Watanabe Yoshiyuki

Yamazaki Tsuyoshi

University of Toyama Repository

Math. J. Toyama Univ.Vol. 26(2003), 45-60On a certain change of Ka¨hler metricsTsuyoshi Yamazaki and Yoshiyuki Watanabe1 IntroductionIt is well known that every Ka¨hler metric ds2 = 2Σgαβ¯dzαdz¯β is locallyexpressible in the formgαβ¯ =∂2ρ∂zα∂z¯β,with respect to local complex coordinates (zα), α = 1, · · · , n, where ρ =ρ(z, z¯) is a real valued function, called (Ka¨hler) potential.The purpose of this paper is to initiate a study on the Ka¨hler metric g˜αβ¯,changing a Ka¨hler metric gαβ¯ as follows:g˜αβ¯ = f(ρ)gαβ¯ + f′(ρ)ραρβ¯,(1.1)2000 Mathematics Subject Classification. 53B35, 53C55.Key words and phrases. Ka¨hler metrics, a change of Ka¨hler metrics, Ricci tensors,constant scalar curvature.46 Tsuyoshi Yamazaki and Yoshiyuki Watanabewhere ρα =∂ρ∂zα, f = f(ρ) and f ′(ρ) =dfdρsatisfyf > 0, f + lf ′ > 0,(1.2)denoting l = gαβ¯ραρβ¯ = ραρα and ρα = gαβ¯ρβ¯. In fact, it can be easilychecked that g˜αβ¯ satisfy the Ka¨hler condition∂g˜αβ¯∂z¯γ=∂g˜αγ¯∂z¯β.(1.3)In particular, if the gαβ¯ is a (locally) flat Ka¨hler metric, that is, gαβ¯ =∂2t∂zα∂z¯β= δαβ, where t = Σzαz¯α, then we haveg˜αβ¯ =∂2f(t)∂zα∂z¯β= f˙(t)δαβ + f¨(t)z¯αzβ,(1.4)denoting f˙(t) =dfdtand f¨(t) =d2fdt2. ( Tachibana and Liu [2], Watanabe[3], [4] and Watanabe and Mori [5]). Now, let gαβ¯ be a Fubini metric (cf.Tachibana [1]) and we study the Ka¨hler metric g˜αβ¯, changed by (1.1). Inparticular, we shall generalize some results in Watanabe [3]. The maintheorem is the followingTheorem 1. Let Dn be Cn or the ball Bn = {(zα) ∈ Cn | Σzαz¯α < r2}around the origin of Cn, whose radius is given by r =2√|k| (k 6= 0). Let(Dn, gαβ¯) be a Fubini space. If the scalar curvature R˜ of the Ka¨hler metricg˜αβ¯, changed by (1.1), satisfies the conditionR˜ = constant,(1.5)then (Dn, g˜αβ¯) is of constant holomorphic curvature or flat.Preliminary facts will be given in Section 2. In Section 3, we shall com-pute the curvature tensors of the Ka¨hler metric, changed by (1.1). InSections 4, 5, and 6, we shall study the Ka¨hler metric g˜αβ¯, changing a Fu-bini metric in the form (1.1). In Section 5, we shall consider such a Ka¨hlermanifold, satisfying the condition:On a certain change of Ka¨hler metrics 47∇˜k∇˜lR˜ij − ∇˜l∇˜kR˜ij = 0,(1.6)where R˜ij is the Ricci tensor of g˜αβ¯. The last section is devoted to provethe main theorem.2 PreliminariesWe agree to adopt the summation convention and the following rangeof indices throughout the paper:1 ≤ i, j, k, · · · ≤ 2n,1 ≤ α, β, γ, · · · ≤ n.Consider a complex n dimensional Ka¨hler manifold with metricds2 = Σgjkdzjdzk,(2.1)where (zα) are local complex coordinates and z¯α = zα¯ (=conjugate of zα).gjk satisfy the conditions:gαβ = gα¯β¯ = 0,(2.2)and (1.3). Then (2.1) becomesds2 = 2Σgαβ¯dzαdz¯β.(2.3)gjk satisfy the corresponding equations to (2.2). The Christoffel symbolsΓijk vanish exceptΓαβγ = gα²¯∂gβ²¯∂zγ,(2.4)and their conjugates. As to the curvature tensor Rijkl, only the componentsof the form Rαβγδ¯and their conjugates are different from zero, and it holdsthatRαβγδ¯ =∂Γαβγ∂z¯δ,(2.5)48 Tsuyoshi Yamazaki and Yoshiyuki Watanabefrom which the Ricci tensor Rij is given byRβγ¯ = Rαβγ¯α = −∂Γαβα∂z¯γ,(2.6)Rβγ = Rβ¯γ¯ = 0.(2.7)The scalar curvature R = gjkRjk is 2gαβ¯Rαβ¯.A Ka¨hler manifold is called a space of constant holomorphic curvature kif its curvature tensor satisfiesRαβ¯γδ¯ = k(gαβ¯gγδ¯ + gαδ¯gγβ¯),denoting k =R2n(n+ 1).3 Curvature tensorsFor a Ka¨hler metric gαβ¯ =∂2ρ∂zα∂z¯β, we denote the curvature tensor, theRicci tensor and the scalar curvature by Rαβγδ¯, Rαβ¯ and R, respectively.We are now going to compute the curvature tensor R˜αβγδ¯, the Ricci tensorR˜αβ¯ and the scalar curvature R˜ of the Ka¨hler metric g˜αβ¯, given byg˜αβ¯ = f(ρ)gαβ¯ + f′ραρβ¯,(3.1)where the dash means differentiation with respect to ρ. As the metric ispositive definite, the function should satisfy (1.2). Then g˜αβ¯ are given byg˜αβ¯ =1f(gαβ¯ − f′f + lf ′ραρβ¯).(3.2)From (2.4), we haveΓ˜αβγ = g˜αλ¯∂g˜βλ¯∂zγ= Γαβγ +f ′f(ρβδαγ + ργδαβ ) +ff ′′ − 2f ′2f(f + lf ′)ραρβργ +f ′f + lf ′ρα∇βργ ,On a certain change of Ka¨hler metrics 49denoting ∇γρβ = ∂ρβ∂zγ− Γαγβρα.After some computations, we have(3.3) R˜αβγδ¯ =∂Γ˜αβγ∂z¯δ= Rαβγδ¯ +ff ′′ − f ′2f2ρ¯δ(ρβδαγ + ργδαβ ) +f ′f(gβδ¯δαγ + gγδ¯δαβ )+f(f + lf ′)(ff ′′ − 2f ′2)′ − (ff ′′ − 2f ′2)(3ff ′ + lff ′′ + lf ′2)f2(f + lf ′)2ραρβργ ρ¯δ+ff ′′ − 2f ′2f(f + lf ′)ρα(ρβgγδ¯ + ργgβδ¯)−f ′(ff ′′ − 2f ′2)f(f + lf ′)2ραρβργρ²¯∇²¯ρ¯δ+ff ′′ − 2f ′2f(f + lf ′)ρβργgα²¯∇²¯ρ¯δ +ff ′′ − 2f ′2(f + lf ′)2ραρ¯δ∇βργ− f′2(f + lf ′)2(ρ²¯∇²¯ρ¯δ)ρα∇βργ +f ′f + lf ′(∇βργ)gα²¯∇²¯ρ¯δ− f′f + lf ′ραρλRλβγδ¯.Next, from (2.6) we have(3.4) R˜βδ¯ = −R˜αβαδ¯= Rβδ¯ −(n+ 1)f ′fgβδ¯ −ff ′′ − 2f ′2f(f + lf ′)lgβδ¯ −(n+ 1)(ff ′′ − f ′2)f2ρβ ρ¯δ−f(f + lf′)(ff ′′ − 2f ′2)′ − (ff ′′ − 2f ′2)(3ff ′ + lff ′′ + lf ′2)f2(f + lf ′)2lρβ ρ¯δ−ff′′ − 2f ′2f(f + lf ′)ρβ ρ¯δ −ff ′′ − 2f ′2(f + lf ′)2ρβρ²¯∇²¯ρ¯δ−ff′′ − 2f ′2(f + lf ′)2ραρ¯δ∇βρα +f ′2(f + lf ′)2(ρ²¯∇²¯ρ¯δ)ρα∇βρα50 Tsuyoshi Yamazaki and Yoshiyuki Watanabe− f′f + lf ′(∇βρα)gα²¯∇²¯ρ¯δ +f ′f + lf ′ραρλRλβαδ¯.Finally, the scalar curvature R˜ is given by(3.5) R˜ = 2g˜βδ¯R˜βδ¯=1fR− 2(n+ 1)(ff′′ − f ′2)f3l−2f(f + lf′)(ff ′′ − 2f ′2)′ − 2(ff ′′ − 2f ′2)(3ff ′ + lff ′′ + lf ′2)f3(f + lf ′)2l2−2(ff′′ − 2f ′2)f2(f + lf ′)l − 2n(n+ 1)f′f2−2n(ff′′ − 2f ′2)f2(f + lf ′)l − 2(ff′′ − 2f ′2)f(f + lf ′)2(ρδ¯ρ²¯∇²¯ρ¯δ + ραρβ∇βρα)− 2f′f(f + lf ′)gβδ¯gα²¯(∇βρα)∇²¯ρ¯δ+2f ′2f(f + lf ′)2gβδ¯(ρ²¯∇²¯ρ¯δ)ρα∇βρα −f ′f(f + lf ′)lR+2(n+ 1)f ′(ff ′′ − f ′2)f3(f + lf ′)l2 +2(n+ 1)f ′2f2(f + lf ′)l+2ff ′(f + lf ′)(ff ′′ − 2f ′2)′ − 2f ′(ff ′′ − 2f ′2)(3ff ′ + lff ′′ + lf ′2)f3(f + lf ′)3l3+2f ′(ff ′′ − 2f ′2)f(f + lf ′)3lρδ¯ρ²¯∇²¯ρ¯δ +4f ′(ff ′′ − 2f ′2)f2(f + lf ′)2l2+2f ′(ff ′′ − 2f ′2)f(f + lf ′)3lραρβ∇βρα − 2f′3f(f + lf ′)3(ρδ¯ρ²¯∇²¯ρ¯δ)ραρβ∇βρα+2f ′2f(f + lf ′)2gα²¯(ρβ∇βρα)ρδ¯∇²¯ρ¯δ −2f ′2f(f + lf ′)2ρλραρβρδ¯Rλβαδ¯.On a certain change of Ka¨hler metrics 514 A Ka¨hler manifold of constant holomorphic curvatureA Ka¨hler metric with non zero constant holomorphic curvature k definedin the complex number n-space Cn is called a Fubini metric. The potentialfunction is given byρ =2klog(1 +k4t),denoting t = Σzαz¯α. Then putting Θ = 1 +k4t, we havegαβ¯ =12Θ2(Θδαβ − k4 z¯αzβ).(4.1)First from (4.1), we havegαβ¯ =12ek2ρδαβ − k2ραρβ¯,andρα =∂ρ∂zα=z¯α2Θ,(4.2)from which we obtaingαβ¯ = 2Θ(δαβ +k4zαz¯β),(4.3)ρα = gα²¯ρ¯² = zαΘ,(4.4)ek2ρ = l′ = 1 +k2l > 0,(4.5)andl = ραρα =t2.Taking account of (2.4), we haveΓαβγ = −k2(δαγ ρβ + δαβργ).(4.6)From (4.1), (4.2) and (4.6), we have∇γρβ = k2ρβργ .(4.7)52 Tsuyoshi Yamazaki and Yoshiyuki WatanabeTaking account of (2.5), we haveRαβγδ¯ = −k2(δαγ gβδ¯ + δαβ gγδ¯).(4.8)Moreover taking account of (2.6), we haveRβδ¯ =(n+ 1)k2gβδ¯.(4.9)Finally, taking account of (4.9), we haveR = n(n+ 1)k.In the sequel, we consider the Ka¨hler metric g˜αβ¯, deforming gαβ¯ in (1.1).Then we compute the followingg˜βδ¯ =f2ek2ρδαβ +(f ′ − k2f)ραρβ¯,(4.10)Γ˜αβγ =(f ′f− k2)(ργδαβ + ρβδαγ ) +2(ff ′′ − 2f ′2 + k2ff ′)f(f + lf ′)ekρρα¯ρβργ ,(4.11)(4.12) R˜αβγδ¯ =(f ′f− k2)(gβδ¯δαγ + gγδ¯δαβ ) +ff ′′ − f ′2f2(δαγ ρβ ρ¯δ + δαβργ ρ¯δ)+Ff(f + lf ′)(ραργgβδ¯ + ραρβgγδ¯)+[1f2(f + lf ′)2{F ′f(f+lf ′)−F (3ff ′+lf ′2+lff ′′)}+k2F(f + lf ′)2]ραρβργ ρ¯δ,where we denote F byF = ff ′′ − 2f ′2 + k2ff ′.(4.13)Moreover, we obtainR˜βδ¯ ={(n+ 1)(k2− f′f)− Flf(f + lf ′)}gβδ¯−[(n+ 1)(ff ′′ − f ′2)f2+Ff(f + lf ′)+lf2(f + lf ′2){F ′f(f + lf ′)On a certain change of Ka¨hler metrics 53−F (3ff ′ + lf ′2 + lff ′′)}+k2 lF(f + lf ′)2]ρβ ρ¯δ,andR˜βδ¯ = µgβδ¯ + µ′ρβ ρ¯δ,(4.14)where µ is a function given byµ = (n+ 1)(k2− f′f)− Flf(f + lf ′).(4.15)Finally, the scalar curvature R˜ is given byR˜ =2nµf+2l(fµ′ − f ′µ)f(f + lf ′).(4.16)5 A Ka¨hler metric satisfying a certain conditionLet Dn be Cn or the ball Bn = {(zα) ∈ Cn | Σzαz¯α < r2} around theorigin O of Cn, whose radius is given by r =2√|k| (k 6= 0). Let (Dn, gαβ¯)be a Fubini space. In this section, let us consider the Ka¨hler metric g˜,satifying the condition (1.6). Then by the Ricci’s formula, we haveR˜hjR˜hikl + R˜ihR˜hjkl = 0,(5.1)from whichR˜αjR˜αikl + R˜α¯jR˜α¯ikl + R˜iαR˜αjkl + R˜iα¯R˜α¯jkl = 0.Thus (5.1) is equivalent toR˜αλ¯R˜αβγδ¯ + R˜βα¯R˜α¯λ¯γδ¯ = 0 (conj.),(5.2)by virtue of (4.12) and (4.14).Now substituting (4.8) and (4.10) into the left hand side of (5.2), we cansee that it reduces to the following.54 Tsuyoshi Yamazaki and Yoshiyuki Watanabe(5.2)′{µ′(f ′f− k2)+lµ′Ff(f + lf ′)+µFf(f + lf ′)− µ(ff′′ − f ′2)f2}(ρ¯λργgβδ¯−ρβ ρ¯δgγλ¯) = 0.Now we assume that n ≥ 2. Then we have{µ′(f ′f− k2)+lµ′Ff(f + lf ′)+µFf(f + lf ′)− µ(ff′′ − f ′2)f2}= 0,(5.3)taking account of f(ρ) ∈ C∞(R). (5.3) givesµ′f(f ′ − k2f)+lµ′fFf + lf ′− lµf′Ff + lf ′− µf ′(f ′ − k2f)= 0.Thus we obtain {f ′ − k2f +lFf + lf ′}(µ′f − µf ′) = 0.(5.4)In the followings, we consider two cases, that is, Case where f ′− k2f+lFf + lf ′= 0 and Case where µ′f − µf ′ = 0.Case : Assume that f ′ − k2f +lFf + lf ′= 0 in an open subdomain ∆n1of Dn. Taking account of (4.13),we havelf ′′f+f ′f− lf′2f2− k2= 0.(5.5)Putting g =f ′f(= (logf)′) in (5.5), we havelg′ + g − k2= 0.(5.6)Putting h = lg in (5.6), we haveh′ − k2h− k2= 0.(5.7)The general solution of (5.7) is given byh = cek2ρ − 1,On a certain change of Ka¨hler metrics 55where c is an integral constant. Therefore the general solution of (5.6) isgiven bylg = cek2ρ − 1,where c is an integral constant. Therefore the general solution of (5.5) isgiven byf = aebk2ρ(1− e− k2 ρk)b−1,(5.8)where a, b are integral constants.If b 6= 1, then f and f + lf ′ satisfyf(0) = 0, (f + lf ′)(0) = 0,because of ρ(0) = 0. Thus in this case the solution does not satisfy thecondition (1.2) where ∆n1 3 O.Next, suppose that b = 1. Then from (1.2), we see that a > 0 where∆n1 3 O, that is,(5.8)′ f = aek2ρ (a > 0),where ∆n1 3 O.Case : Suppose thatµ′f − µf ′ = 0(5.9)holds in a subdomain ∆n2 of Dn. First by (4.10) and (4.14) we see that thepotential function satisfying (5.9) gives an Einstein metric. By (4.15), wehave(5.10)Ff(f + lf ′){(n+ 2 +k2l)f2 + nlff ′}+l(f + lf ′)2{F ′f(f + lf ′)− F(3ff ′ + lf ′2 + lff ′′ +k2lff ′)}= 0.56 Tsuyoshi Yamazaki and Yoshiyuki WatanabeSettingσ =Ff(f + lf ′),(5.11)we havelfσ′ +{(n+ 2 +k2l)f + nlf ′}σ = 0.(5.12)If ∆n2 contains the origin O, putting l(0) = 0 in (5.12) we haveσ(0) = 0,(5.13)because of f(0) > 0.If l > 0, then multiplying (5.12) by ln+1fn−1, we haveln+2fnσ′ +(n+ 2 +k2l)ln+1fnσ + nln+2fn−1f ′σ = 0,which implies(ln+2fnσ)′ − (n+ 1)k2(ln+2fnσ) = 0.Thus the general solution is given byln+2fnσ = ce(n+1)k2ρ,(5.14)where c is an integral constant. Since the function f satisfies (5.12), by theargument of continuity of the left hand side of (5.14), we can conclude thatc = 0 if ∆n2 contains the origin O. Thus we haveσ(ρ) = 0,(5.15)together with (5.14) if ∆n2 3 O. Since(f ′ − k2ff2)′=Ff3, we havef ′ − k2f − Cf2 = 0 (C = constant)(5.16)together with (5.11). (5.16) has the general solutionf2 =1ce−k2ρ + d,(5.17)On a certain change of Ka¨hler metrics 57where constants c(> 0) and d satisfy the condition c+ d > 0, that is, whenO ∈ ∆n2 ,f2 =1ce−k2ρ + d(c > 0, c+ d > 0).(5.18)Note that in the case of (5.18) the corresponding Ka¨hler manifold (∆n2 , g˜)is of constant holomorphic curvaturedk2.Finally, the function f given by (5.8)′ does not satisfy (5.17) with d 6=0 and can be smoothly connected only the solution of (5.18) for d = 0.Conversely the solution of (5.18) for d 6= 0 or a 6= 1cdoes not satisfy (5.8)′.Thus we obtain the followingTheorem 2. Let Dn be Cn or the ball Bn = {(zα) ∈ Cn | Σzαz¯α < r2}around the origin of Cn, whose radius is given by r =2√|k| (k 6= 0). Let(Dn, gαβ¯) be a Fubini space. If the Ka¨hler metric g˜αβ¯, changed by (1.1)satisfies the condition (1.6), then it is of constant holomorphic curvatureor flat.6 Proof of the main theoremBy assumption R˜ is constant in (4.16). Multiplying (4.16) by f(f+ lf ′),we haveR˜2f(f + lf ′) = lµ′f + nfµ+ (n− 1)lf ′µ.(6.1)Putting l = 0 in (6.1), we haveR˜2f(0)− nµ(0) = 0,(6.2)because of f(0) > 0. For l > 0, we multiply (6.1) by ln−1fn−2. Then wehaveR˜2(fnln−1 + lnfn−1f ′) = lnfn−1µ′ + nln−1fn−1µ+ (n− 1)lnfn−2f ′µ.58 Tsuyoshi Yamazaki and Yoshiyuki WatanabeBy ( R˜2nlnfn)′=R˜2(fnln−1 + lnfn−1f ′) +nk2( R˜2nlnfn)and(lnfn−1µ)′ = lnfn−1µ′ + nln−1fn−1µ+ (n− 1)lnfn−2f ′µ+ nk2(lnfn−1µ),we obtain ( R˜2nlnfn − lnfn−1µ)′=nk2( R˜2nlnfn − lnfn−1µ),from whichR˜2nlnfn = lnfn−1µ+ Cenk2ρ,where C is an integral constant. Then we havelnfn−1( R˜2nf − µ)= Cenk2ρ.(6.3)But we can see that C = 0, taking limit of the left hand side of (6.3) of las l tends to 0. Therefore we haveR˜2nf2(f + lf ′) + (n+ 1)(f ′f− k2)f(f + lf ′)(6.4)+l(ff ′′ − 2f ′2 + k2ff ′)= 0.Putting l = 0 in (6.4), we havef ′(0)− k2f(0) +R˜2n(n+ 1)f(0)2 = 0.(6.5)Multiplying (6.4) by fn−2ln, we haveR˜2n(fn+1ln+ fnln+1f ′) + (n+1)fn−1lnf ′+ ln+1fn−1f ′′+ (n− 1)fn−2ln+1f ′2−(n+ 1)k2fnln − nk2fn−1ln+1f ′ = 0,On a certain change of Ka¨hler metrics 59from which{ln+1fn−1(f ′ − k2f +R˜2n(n+ 1)f2)}′=(n+ 1)k2ln+1fn−1(f ′ − k2f +R˜2n(n+ 1)f2),that is,ln+1fn−1(f ′ − k2f +R˜2n(n+ 1)f2)= C˜e(n+1)k2ρ,(6.6)where C˜ is an integral constant. But we haveC˜ = 0taking limit of the left hand side of (6.6) of l as l tends to 0. Thus we havef ′ − k2f + kf2 = f ′ − k2f +R˜2n(n+ 1)f2 = 0,(6.7)together with (6.5).Thus the general solution of (6.7) is given byf =1ae−k2ρ + b,where constants a(> 0) and b satisfy the condition a + b > 0. Now, ifb = 0, then the corresponding Ka¨hler metric is flat. Next, if b 6= 0, thecorresponding Ka¨hler metric is of constant holomorphic curvature and theholomorphic curvature isR˜2n(n+ 1)(cf.(4.12)) because of an elementarycalculation. Thus as Theorem 2, this proves the main theorem.References[1] S. Tachibana, On the characteristic function of spaces of constantholomorphic curvature, Colloq. Math., 26(1972), 149-155.[2] S. Tachibana and R.C. Liu, Notes on Ka¨hlerian metrics with vanishing60 Tsuyoshi Yamazaki and Yoshiyuki WatanabeBochner curvature tensor, Ko¯dai Math. Sem. Rep., 22(1970), 313-321.[3] Y. Watanabe, Ka¨hlerian metrics given by certain smooth potentialfunctions, Kodai Math J., 5(1982), 329-338.[4] Y. Watanabe, Unitary-symmetric Ka¨hlerian manifolds and pointedBlaschke manifolds, Tsukuba J. Math., 12(1988), 129-148.[5] Y. Watanabe and H. Mori, Geometry on unitary-symmetric Ka¨hlermanifold, Math. J. Toyama Univ., 16(1993), 135-193.Department of MathematicsFaculty of ScienceToyama UniversityToyama, 930-8555, Japan(Received May 7, 2003)

On a certain change of Kähler metrics

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On a certain change of Kähler metrics

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