir0i+r00|ibi-r0i|0bi)+r0is0i[(n-1)c-15A1A22A3+c-16A12A22]+s0|0[(n-1)c-17A1A22A3+c-18A12A22]+c-19A25s0is0i+α{rs0c-20A12A22+sis0i[(n-1)c-21A1A24A3+c-22A12A24]}+α[c-23A1A22(3sir0i-2s0rii+2ris0i-2s0|ibi+si|0bi)mmi×c-23A1A22+c-24A22s0|ii]+α2(c-25A1A24sisi+c-26A24sijsji),
where
(32)A1=1+4kB-5ks2,A2=-1+ks2,A3=3+3εs+6ks2-k2s4
and c-i(i=1,2,…,26) are polynomials of s.
Plugging (31) into (30) and multiplying it by A14 yield
(33)Rαii(1+4kB-5ks2)4+Tii(1+4kB-5ks2)4-K(n-1)F2(1+4kB-5ks2)4=0.
It is easy to see that
(34)αRii(1+4kB-5ks2)4-K(n-1)F2(1+4kB-5ks2)4≡0mod(1+4kB-5ks2).
Hence,
(35)Tii(1+4kB-5ks2)4≡0mod(1+4kB-5ks2).
From (31) we find that
(36)r002α2c-2+r00s0αc-4A22+c-10A25s02≡0mod(1+4kB-5ks2).
By Lemma 9, we need to divide two cases.
Consider (1) k>0.
In this case, we know that k∈(0,(1/B)), where B:=b2=∥β∥α2. Hence, ((1+4kB)/5k)>0.
For
(37)1+4kB-5ks2=-5k(s2-1+4kB5k)=-5k(s-1+4kB5k)(s+1+4kB5k),
we get
(38)r002α2c-2+r00s0αc-4A22+c-10A25s02≡0mod(s+1+4kB5k),r002α2c-2+r00s0αc-4A22+c-10A25s02≡0mod(s-1+4kB5k).
Simplifying (38) by Maple yields
(39)(r00α-8k(5(1+4kB)/k)(2kB-7)+75ε24(kB-1)2s0)2≡0mod(s+1+4kB5k),(r00α+8k(5(1+4kB)/k)(2kB-7)-75ε24(kB-1)2s0)2≡0mod(s-1+4kB5k).
Since k∈(0,(1/B)), 2kB-7≠0. Moreover, we note that ε≠±(8k/75)(5(1+4kB)/k)(2kB-7). Otherwise, by Lemma 9, F is not a Finsler metric.
From (39), and Zhou’s Lemma 4.1 [2], we obtain that
(40)r00+5k[8k(5(1+4kB)/k)(2kB-7)+75ε]24(kB-1)21+4kBαss0=σ1(1+4kB-5ks2)α2,r00+5k[8k(5(1+4kB)/k)(2kB-7)-75ε]24(kB-1)21+4kBαss0=σ2(1+4kB-5ks2)α2,
where σ1=σ1(x) and σ2=σ2(x) are smooth functions on a manifold Mn.
It is easy to see that
(41)ss0≡0mod(1+4kB-5ks2).
The above equation holds if and only if
(42)s0=0.
Therefore, we have
(43)r00=σ(1+4kB-5ks2)α2,
where σ=σ(x) is a smooth function on a manifold Mn.
Consider (2) k<0.
In this case, we know that k∈(-(1/4B),0), where B:=b2=∥β∥α2. Hence, ((1+4kB)/5k)<0. Moreover, 1+4kB-5ks2 is irreducible with respect to s.
Simplifying (36) directly by Maple yields
(44)[r00α+8(2kB-7)(5ks+5k(5ks2-4kB-1))-75ε24(1-kB)2s0]×[8(2kB-7)(5ks-5k(5ks2-4kB-1))-75ε24(1-kB)2s0r00α+8(2kB-7)(5ks-5k(5ks2-4kB-1))-75ε24(1-kB)2s0]≡0mod(1+4kB-5ks2).
By Lemma 9, we note that 5ks2-4kB-1<0. The above equation holds if and only if
(45)r00+8(2kB-7)(5ks+5k(5ks2-4kB-1))-75ε24(1-kB)2αs0=σ3(1+4kB-5ks2)α2
or
(46)r00+8(2kB-7)(5ks-5k(5ks2-4kB-1))-75ε24(1-kB)2αs0=σ4(1+4kB-5ks2)α2,
where σ3=σ3(x) and σ4=σ4(x) are smooth functions on a manifold Mn.
In any case, it is easy to find that s0=0. Hence, r00=σ(1+4kB-5ks2)α2, where σ=σ(x) is a smooth function on a manifold Mn.
Plugging (31) and s0=0 into (30) and multiplying it by A25, we have
(47)s0is0ic-19≡0mod(-1+ks2),
where
(48)c-19=-29(20k5s8-160k4s6-48εk3s5+360k3s4mmi+240εk2s3+45ε2ks2+36k-9ε2).
Simplifying (47) by Maple, we have
(49)s0is0i(192εks+256k+36ε2)≡0mod(-1+ks2).
So
(50)s0is0i=0.
Lemma 11.
Let F=α+εβ+(2kβ2/α)-(k2β4/3α3) be a Finsler metric on an n-dimensional manifold M, where ε is a constant and k is a nonzero constant. Suppose that F is of constant flag curvature; then β is closed and K=0.
Proof.
By Lemma 10, we have
(51)rij=σ[(1+4kB)aij-5kbibj],si=0,skisjk=0,
where σ=σ(x) is a smooth function on an n-dimensional manifold M.
Thus, by some computations, we have
(52)r0=σβ(1-kB),r=σ(1-kB)B,r00|0=σ0α2(1+4kB-5ks2)+2kσα(4αr0-5sr00),r00|jbj=σbα2(1+4kB-5ks2)+2kσα(4αr-5sr0),rj0|0bj=σ0(1-kB)β+kσ(3βr0-5Br00),rii=σ[(1+4kB)n-5kB],ri0r0i=σ2α2[(1+4kB)2-5ks2(2+3kB)],rj0=σα[(1+4kB)lj-5ksbj],rj=σ(1-kB)bj,rj0|0=σ0α[(1+4kB)lj-5ksbj]+σkα(8r0lj-5r00αbj-5srj0+5ss0j),r00|j=σjα2(1+4kB-5ks2)+2σkα(4rjα-5srj0-5ss0j),r0i=σα[(1+4kB)li-5ksbi],rji=σ[(1+4kB)δji-5kbibj],
where σ0:=σxiyi,σb:=σxibi,r0:=riyi,lj:=(yj/α),li:=aijlj=(yi/α),(aij):=(aij)-1.
Case 1 (ε≠0). Plugging the above equations into the expression of Tji in Proposition 3 and simplifying it by Maple, we have
(53)Tji=(T1A3σ2+T2σ0)α2A3δji+[T3A32ασ2+(T4A3α+T5)σ0A1]αA3lilj+[T6A32ασ2+(T7A3α+T8)σ0A1]αA3libj+αA22A3(T9σ+T10)lis0j+T11A3α2liσj+4αk(5αks2-4kB-1A1σ0+αksσ2)bilj+4αk2[5s(1-α)A1σ0-ασ2]bibj+ασT12A22bis0j+4kα2biσj-20k2sA3A23(slj-bj)s0i+T13A25s0is0j+T14αA22(2s0|ji-sj|0i)+T15A23s0|0ilj+T16A23s0|0ibj,
where Ti(i=1,…,16) are polynomials of s, A1=1+4kB-5ks2, A2=-1+ks2, and A3=3+3εs+6ks2-k2s4.
Because F has constant flag curvature, it is equivalent to
(54)αRji+Tji=KF2(δji-yiFFyj).
Substitute (53) into (54) and multiplying it by A25, we find
(55)s0is0jT13≡0mod(ks2-1),
where T14=-(1/3)(20k5s8-160k4s6-48εk3s5+360k3s4+240εk2s3+45ε2ks2+36k-9ε2).
By the same way as Lemma 10, we conclude s0i=0 or s0j=0; that is, β is closed.
Case 2(ε=0). By the same approach as the above, we still conclude that β is closed.
Now we will prove K=0.
Let ϕ=1+εs+2ks2-(k2s4/3). If F=αϕ(s), s=β/α, has constant flag curvature K, then F has constant Ricci curvature. That means that Ric=(n-1)KF2. Hence, F is an Einstein metric with Ricci constant K.
Note that ϕ=1+εs+2ks2-(k2s4/3) is a polynomial in s of degree 4. According to Theorem 1.1 (see [6]), we know that it is Ricci flat. So K=0.
Proof of Theorem 1.
By Lemma 10, we have
(56)rij=σ[(1+4kB)aij-5kbibj],
where σ=σ(x) is a smooth function on an n-dimensional manifold.
Moreover, by Lemma 11, we know that β is closed; that is, sij=0, Hence,
(57)bi|j=σ[(1+4kB)aij-5kbibj].
If σ=0, then bi|j=0; that is, β is parallel with respect to α. By the proof of Lemma 11, we find that the sectional curvature of α equals 0. That means that α is flat. Therefore, F is locally Minkowskian.
If σ≠0.
By (57), we obtain that F is a Douglas metric. That means that the Douglas curvature tensor D=0.
Since F has constant flag curvature, the Weyl curvature tensor W=0. By Theorem 8, we know that F must be locally projectively flat. Hence, by Theorem 1.3 (see [3]), F is locally Minkowskian.
Conflict of Interests
The author declares that there is no conflict of interests regarding the publication of this paper.
Acknowledgment
The author is grateful to the referee for the useful comments.
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