佐々木幾何についての注意 : Myersの定理と標準擬似接続の応用

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(1)Title. 佐々木幾何についての注意 : Myersの定理と標準擬似接続の応用. Author(s). 長谷川, 和泉; 清野, 傑. Citation. 北海道教育大学紀要. 第二部. A, 数学・物理学・化学・工学編, 32(1) : 1-7. Issue Date. 1981-09. URL. http://s-ir.sap.hokkyodai.ac.jp/dspace/handle/123456789/6068. Rights. Hokkaido University of Education.

(2) Journal of Hokkaido University of Education (Section II A) Vol. 32, No. 1 September, 1981. *!«ittSE»*â® ? 2 SRA) ^ 32 S ^ 1 -f "Sffi 56 ^ 9 fl. Some Remarks on Sasakian Geometry Applications of Myers' Theorem and the Canonical Affine Connection. Izumi HASEGAWA and Masaru SEINO* Mathematics Laboratory, Sapporo College, Hokkaido University of Education, Sapporo 064 'Department of Mathematics, Hokkaido University. :^jiiâ • : ^^-fpr^-^-c^^ - Myers y)^M b m-WmW<?~)&» •. WS.Sili^^WfSi^^ *-ib?f»ia^'^»^^s. Abstract The main purpose of this paper is to apply Myers' theorem to the Sasakian manifold. The second purpose is to apply the canonical affine connection and to argue an analogy between Kaehlerian geometry and Sasakian geometry.. Statement of results.. The following result is well-known : THEOREM A (S. B. Myers[ 3 ]). In a complete Riemannian manifold M, if the infimum of eigenvalues of Ricci tensor is positive, then M is compact. One of the purposes of the present paper is, by using Theorem A, to show THEOREM 1. In a complete Sasakian manifold M, if the infimum of eigenvalues of the Ricci tensor is greater than — 2, then M is compact.. On the other hand, one of the present authors has proved the following THEOREM B ([5]). If a compact Sasakian manifold M with vanishing contact Bochner ciiruature tensor is p.-holomorphically pinched with p. > 0, then the second Betti number of M vanishes.. We show the following THEOREM 2. If a rf(^5)—dimensional Sasakian manifold with vanishing contact Bochner curvature tensor is /.t-holomorphically pinched with p. > 0, then the infimum of eigenvalues of the Ricci tensor is greater than —2. Hence, combining Theorem 1, Theorem B and Theorem 2, we have the following. (D.

(3) Izumi HASEGAWA and Masaru SEINO. THEOREM 3. If a d(^_ 5}-dimensional complete Sasakian manifold M with vanishing contact Bochner curvature tensor is fi-holomorphically pinched with // > 0 , then M is compact and the second Betti number of M vanishes. The Second purpose of this paper is to apply the canonical affine connection ([ 6 ]) and to argue an analogy between Kaehlerian geometry and Sasakian geometry. Let M, 7 and Ri= (Rkjih) be a Sasakian manifold, the canonical affine connection and the curvature tensor of F on M, respectively. We can demonstrate the following theorems.. THEOREM 4. In a compact Sasakian manifold M, if the relation |7<""7?i=0. holds, then M is locally </>-symmetric, where 7{w denotes the m-th covariant differentiation for. v.. THEOREM 5 (cf. [ 4 ]). In a compact Sasakian manifold M with constant scalar curvature, if the relation F mVl RkJi"-VtVn,R,.J,"-=LQ. holds, then M is locally cfi-symmetric.. § 1. Preliminaries.. ( i ) Let M be a rf-dimensional Riemannian manifold with a positive definite Riemannian metric gjr Moreover, let {/,}, Vs, Ki,ji", Kji and K be the Christoffel symbols formed with gj,, the operator of covariant differentiation induced from {/'j} , the Riemannian curvature tensor, the Ricci tensor and the scalar curvature, respectively.. M is called Sasakian if there exists a unit Killing vector field 77' satisfying (1.1) V h V j rj ,= rijgm- rj ,gi,j.. If we put (f>,'':= 7; nh then {(f>ih, rj', gjs) gives an almost contact metric structure to M and hence. M with (<f>ih, r]t, gji) is orientable and odd dimension d=2n+l. (ii) We define two values H and L by H=sup {p(X,rf,X) ; XeDp. PeM}, L=inf {p(X.4>X) ; XeDp, PeM}. where p denotes the sectional curvature given by. (1.2) p(X, Y)=KW,.X" Y'Y'X", for any orthonormal vectors X and Y and D is the 2n-dimensional distribution orthogonal to ril. Let us assume that they exist and H> —3. In this case M is said to be //-holomorphically. pinched, fi being (L+3)/(ff+3). (iii) A D-homothetic deformation of the metric ([7] , [ 2 ]) : gji—"*gji is defined by (1.3) *gj.= agj,+ a(a - 1)%-?7,,. (2).

(4) Some Remarks on Sasakian Geometry. for a positive constant a. We have (1.4) *^,,-X,,- 2(ff - !}&,+2(a - l)(na + n+1)%^,, *Kj, being the Ricci tensor of *gj,. If (</>,'', fj', &•;) is Sasakian, then (*<f>ih, *r)t, *gj,) is Sasakian, where we put *^h= </>,", *rii=a-î.. As there exist the relations a^X3X'^ *&,XJX'^ a2g,,X}Xi (a > 1),. (1.5) a2g,.XJXi^g,.XJX{âg,,XJXi (a<l), tg^XJX{=g,,XjX- (ff-1),. for any vector X' in M, M is complete (resp. compact) with respect to *gj, if and only if M is complete (resp. compact) with respect to gji.. (iv) An affine connection 7 is called a canonical affine connection if the coefficients yjth of 7 are given by (1.6) y,," = {/;} - ^hri, - 4>ih-ns + ^.Tf". The canonical affine connection |7 satisfies (1.7) ^gjiÔ, F^,=0, F^,A=0 Let Ri,ji , Rji and R be the curvature tensor, the Ricci tensor and the scalar curvature of V , respectively. Then we have ^kjih— —^Jkih— —l\-hjhi— îhhj,. Rknh+Rjikh-\-Rikjh=Q (the first Bianchi identity),. (1.8) Rws<^<f>HS=Rwh, rihRwh=Q, f!JRj,^0,. f7iRi,j,h+ (7kRjtih+ ^Rikih=OWê second Bianchi identity),. r)1 V i RkHh=Q, T} " 7 ^,,=0, n- V , R=0.. M is said to be locally ^-symmetric if (7 iRnjih=0. (v) The contact Bochner curvature tensor Bi.jsh is given by (1.9) Buii,=Ru,h+gi,hLji-gjhLi.i+Li,i£ji-Ljhgki+</>khMj{. - 4>shMki + M^<f>j, - Mjhî - 2 <f>^M,,, - 2Mu<f>.h,. (3).

(5) Izumi KASEGAWA and Masaru SEINO. where L,,=- ^^ (R,,- ^^ &,.), M,..=-L,^,(=- ^^ (S,,- ^^-^,), S;,= - Rjt<f>i' and &•,•=&>•- w,- S,-, is closed in the sense that. (1.10) K,.Sji+ F,S,/,+ F;S/,,=0. And, if R is constant, then we have p" Sji=0 and. (1.11) (A5),,.=(?+<?rf)S),,=0, where (d T) „,..„= (7 , T „..,,+ F „ T,,..,» +•••+ (7 ,,T ,,,..,,_, (1.12) (5 D,,.,, = FT „,...,,. for any r-form T ,,...,y.. As there is the relation (1.13) R=K+2n, R is a constant if and only if K is a constant.. § 2 . Proofs of Theorem 1 and Theorem 2.. ( i ) PROOF of THEOREM 1. Let 6 be the infimum of eigenvalues of the Ricci tensor Kj,, then 0+2>0 by our assumption. If 8>0, then, by Theorem A, M is compact. Hence we may. assume that 0^0>-2, that is, 0<((9+2)/2^1. First of all, we consider a quadratic function f ft) with the variable t given by (2.1) fft)=2nt2-{-2t-{9+2). Since the inequalities. (2.2) f(0)= -(0+2)<0, /(ff—2)=^+2)2>0 hold, f{t)=Q has two real solutions ^ and fc (A<0<fe<"-7T'^l). Therefore, there exists a positive number a such that. (2.3) fc^ff<^^l, and (2.4) f(a)=2na'2+2a-(0+2)^Q. For such a we consider the D-homothetic deformation (1.3). If X' 'is a unit vector with respect to gji, then we decompose it as follows, (2.5) Xi=aXD+b}f',gjiXiXD=l, rn X'D=O, a2+b2=l.. Then we have. (4).

(6) Some Remarks on Sasakian Geometry. (2.6) * Kj,XJX' = a2 Kji XiXo+ b2*K,, r) j r, •. ^(l-b2)(e+2-2a)+2nb2a2 =0+2-2a+(2na2+2a-0-2)b2 ^0+2-2a!. that is, (2.7) 'fKji*XJ*X'> a-l((G+2)-2a)/(l+(a-l)bs) >a-l((0+2)-2a) >0, where *X'=X''/(a+a(a—l}b2)112 is a unit vector with respect to *gji. Therefore, if M is complete with respect to gj,, then M is complete with respect to *gj,, whence, by Theorem A, M is compact with respect to *gj, and gj{. (ii) PROOF of THEOREM 2. An orthonormal basis {e,}={eo=^, e<», eo* :=êo,} (a,ft= 1,2,' ••,n) is called a <^-basis. Then we have. LEMMA 1([5]). Let M be a d(>S)-dimensional Sasakian manifold. If M has vanishing contact Bochner curvature tensor, then we have p(ea,ea*)+p(eft,ept)=Sp(e«,ep)-6,(ot^/3), for every ^-basis {e,}.. If we take a <^-basis {e,} consisting of the eigenvectors of the Ricci tensor Ks=(Kji), then, by Lemma 1, we have. p{&a, en):=P(ea, 0ft*)=(p(ea,et,*)+p(ep,en*)+6)/8^(L+3)/^>0, (a 1=0). It follows that K2(ea,0a)=K2(ea*,ef*)= ^ (p(ea, eti)+p(ea, e/ ))+p(ea, ef )+p(ea, eo) Let X be a unit vector and decompose it as follows, X=ae«+^ (aaea+aate^), fl2+S ((a")2+(aa*)2)=l. a.. a. Then we have K^X,X)^2nal+{-^—(L+3)-2)(l-a2). Here we consider two cases :(!)£<! and (2) L ^ 1. Case (1) : If L<1, then we have. K^X,X)^^±il-L)a2+"y-(L+3)-2. >n^W)-2. (5). ^(%+1)(£+3)-2..

(7) Izumi HASEGAWA and Masaru SEINO. Case (2) : If L^l, then we have K^(X,X)>2n, which completes our proof.. § 3 . Proofs of Theorems 4 and 5.. ( i) The proof of Theorem 4 is clear by the following Lemmas 2 and 3. LEMMA 2 (Green's theorem). In a compact Sasakian manifold, we have (3. 1) /, V,vida=Q, 'M. for any vector field v', where da is volume element.. PROOF. Lemma 2 follows from the relation (3.2) ViV'=ViV'. LEMMA 3. In a compact Sasakian manifold, if a tensor field T „...;,- satisfies (3.3) FT,r.,..,,=0,. then we have (3. 4) F,r,,...^=0.. PROOF. By the usual calculation, we have (3.5) yF'F,(T,,..,,r"'-'r)=(F,r<,...,,)(i7!r"-"-),. which, together with Lemma 2, shows Lemma 3.. (ii) Since the proof of Theorem 5 is analogous to that in a Kaehlerian case of [ 4 ], we only give the outline. LEMMA 4. In a compact Sasakian manifold M, we have + f Vi R,.}U, V IXWH da = f..(( F ,,7?,,- F,7? *,)( F "^JI- V3Rhi) M. ~. •'. M. -(F"F;7?,,,,,,-F,F"^,»,)7? "jil}d a.. LEMMA5. In a Sasakian manifold we have ( A S)j,= V " V ,.Sj,+2RjiSit- Rj.i.hS"",. (3.7) ijtlth'J — —^^kjih^l ,. R,.^SI"'</,.t=R,^R'1".. LEMMA 6. In a Sasakian manifold, if the relation (3-8) V,.VjRi,,-Vj7hRih=0. (6).

(8) Some Remarks on Sasakian Geometry. holds, then we have (3. 9) R,tRit=Ri.wRhh.. In a compact Sasakian manifold with constant scalar curvature, if the relation (3. 10) ?•„, 7iRw- Vi Vm Rhnh=Q. holds, then, by contracting with k and h, we have (3. 11) 7 mViRn-ViV ,nRj,=0, and, by Lemma 6, (3. 12) RjtR,t=R^Rhh. On the other hand, by constancy of the scalar curvature, we have. (3. 13) (AS),, =Q, that is, (3. 14) V " V /, S,, = - 27?,< R,t+ R,;,,, S"",. and thus (3.15) F/'F*7?,,=0,. which together with Lemma 3 gives us (3-16) V *7?,,=0. Therefore, by Lemma 4, we see that the manifold is locally <^-symmetric.. References. [ 1 ] A. LICHNEROWICZ : Geometrie des groupes de transformations, Dunod, Paris. 1958. [ 2 ] M. MATSUMOTO & G. CHOMAN : On the C-Bochner curvature tensor, TRU Math. 5 (1969), 21-30. [ 3 ] S. B. MYERS : Riemannian manifolds with positive mean curvature, Duke Math. J. 8 (1941), 401—404. [ 4 ] Y. OGAWA : A condition for a compact Kaehlerian space to be locally symmetric, Natur. Sci. Rep.. Ochanomizu Univ. 28 (1977), 21-23. [ 5 ] M. SEINO : On vanishing contact Bochner curvature tensor, to appear in Hokkaido Math. J. [6] N. TANAKA : On non-degenerate real hypersurfaces, graded Lie algebras and Cartan connections,. Japan J, Math. 2 (1976), 131-190. [ 7 ] S. TANNO : The topology of contact Riemannian manifolds, Ilinois J. Math. 12 (1968), 700-717. Izumi HASEGAWA Masaru SEINO Mathematics Laboratory Department of Mathematics Sapporo College Hokkaido University Hokkaido University of Education. ;7).

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