η : 平行な接触Bocher曲率テンソルをもつ佐々木多様体
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(2) Journal of Hokkaido University of Education (Section II A) Vol. 29, No. 2 February 1979. ;)b?Sig^t^^ffiS (^2SRA) ® 29^ ^2^- Bg^D54^2fl. Sasakian Manifolds with ^-parallel Contact Bochner Curvature Tensor. Izumi HASEGAWA Mathematics Laboratory, Sapporo College, Hokkaido University of Education Sapporo 064. :^JI|^: ^-^T^^^Bochner?^r>y^$-^-^fe^^^^? -ft?®iI^t^^tLtft^?&^^g ^. Abstract For a Riemannian manifold, E. Glodek [ 2 ] proved the following THEOREM. A conformally symmetric Riemannian manifold of dimension w(S4) 25 conformally flat or its scalar curvature is constant. For a Kaehlerian manifold, M. Matsumoto and S. Tanno [ 5 ] proved an analogous theorem. In this paper, we obtain an analogous theorem for a Sasakian manifold. In a Sasakian manifold, if the contact Bochner curvature tensor is parallel, then the contact Bochner curvature tensor vanishes. Therefore we need the another notion for a Sasakian manifold.. § 1. Sasakian manifold In this section, we would like to recall the definition and the fundamental properties of a Sasakian manifold for later use.. Let M be a (2n+l)-dimensional differentiable manifold covered by a system of coordinate neighborhoods {U\ x'} in which there are given a tensor field Vi of type (1,1), a vector field ^' and a 1-form -7, satisfying (1. D vi<p,a=-8f+7/^, f^a=0, r/a<pf=0 and Va^a=l, where the indices a, b, •••, h, i,j, ••• run over the range {1, 2, •••, 2n+l}.. Then M is said to have an almost contact structure (<p, ^, v) and is called an almost contact manifold.. If the Nijenhuis tensor (1. 2) Nu'l=p.a9apj'-fj'9a<p.k-(9i<p?-9jpnpS formed with ip,' satisfies. (1. 3) N,j'+0^-9^.)^=Q, where 9{=9/9x',then the almost contact structure is said to be normal and M is called a normal almost contact manifold.. (1).
(3) I. HASEGAWA. Suppose that in an almost contact manifold M there is given a Riemannian metric gu satis-. fying (1.4) gab<Pa<f>^=gu-rnrji and ^i=g'a7]a.. Then the almost contact structure is said to be metric and M is called an almost contact metric manifold. In view of the second equation of (1. 4), we shall write 9' instead of i£I in the sequel. Henceforth the covariant and contravariant tensors are identified in the canonical manner. In an almost contact metric manifold, the tendor field cpu = <p?gaj is skew-symmetric. If an almost contact metric structure-satisfies. (1. 5) ?'»-(1/2)(^,-^,), then the almost contact metric structure is called a contact metric structure. A manifold with a normal contact metric structure is called a Sasakian manifold. In the following, let Rhijn, Rij, R and Fi denote the Riemannian curvature tensor, the Ricci tensor, the scalar curvature and the operator of covariant differentiation with respect to 9u. respectively. It is well known that in a Sasakian manifold of dimension 2n+l we have the following identities: (1. 6) ViHl=Vi, (1. 7) Vi<P^= -guri"+8f7],, (1. 8) r]aRa,jk=rikgij-rijgik,. (1. 9) riaRai=2nr/i, (1. 10) Rh if Vak—R hi& Vaj = VuQhh — fihQ hj + gijfhk —gih fhj, (1. 11) <pal'Raw=Rafaj-(.7.n-\)<pu,. (1. 12) <pe"'RaM=-2R.a<pu+2(2n-l)p,j,. (1. 13) R?<PaJ+R?Vai=Q. For a Sasakian manifold, it seems that the usual parallelism is not essential. For example a locally symmetric Sasakian manifold is of constant curvature 1. Furthermore a conformally sym-. metric Sasakian manifold of dimension 2n+l (1^5) is of constant curvature 1 since this manifold is conformally flat or locally symmetric [6] . DEFINITION. If a tensor r,i...,p""'J' of a Sasakian manifold At satisfies (1. 14) Vial-Vi?'''P^-V^''<pfVcTa,:.a,l'r"l"'=Q,. then we say that the tensor T,,...,,-"'"" of M is ^-parallel. If the Riemannian curvature tensor of a Sasakian manifold M is ^-parallel, then we say that M is locally ip-symmetric.. (1. 6), (1. 8), (1. 10) and the second Bianchi identity (1. 15) 7iRhijK+ 7hRiui,JT 7.Rwk= 0 give (1. 16) ri"7aRhUk= O.. From this fact and simple computations, we have the following LEMMA 1. A Sasakian manifold Mis locally (p-symmetric if and only if the Riemannian curvature tensor of M satisfies VlRhtjh = — <P?( rihRaijk-\- rjiRhaJk+ fjjRhiah-\- VkRhiJa). (L 17) +7/h(<pikgij-pugn.)+'ii(fisghk-vikgi,j) + 9j( VliQhk— Vlhgik) + rjk(. vihgii — <pngv).. (2).
(4) ^-parallel Contact Bochner Curvature Tensor. § 2. Contact Bochner curvature tensor. Let M be a Sasakian manifold of dimension 2%+1. As an analogue of the Bochner curvature tensor of a Kaehlerian manifold, we define the contact Bochner curvature tensor Bi,,ji, of M. by Bhijk = RhUk— 1)1 J~_s_r)\ (Rij-ghk — Rikghj+ gijRhk — gikRhj— RijVhVJk + RtkVhVj — V,7/jRhk+ r/i!/kRu+Hu<p/,k— Hii,<phj+ VijHhk. — <pikHhj—2Hhi<pjh—2<f>h{Hjk). + 4(n+l)^2) ^^-^"^ 4Rn++^tn2) ^^-V^-^^} R+2n. ,. _. ._. _. _....,.. s. I,. where Hij=Ria<f>aj.. We can easily verify that the contact Bochner curvature tensor satisfies the following identitles: (2. 2) Bhi]k=—Bihjk=—BhikJ=Bjkhi, Baija = 0 ,. (2. 3) Bhijk+Bhjkl+Bhl!iJ= 0 , (2.4) BwaVak=Bhtka<Pa,, <PC"'BaM=<Pat'BaUI,=Q,. (2. 5) vaBa»k= O . From (1. 16) and simple computations, we heve (2.6) va7aB^=0.. From this fact and simple computations, we have the following LEMMA 2. The contact Bochner curvature tensor Bhijk is T]- parallel if and only if it. satisfies (2. 7) ViBhuh=-<P?(v]hBaijk+r)iBhajkJt-rijBhiak+r]kBhua}. We can easily verify that ^-parallel contact Bochner curvature tensor satisfies (2.8) F,CB^5a6cd)=0. The ^-parallel contact Bochner curvature tensor satisfies F "Baijk = 0 , whence we have the. following LEMMA 3 . In a Sasakian manifold with ^-parallel contact Bochner curvature tensor, there. exists the following identity:. (2. 9) r^y-F^,*=^^y{(F^)(^,,-9,^,)-(F,7?)(^-9,^) + ( 7 aR )( fakfij— Pajfik- 2<Pai<P,k)} -(rjkHu-rijHik-2i]iH,k)Jr'in{rik<pij-rij<pik-2,r]i(pjk).. If a Sasakian manifold M is locally p-symmetric, then the contatact Bochner curvature tensor. of M is naturally ^-parallel. If the contact Bochner curvature tensor of a Sasakian manifold M is parallel, then M has the vanishing contact Bochner curvature tensor.. (3).
(5) I. HASEGAWA. § 3. Theorems THEOREM 1. Let M be a Sasakian manifold with rj-parallel contact Bochner curvature tensor. Then M is a manifold with vanishing contact Bochner curvature tensor or its scalar curvature is constant.. PROOF. ' (2; 7) and the Ricci identity give RmlhaBauk+RmltaBhajk+RmlJaBhiak+RmlkaBhija. =r/t(7/hBmljk+ ViB hmjk+7/jBhimk+VliBhUm) (3. 1) -Vm(VhBnjk+r/iB hi,k+ r/jB huk+ rfkBhui) -\-<P?'{<PmhBa{Jh-\-VmiBhaJk-\-<PmjBhiak-\-<PmkBhija) — fma(<PthBat.ik+ <PuBhajk+ fljBhiak+ VlkBhija).. Operating p"»to (3. 1) and using (2. 7), we obtain ( F''7?6»a)5ay*+( VBkUa)Bha^+{ f7l'Rl,Ua)BHiak+( VbR^a)Bwa (3. 2) =(Hia-2n<f>ia)(r/HBauk+r!iBwk+^BHiak+VkBwa).. Contracting (3. -2) with glh and substituting (2. 9), we have (3.3) (Fs^?)5au*=0. On the other hand, substituting (2. 9) into (3. 2) and using (3. 3), we obtain. ( ^^R)Bwk+( 7iR)Bwk+( ^R)BHuk+( V^BMH (3. 4) =S(n+l)))t(HKBaw+H?BHajk+H?Bh.ak+HSBwa) —( 7 R)vf(<Pl,hBaijh-}-<PbtBhaJk-{-VwBhiak-\- VbkBhija).. Transvecting (3. 4) with F hR and using (3. 3), we have (3. 5) (VaR)(VaR)BHw=9,{n+\)(VI'R)r)hHSBaw. Transvecting this with r)h, we have. (3.6) (F67?W5aM=0. From (3. 5) and (3. 6), we have (3. 7) (FaR)(^aR)BHi,k=0. From (2. 8) and (3. 7), we have Bhuk= 0 or 7iR= 0 . If ViR= 0 holds, then the scalar curvature is. constant.. Q.E.D.. At the meeting of Mathematical Society of Japan, which was held in the spring of 1978, T. Ikawa announced the following LEMMA 4. If a Sasakian manifold-M with ^-parallel contact Bochner curvature tensor has the constant scalar curvature, then M is locally <p-symmetric.. From THEOREM 1 and LEMMA 4, we obtain the following THEOREM 2. If a Sasakian manifold M has the T) -parallel contact Bochner curvatnre tensor, then M is a manifold with vanishing contact Bochner curvature tensor or M is locally <p-symmetric.. (4).
(6) ly-parallel Contact Bochner Curvature Tensor. References. '•. [ 1] Bochner, S. (1949) : Curvature and Betti numbers II. Ann. of Math., Vol. 50, 77-93. j [ 2 ] Glodek, H. (1971) : Some remarks on conformally symmetric Riemannian spaces. Colloq. Math., Vol. 23, ! 121-123. [ 3 ] Matsumoto, M. (1969) : On Kahlerian spaces with parallel or vanishing Bochner curvature tensor. Tensor, N.. S.,. Vol.. 20,. 25-28.. !. [ 4 ] Matsumoto, M. and Chuman, G. (1969) : On the C-Bochner curvature tensor. TRU Math. Vol. 5, 21-30. i [ 5 ] Matsumoto, M. and Tanno, S. (1973) : Kahlerian spaces with parallel or vanishing Bochner curvature tensor. Tensor, N. S., Vol. 27, 291-294. [ 6 ] Miyazawa, T. (1978) : Some theorems on conformally symmetric spaces. Tensor, N. S., Vol. 32, 24—26. I [ 7 ] Tachibana, S. (1967) : On the Bochner curvature tensor. Nat. Sci. Rep. of Ochanomizu Univ., Vol. 18,. 15-19.. (5). j.
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