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(2) (^ 2 SPA) m 1 ^- [)3ft]61^10^. Journal of Hokkaido University of Education (Section II A) Vol. 37, No. 1 October, 1986. On p-th Sasakian Manifolds. Izumi HASEGAWA, Yukihiko OKUYAMA* and Tomoji ABE** Mathematics Laboratory, Sapporo College, Hokkaido University of Education, Sapporo 064. ''' Higashi-Tsukisamu Junior High School, Sapporo. ** Sapporo-Kosei Senior High School, Sapporo.. pW&^^^i^-^^r :^U % ^-A Lij ^ ^*-N §P ^ r:** ^^M^^^^tL^^-^Kl^^i * +Lt8TtTA^^^4'^^ **. Abstract In this paper, we should like to investigate the p-tb Sasakian manifolds. In § 1, we give the definition of the p-th, Sasakian manifold. For the special case p= 0, a p-th Sasakian manifold is a. Kaehlerian manifold, and a 1-st Sasakian manifold is the usual Sasakian manifold itself. We are therefore able to discuss these manifolds in the same category. In § 2, we shall examine the Nijenhuis torsion tensor of almost />-contact manifolds and give the fundamental properties of p-th Sasakian manifolds. § 3 is devoted to the study of the 0-sectional curvature of p-th Sasakian manifolds and p-th Sasakian space forms. In § 4, we define the ^-contact Bochner curvature tensor. § 5 is devoted to the study of the hypersurfaces in p-th Sasakian manifolds. In § 6, we give some typical examples of p-th Sasakian space forms.. (1.
(3) 2 Izumi HASEGAWA, Yukihiko OKUYAMA and Tomoji ABE § 1. p-th Sasakian manifolds.. Let M be a (2n+p)-dimensional differentiable manifold of class C°. If there exist 2p+l tensor fields (*, ^i, ••••••, &, '7i, ••••••> ??p on M with types (1,1), (1,0), ... , (1,0), (0,1), ... , (0,1). respectively which satisfy the following equations T]a(^}=S-al, (a, 6=1, ••••••, P), (1.1) </>^a=0 (a=l, ••••••, P),. <f>2X=-X+I]ria(X)Sa a=l. for tangent vector fields X and Y on M, then A/ is said to admit an a/mosf^-cowtocf sfrwctore {<!>, fn ••••••, ^p, ?7i, ••••••, 7)p) and M is called an almost p-contact manifold. In this case, the. relations (1.2) r}a(^>X)=0 (ffl=l, •••••:, P) and (1.3) rank(A=2n. hold good. We consider a product manifold MXR", where R" denotes a ^-dimensional Euclidean p. space. A vector field on M X J%" is then given by X+'Zjfa3a, where X is a vector field tangent a=i. to M, 3a:=9/3ia the natural frame fields of R" and /a(a==l, ••••••> p) functions on MX R • We define a linear map J on the tangent space of MXRP by. (1.4) J(X+ E/a3a)=^- E/^a+ S r]a(X)3a. a=l~. a=I. a-I. Then we have J2=—l and hence J is an almost complex structure on MXR". The almost. complex structure J is said to be integrable if its Nijenhuis torsion Nj vanishes. The Nijenhuis torsion NF of a tensor field F of type (1,1) is a tensor field of type (1,2) given by Nr(X, Y):=F2[X, Y]+[FX, FY]-F[FX, Y]-F[X, FY]. If the almost complex structure J on MX R" is integrable, we say that the almost ^-contact structure (<t>, ?i, ••••••, ^p, ?7i, ••••••, 7?,,)' is normal (cf. [9]).. Suppose that in an almost ^-contact manifold M there exists a Riemannian metric 3= < , >. satisfying (1.5) <X, ^a>=ria(X) (a=l, ••••-, P), <<AX, Y>+<X, <pY>=0. Then the almost ^-contact structure is said to be metric and M is called an almost p-contact Riemannian manifold. If an almost ^(•Sl)-contact metric structure (^, ^i, ••••••, ^p, T?), ••••••, ??p, < , » on. M satisfies (1.6) 2Ca<^>X, Y>=dr;a(X, Y) (a=l, ••••••, p) for some nonzero constant Cn (a=l, •....•, p), then the structure is called a homothetk p-contact metric structure. M with such a structure is called a homothetic p-contact Riemannian manifold. Especially if Ci="-'"=Cp=l in (1.6), then the structure is called a p-contact metric structure and M with such a structure is called a p-contact Riemannian manifold. If an almost 0-contact metric structure, i.e., almost Hermitian structure, (</>, < , » on M satisfies d<&=?0 where 9{X, Y)==<<f>X, Y>, then the structure is called a 0-contact metric structure, i.e., an almost Kaehlerian structure..
(4) On p-tb Sasakian Manifolds 3 If the structure vector fields ?i, ••••••, ?p of ^-contact Riemannian manifold (resp., homothetie ^-contact Riemannian manifold) M are the Killing vector fields, that is, (1.7) Lfaff=0 (a=l, ....... p). hold good where Lta denotes the operator of Lie differentiation with respect to ^a, then the structure is said to be a K-p-contact metric structure (rosy., homothetic K-p-contact metric structure) and M is called a K-p-contact Riemannian manifold (resp., homothetic K-p-contact Riemannian. manifold). If a K-^-contact metric structure (resp., homothetic K-^-contact metric structure) on M is normal, then the structure is said to be a p-th Sasakian structure (resp., homothetic p-th Sasakian. structure) and M is called a ^>-? Sasakian manifold (resp., homothetic p-th Sasakian manifold),. § 2. Fundamental properties of p-th Sasakian manifolds.. In this section, we first seek to express the condition of normality in terms of the Nijenhuis torsion tensor N^, of </> on an almost ^-contact manifold M(cf. [1, 9, 15]). Since Nj is a tensor. field of type (1,2) on MXR" it suffices to compute NAX, Y), NAX, 3J and iW«, 3,) for vector fields X and Y on M. From (1.4); we obtain. NAX, Y)=Nt(X, Y)+^dT]a(X, Y)^+^((Li,.^a)Y-(Li,v7]a}X)3a, a=l. a=l. (2.1) NAX, 3^=(L^)X+^((L^^X)3,, 6=1. NA9a, 3b)=[^, ^]. Here we define the tensors TV", N{9a, Niva and N{wa,i, (a, 6=1, ••••••, p) respectively by. N"{X. Y):^Nf(X, Y)+Sdrja(X, Y)^. a=l. ?\(X, Y) : =(L^ria)Y-(L^ria)X, and. ?\(X):^(L^)X ?\,(X):=(L^)X.. It is clear that the almost complex structure J is integrable, that is, M is normal, if and only if these tensors and [^a, Sb] vanish. LEMMA 2.1. If Niv=0, thenN12)<,=A?13)a=A?l\t=[^,^]=0. PROOF. If N(ll==0, then we have. (2.2) [?<,, X]+f4[^, ^]-E(?a(^(X)))^=0. C-l. Applying li, to this equation, we have. n^a, x])-U^(x))=o which shows that N{va.b=0. From this equation we also have. (2.3) ri,([Sa, (AX]) =0. Applying <f> to (2.2) and using (2.3), we see that </>(L^X)-L^('f>X)=0 from which Ni31a=0, Furthermore, from N'"=0, we obtain.
(5) 4 Izumi HASEGAWA, Yukihiko OKUYAMA and Tomoji ABE. 0=N^(</>X, Y)+Sd7?o(0X, Y)^. =-[<6J, Y]-[X, (47]+S i^0)[^, ^y]-(^Y(^W))^t "-• _p_. -</>[</>1X, Y]-^>[<f>X, </>¥]+ r,(</>X(r),{Y)))^. b-1. Applying ria to this equation and using (2.3), we get <l>X{r)a{Y})-7,a(['f>X,, Y])-^,Y(r]o(X})+T]a([<f>X, Y])=0. Thus we have JV21a=0 (a=l, ••••••, p). Finally, from (2.2) we easily obtain [^a, Si>]=0 (a, b =1,. ••••••,. p).. .. .. ,. Q.E.D.. In view of Lemma 2,1 we have PROPOSITION 2.1. The almost p-contact manifold M is normal if and only if Ntv=0. We next consider the case of a homothetic jd-contact metric structure, LEMMA 2.2. Let M be a homothetic p-contact Riemannian manifold with the structure (<f>, ^i, ••••••, &>• 7?I, ••••••, ??fl, < , >). Then Nwa and A?14)a,;, (a, 6=1, ••••••, p) vanish. Moreover. Nt9a and [^a, ^>] (a, 6=1, ••••••, p) vanish if and only if ?a(a=l, ••••••, p)are Killing vector. fields with respect to < , >. PROOF. We have dria(<f>X, Y)+dria{X, </>Y)=2Ca<</>lX, Y>4-:2Ca<0Z, (;iy>=0, This is equivalent to Nt2}a'=Q. On the other hand, we have 0=2Ca<^a, X>. =dr]^a, X} =U^{X))-ri^a, X]). Therefore we have N141a,t>=0 (a, b=l, ••••••, p).. Next, from (2.4) we see that. (L^< , »(X, ^)=Sa<X, ^>-<[^, X], ^>-<X, [Sa, ^]> ==<X, [?<,, ^]>. Furthermore, from (2.4) we have. Q=X(tL^}(Y))-Y{(L^)(X))-(L^^)([X, Y]) =(Lfad7?,)(Z, Y). =^(d^(X. Y))-d^([^, X], Y)-d^(Z, [^a, Y]) =-2c,t(Lfc< , »(X, <f>Y)+<X, N{\{Y)>\.. Thus ^a(a=l, ••••••, p) are the Killing vector fields if and only if N^a=0 and [^, ^]=0 (a, 6 =1,. ••••••,. p).. Q.E.D.. PROPOSITION 2.2. Let (<f>, Si, ••••••, Sp, ri\, ••••••, rjp, < , » be an almost p-contact metric structure on M. In order that (</>, ^, •••••-, (;p, 7?,, ••••••, rjp, < , » is a homothetic. K-p-contact metric structure it is necesshry and sufficient that (2,5) V^a=Ca<l>X (0=1, ••••••, p) for some. nonzero constant Cg.. PROOF. We have dT]a(X, Y)=<7.^a, Y>-<7r^, X>. and.
(6) On p-tb Sasakian Manifolds 5. (Lfa< , »(X, Y)=<y^a, Y>+<X, 7fcY>. Therefore the structure is homothetic K-^-contact if and only if (2.5) holds good. Q.E.D.. If we put <P(X, Y):=<^, y>, then we have the following property. LEMMA 2.3. For an almost p-contact metric structure (efi, {;„ ••••••, ^,, r)^ •••••; rjp,<, ». of M, we have. 2<(7^}Y, Z>=d<P(X, Y, Z}-d^(X, <pY, ^,Z)+<Nt"(Y, Z), <t>X> (2-6) +SN12)<,(y, Z)r]a(X}+^(df]a^Y, X)ria(Z)-dr,n(</>Z, X)rja(Y)). arl. a=l. PROOF. The Riemannian connection V with respect to <. , /> is given by. 2<7xY, Z>^X<Y, Z>+Y<X, Z>-Z<X, Y>-<X, [7, Z]>+<Y,[Z, X]> +<Z, [X, Y]>. On the other hand, d<Si is given by. d$(X, Y, Z)=XWY, Z))+YWZ, X))+ZWX, Y)) +f(X, [Y, Z])+$(Y, [Z, X])+$(Z, [X,Y]). ' From these equations and the definition of <? we have (2.6). Q.E.D. From Lemma 2.2 and Lemma 2.3, we have the following LEMMA 2.4. For a homothetic p-contact metric structure ((/>, ^,, ....... ^^ ^, ...... ^ ^p < , >). of M, we have. (2.7) 2<(17,^)Y, Z>=<Ar(»(Y, Z), ^>+E2c<.<(*2y, X>r]o{Z)-<</>lZ, X>r}a(Y)). a=i. Especially we have (2.8) 7ft,0=0 (a=l, ••••••, p).. From Lemma 2.4, we have the following LEMMA 2.5. For a homothetic p-th Sasakian structure of M,. we have. (2.9) (7.^)Y=-^Ca(7]a(Y)</,2X+<<f>X, <t>Y>^}. a=l. PROPOSITION 2.3. An almost p-contact metric structure (</>, ^i, ••••••, ^p, 771, ••••••, ?;p, < , ». on M is a homothetic p-th Sasakian structure if and only if (2.5) and (2.9) hold good. PROOF. Let M be a homothetic p-th Sasakian manifold. Then, from Proposition' 2.2 and Lemma 2.5, we have (2. 5) and (2. 9). Conversely, we suppose that (2. 5) and (2. 9) hold good on M. Then we have. <N<"(X, V), ^>=<^X, </>¥], ^a>+dria(X, Y) ^^ =<(r^)r, ^>-<.(7^^}x, So>+<Y, y,Sa>-<x, F,.^> ^-Ca<^X, </>Y>+Ca<</>2Y, 't>X>+Ca<<t>X, Y>-Ca<</>Y, X> =0.. Using Proposition 2.2 and Lemma 2.4, we have. (2.ii) <w<"(x, y), </>z>=o. Therefore, from (2. 10) and (2. 11) we have Niv=0, which shows that M is a homothetic p-th Sasakian. manifold.. Q.. E.. D.. PROPOSITION 2. 4. A homothtic p-contact metric structure (if>, ^, •••••., ^p, ?7i, ••••••, r}p,,< , ». on M is a homothetic p-th Sasakian structure if and only if (2. 9) and [^a, ^i>]=0 (a, b=l,•••••-, p). (5).
(7) 6 Izumi HASEGAWA, Yukihiko OKUYAMA and Tomoji ABE hold good. PROOF. Since the necessity is clear, we prove the sufficiency. From (2. 9) we have (2.12) <t>V^a=Ca't'tX (a=l, ••••••, p).. Since the structure is homothetic ^-contact metric, we have (2.13) <V^a, Y>-<X, [7,.^>=2Co<0Z, 7> (a=l, ••••••, p).. Therefore, from (2. 12) and (2. 13) we obtain (2.14) ^^a=Ca</>2X (0=1, ••••••, P).. On the other hand, from [^a, ^]=0, we have. ^=f7^a, from which we obtain (2.15) <Ffc^, ^c>=0 (a, b, c=l, ••••••, p).. From (2. 12), we have (2.16) <!7.<,^, X>=0 (a, b=l, ••••••, p).. From (2. 15) and (2. 16) we have (2.17) F^t,=0 (a, 6=1, ••••••, p).. Therefore, from (2. 14) and (2. 17) we have (2. 5). By virtue of Proposition 2.3, M is a homothetic p-th. Sasakian. manifold.. Q.. E.. D.. §3. p-th Sasakian space forms. Let M be a (2n+p)-dimensional ^>-th Sasakian manifold with structure (</>, ^i, ••••••, ^p, >7i, ••••••, rip, < , >). We have the following identities: (3.1) F,.?a=^X (a=l, •••-, p), (r,.^y=-(E??a(y))^^~<<*x, ^syXE^a), n=. l. a-1. (3.2) R(X,Y)^(^ri,(X))</>2Y-(^r],(Y))<f>lX (a=l, ••••••, p), b=l. '. '. 6-1. R(X,Y)<pZ=R{X, Y)Z-p<</>X, Z>(A2y+p<fiy, Z>^X (3^3). +\p<<f.X, </>Z>+(S7]a{X))(~Sri^ZW<f,Y ,^,»,-,, ,^. .,-.,-.,--.. -lp<</>Y, (*Z>+(S?7a(Y))(E»7o(Z))W a-l. !>-l. +}<</>X, z>(E^(Y))-<(*y, z>(E??aUO)KE^), a==l. a-1. 6==l. R{X,Y)Z=-<f>R(X,Y}<f>Z+\p<<f>X, </>Z>+(^r,a(X))CSrj,(Z})}</,SY a=l. ft-1. -\p<</>Y, ^Z>+(E^(Y))(E^(Z))1^ a»l. b-1. (3'4) -P<X, </>Z></>Y+p<Y, <j>Z><t>X. -\<<1>X, <t>Z>{T,TlaW)-<<t>Y, ^>(S^(X))1(S^) and. a=l. a==l. b=l. <R(<t>x, fiy)^z, </>W>=<R(X,Y)Z, w>+<</>x, 0z>(E??a(y))(E^W) a==i b=i -<4>x, (W>(S^(y))(l]^(z)) a-l. b=l.
(8) On p-th Sasakian Manifolds 7. +<{4y, fW>(E>7a(x))(a^(z)) a=l. 6-1. -<<!>¥, </>Z>(^ri,(X))(r,T!,(W)) a-l ' !>=l. where R denotes a Riemannian curvature tensor of M defined by R(X,Y)Z : =(WZ-[7,.|7<Z-|7|.<, >,Z.. A plane section in a tangent space Tx{M} {x^ M) is called a yi-section if there exists a unit vector X in Tx(M} orthogonal to ?a(a=l, ••••", p) such that \X, ^A'l is the orthonormal. basis of this plane section. The sectional curvature K(X, <fiX) : =<R(X, <f>X}<t>X, X> is then called a </>-sectional curvatwe ; this will be donated by H[X). We shall show that on a p-th Sasakian manifold the <t> -sectional curvatures determine the Riemannian curvature tensor completely.. We put P(X, Y, ;Z, W):=<X. WX</>Y, Z>-<X, ZX^SY, W> +<y, zx</>x, iy>-<y, w><if>x, z>. Then we have. P(X, Y;Z, W)=-P(Z, W;X, Y). If \X, Y\ is an orthonormal pair orthogonal to ^a(a=l> •"•••, p) and if we put <X, 'f>X>=.:cos 0, O^Q^Tt, then P(X, Y;X, <f>Y)=-smse. We now put. B(X,Y}:=<R(X,Y)Y, X> and for X orthogonal to Sa (a=l, ••••••, p). D(X) : =B(X, <f>X}. Then we have. (3.6) D(X+Y)+D(X-Y)=2\D(X)+D(Y}+2B(X, </>Y)+2<R(X, <f>X)<ftY, Y> -2<R(X, <t>Y}Y, i>X>\. From (3. 6) and D(X)=D(<f>X), we have. (3,7) D(X+</>Y)+D(X-<f>Y)=2\D(X)+D(Y)+2B(X,Y)+2<R(X, ^X)<f>Y, Y> +2<R(X, Y)'/>Y, <1>X>\. LEMMA 3. 1. For tangent vectors X and Y orthogonal to Sa (a=l, ••••••, p) we obtain. (3.8) B(X,Y)=^[3D(X+<^Y)+3D(X-<f>Y)-D(X+Y)-D(X-Y)-AD(X)-iD(Y) -24p P{X,Y;X,<t>Y}\. PROOF. From (3. 6) and (3. 7), we have. 3D(X+ 4, Y) +3D(X- <1> Y) -D(X+ Y)-D{X- Y) -4£>(X) -4£>( 7) =12B(X,Y)-'IB(X, <ftY)+S<R(X, <f>X)i/,Y, Y> +12<R(X, 7)07, <t>X> -4<fi(X, <i>Y)<l>X, Y> From (3. 3) and the 1-st Bianchi's identity, we have. <R(X, </>X)<f>Y, Y>=B{X,Y)+B(X, </>Y)+2p P(X,Y;X, <f>Y). Similarly we obtain <R(X, Y)<f>Y, </>X>=B(X, Y)+pP(X,Y;X, <f>Y).
(9) 8 Izumi HASEGAWA, Yukihiko OKUYAMA and Tomoji ABE. and. <R(X, (67)^, Y>=B(X, </>Y)+pP(X,Y;X, </>Y).. From these equations we derive our assertion. Q. E. D.. We notice that D(X)==H(X) if and only if X is a unit vector and B(X, Y) =K(X,Y) if and only if \X, Y| is an orthonormal pair. LEMMA 3. 2. Let \X, Y\ be an orthonormal pair orthogonal to ?a (a=l> ••••'•, p). If we put <X, (/>Y>=:cos0 {O^Q^jt), then. ^ M y)=i{3(l+cos6rH( ii ^J 11 )+3(l-cosff)2H(n1^T -H(j^w)-H(-j^r}-H(x}-H(Y)+6p sin201 ^. PROOF. Since D(Z)= || Z \\*H( „ ^ „ ) for any tangent vector Z, we see that ,„/ X+^Y £>(X+(4Y)=4(l+cos(9)2H X+<f>Y Z)(X-fSy)=4(l-cos^)2ff. X-</>Y. X-<pY\. D(x+Y)=w(\ixxtYY\\ ) and z)(z-y)=4H(^fl From these equations and Lemma 3. 1, we have (3. 9). Q. E. D. PROPOSITION 3. 1. The <p -sectional curvatures determine the Riemannian curvatwe tensor of a. p -th Sasakian manifold. PROOF. Since the sectional curvatures of a Riemannian manifold determine the Riemannian curvature tensor, it suffices to show that the sectional curvatures are uniquely determined by the <t> -sectional curvatures. Let iZ, 71 be an orthonormal pair. We put. Z= : aX'+^r]a W^a and 7= : /3Y'+ E ^(V)?., where a='/l- {,T]a(X)2, P=^\—T,r]a(YY and X', Y' denote the unit vectors orthogonalto ^(a=l, a::-, p). Since ^>T=a</>X', <^Y=/3</>Y', <R(X',Y')Y;X/>=(1-<X', Y'>2) K(X',Y') and <R(^X', </>Y')<f>Y')<f>Y', </>X'> =K(X', Y'), from (3. 5) we obtain. ?Y)=|i-E^(X)2-E^(y)2+(S^(x)2)(S^(Y)2) a=l. a=l. a=l. b^l. -(F,ria(X)ria(YWK(X',Y') a=l. +2(S^(^)^(Y))(E^(X))(S^(Y)) a=l. b=l. c-l. +(i-E^(^)2)(E^(y))2+d-E^(y)2)(E.^(^))2. On the other hand, by Lemma 3. 2, K{X , Y ) is determined by ^-sectional curvatures. This completes. the. proof.. Q.. E.. D..
(10) On p-th Sasakian Manifolds 9 We now prepare a well-known algebraic lemma for a quadrilinear mapping on a real vector space. LEMMA 3. 3. Let V be an n-dimensional real vector space and T : V* —> R a quadritinear. mapping with the following properties : (D m,, Xz, x,, x^=-T(Xz, X,, X,, X^)=-T(X,, X,, X,, X,), (2) m., Xz, A-,, xj+r(^,, X,, X,, Xz)+T(X,, X,, X,, X,}=0, (3) T(X,, X,, X,, X,)=0. Then T=0.. THEOREM 3. 2 (cf. [5]) . If the ^-sectional cwvature at any point of a p-th Sasakian nmnifold of dimension ^ 4 + p is indepedent of the choice of <f>-section at the point, then it is constant on the Manifold and the curvature tensor is given by. R(X,Y)Z=^-(c+3p)\<</>X, </>Z>'f>lY-<rf>Y, <1>Z>^X\ +_L(c-p)|<^y, Z>'f>X-<<f>X, Z><f>Y+2<X, </>Y></>Z\ (3.10) ' 4. +(S^m)(E^(z))^r-(E^(Y))(S^(z))^ a-I. 6-1. a-l. b=l. +<<t,Y, ^ZXS^(X))(,E^)-<^, (AZ>(E^(y))(.E^), a=l. '. b==l. a=l. &==l. where c is the constant <1>-sectional curvature. PROOF. Using Lemma 3. 3, we see that R(X,Y)Z is of the given form. We prove that a function c is constant. In this case the Ricci tensor S and scalar curvature P are given by. (3.11) S(X,Y)=n(c+3PHC~p <-^X, ^Y>+2n(i^(X))(f]^(Y)) a-I. 6=1. and. (3.12) />=(n+l)c+n(3n+l)p. Therefore we have. (n-l)dc+Sdc(^)^=0. t>=I. Applying this to ?a, we have dc(^a)=0 and hence dc=0 for n>l, as desired. Q. E. D. A p- th Sasakian manifold M of constant ^-sectional curvature c will be called a p-th Sasakian space form and denoted by M[c] . COROLLARY. Under the assumption of Theorem 3.2, we ha^ie. fl(x,m=4^%-t<^, ^z>^2y-<^y, ^>^2xi ^_p-2n(2n+l)p ^^y, Z><pX-<<f>X, Z><j,Y+2<X, <f>Y><f>Z\ (3.13) ' 4n(n+l). +{i^(X)(ir!,(Z))^Y-(iT]a(Y))(Zr)^Z})^X a»l. b=l. p. +(S^(X)X(sy, ^>(S?<,)-(E^(Y))<(iX, (SZXS^) a«l. •. b-1. p. a=l. p. (3.14) S(X.Y)=(-^—P)<'ftX, ^y>+2n(I^(Z))(E^(y)). 9. °=i.
(11) 10 Izumi HASEGAWA, Yukihiko OKUYAMA and Tomoji ABE From the simple calculation, we have the following LEMMA 3. 4. The Ricci tensor S of (2n+p)-dimensional p-th Sasakian manifold demonstrates. the following: (1) S(X, ^)=2nE^W (a=l. ••••••. P). b=l. (2) S(</>X, 0y)=S(X,y)-2n(E??aW)(E^(y)), a^i b=l (3) S(X, rf>Y)=-S(</>X,Y). If the Ricci tensor S of a p-th Sasakian manifold M is of form. S(X, Y)-=f<4>X, (*y>+/t(S7?a(X))(S^(Y)), a=l. fa=l. / and h being functions on M, then M is called a (??i, ••••••, rip) -Einstein manifold. In this case, / and h are necessarily constant (f=~^~~P and /i=2n). From Theorem 3. 2 and Lemma 3. 4, we derive the following PROPOSITION 3. 3. Ap-th Sasakian space from is necessarily (fji, ••••••, n?) -Einstein,. § 4. p-contact Bochner curvature tensor.. Let M be a (2n+p)-dimensional p-tb Sasakian manifold with the structure (</>, ^i, ••••••, ^p, rit, ••••••,r]p,< , ».We define a ^-contact Bochner curvature tensor B(cf. [4, 11]) by. (4.1) B(X,Y)Z^R(X, DZ—g^^y-K^, ^Z>Q<f>lY+S(^X, <f>Z)^Y -<(*7, </>Z>Q<f>2X-S(^,Y, <f>Z)^X-<^,X, Z>Q(A7-S((*X, Z)^y +<</>¥, Z>Q<*X+S((AY, Z)<fiX-2<<f>X, Y>Q'f>Z-2S(if>X, Y)i/>Z\. 4^+!?)Tn+2)t<^' ^>^^-<^^ Z>^-2<^X, y>^71 ~ 4(2i+lT(nT2) !<^' ^>^2y-<<*y- ^>^2^1. +(S ^(Y))(I^(Z))02X-(E ^(X))(E ^(Z))^2Y a-l. t)=l. b^l. b=l. +(S7?a(y))<^, (AZ>(S^)-(E7?a(A'))<(*y, (4Z>(£^) a=i. b=i. where <QX,Y> :=S(X,Y). From Proposition 3. 3 and a simple calculation, we drive the following PROPOSITION 4. 1. Let M be a (2n + p)-dimensional p-th Sasakian manifold with structure (<f>, ^,, ••••", Sp, 1,, ••••••, Tip, < , », Then M is a p-th Sasakian space form if and only if M is an. ?" ••••••, r] p) -Einstein manifold with vanishing p-contact Bochner curvature tensor.. § 5. Hypersurfaces in a p-th Sasakian manifold.. LEMMA_5._1. Let M be a (2n+2+p)-dimensional almost p-contact Riemannian manifold with structure {</>, ^, ••••••, ^, ^, ....... ^,< , » and M an orientable hypersurface in M tan-. (10).
(12) On p-tb Sasakian Manifolds 11 gent to Si, ••••••, ^. Then M admits an almost (p+D-contact metric structure. PROOF. Let e be a unit normal vector field on M . We put Sp+i :==—(*£, rjp+,(X) : =<X, ^p+i>, </>X : =<fiX—rjp+t(X)£ for any tangent vector Z of M and, for simplicity, ^n : =^a, ria:=rja (a=l, ••••••, p) on M. Then (0, ^i, ••••••, ?p+i, T?,, ••••••, ^p+i, < , » is an. almost (p+D-contact metric structure. Q. g. D. PROPOSITION 5. 1. Let M be a (2n+2+p)-dimensional p-th Sasakian manifold with structure (</>, ?i, ••••••, ^p, rj\, ••••.., rjp, < , » and M a totally p-contact umbilical hypersurface with constant mean curvature o»,_Li _!_„ • Then M admits a (p+1 )-th Sasakian structure.. PROOF. From Lemma 5. 1, we see that M admits an almost (p+D-contact metric structure. Let e be a normalized mean curvature vector on M , that is, e := "'^ ^ ., v //, where ^ donates the mean curvature vector on M in M. The second fundamental from o is then represented by. (5.1) a{X, Y)=<^X,^Y>e+I]riaWa{Y, ?„)+ E ria(Y)a(X. ^) for tangent vector X and Y of M.. a=l. a-l. From 7s^a=</>X, we have (5.2) y.^a=</>x and (5.3) a{X, ^)=??p+.We where [7 denotes the Riemannian connection of M and V the induced Riemannian connection of M . Here we put h(X, Y)e:=a{X, Y) and < AX, Y> : =h(X, Y). We can then rewrite (5. 1) as follows :. (5.4) h(X,Y)=<X:Y>-^^(X)ria(Y)+(^riaW)ri^(Y)+(F,7)^Y))7i^(X) a=i. or. a=i. a==i. (5.5) ^=x-s»7a(x)^+(a^m)^,+^,(x)(E^). a==i. On the other hand, from. (V,^Y=-(^r)a(Y))^X-<^X, (6YXI3^a), we obtain. O=TI. •. a=. (5.6) <I7<^,, Y>+h(X, <t,Y\=Q and. (5.7) [7^}Y=n^(Y)AX-h{X, Y)^>-(S r,a(Y))^X-<-ftX. ^YXE^). From (5. 4), (5. 5) and (5. 7), we obtain. (5.8) (V^}Y=-^^(X)^Y-<^X, ^VXE^). From (5. 6) and (5. 8), we have (5.9) 7^^=<t>X. Therefore, using Proposition 2. 3, M admits a (p+l)-th Sasakian structure. Q.E.D.. THEOREM 5. 2. Let M[—3p] be a (2n+2+p)-dimensional p-th Sasakian space form with constant <f>-sectional cwvature —3p and M a totally p-contact umbilical hypersurface with constant mean curvature r.^.jiil^' Then M admits a (p+D-th Sasakian structure with constant i> -sectional curvature 4—3 (p+1). PROOF. From Proposition 5. 1, we can see that M admits a (p+D-th Sasakian. (11).
(13) 12. Izumi HASEGAWA, Yukihiko OKUYAMA and Tomoji ABE. structure. The Gauss-Weingarten formulas imply. (5.10) R(X, Y)Z-=R[X, Y}Z+h(X, Z)AY-h(Y, Z)AX +\(7,h)(Y,Z)-(7,.h}(X,Z}\£, where R denotes the Riemannian curvature tensor of M and R the Riemannian curvature tensor of M . Using Theorem 3. 2, we have. R{X,Y}Z=p\<^X, Z>^Y-<^Y, Z>^X+2<</>X, Y><f>Z\ (5.11) +(£;?7a(^))(E'?i,(Z))^Y-(E?<,(Y))(E^(Z))<*2^ a=i. p. +(S Tla(X)X^Y, 0Z>(S ^)-(E 7?a(y))<^, (*7>(E^). a=l,. •. •. i>=l. a=l. 6=1. From (5. 10) and (5. 11), we obtain R{X, Y)Z=<</>X,</>Z></>SY-<<f>Y, </,Z>^X +p|<(6X, Z></>Y-<</>Y, Z><i>X+2<if>X,Y>-f>Z\. +(El^w)(sl^(z))^y-(s^(y))(E^(z))^x o^"i'"'. '''. ^i. ". '. a-i. ^=l. +(sl^(x))<0x,^z>(s^)-(i;^m)<? ^>(g^). Therefore M admits a (p+l)-th Sasakian structure with constant ^-sectional curvature 4 -3(p+l).. Q.E.D.. § 6. Examples of p-th Sasakian space forms.. (1) /i2n+p[-3p]. Let (a;', ••••••, x", y\ ••••••, y", z\ ••••••, zp)= { x\ ••••••, x^") be. the cartesian coordinate system on R2n+tl. We put ^: =28/3^° (a=l, ••••••, p), 7}a:=±(dza-i,y'dxt} (a=l, ••••••, p),. (6.1). t=l. 4>X : =- t,X"3/3x'+ E X'a/3yi-(S X/<y')(S. 3/3^°), i-^i. ;»!. -. (-]. a=l. p. g=< , > : =t,T)a<8>T]a+l-t,(dxt<9dxt+dyt'8>dy'}, r)a!8>r]a+' a=l. t. t=l. where x= E X{3/3xl+ Z; X't3/3y'+ EX'/a3/3-?a. (=1. i-i. a=l. With this structure (</>, ^i, ••••••, ^p, T?,, ••••••, ?;p, < , », A2n+fl is a ^-th Sasakian manifold.. The Riemannian metric 9=< , > has the component matrix 9u 9u* 9w. (6.2) (<7,. Su+py'y' o -yl 0. 9i*j 9i*j* gi*v. 5-». 0. -yj o. <5~n,. 9 ec 1 g a'I* 9a'b'. where g^=0/3xA, 3/3x"> (l^A, /u<2n+p), i*=n+i (l<i<n) and a'=2n+a (1^ a^p), etc. The inverse matrix is given by. (12).
(14) 13. On p-th Sasakian Manifolds 9tl 9"* 9W \. (6.3) (gw)=. 9lv] 9'^ 9itb'. ^» 0 0 ^». =4. y] o ^+S(i/k. ga'' ga3f g"'1'' ). ;=1. In the following, we shall calculate the Christoffel symbols and the curvature tensor. We put _ 1 / 9(7ue i 9ffA£ _ _9.i. '' £}=t[~^~+~^~~^)-. We can verify that. [i*j, h]=^(S,,yJ+y,,yh),. [ij. h*]=-^(S^+^yl), (6.4) [1*7, a']=[i*a/,n=-^^,. [ia', 7*]=-g-^,. the other components are zero. From (6. 3), (6. 4) and | , K,. \ :=S'K£ [A//, e], we have '". \i'hi}=^"' ^*.|=-f(^.^+<W), A* }_!_ (•" a/}=t^'. (6.5) ,.a/.) =^(pyiyl-s^, .<.1=_J_,,< i*~ b'\=~~2y' y,. h l__J_. [»•* ffl'J=-Y°"" I. the other components are zero.. After some straightforward calculations, we obtain the independent components of the Riemannian curvature tensor R as follows (c/. [7]) : R^=^(S^ytyj+S^hylc-S^ylyk-S^hyj),. Rkt*j^=J^(p^uyhy':-^^^-s^s^,. Rh* (*A-=-[g(l5'hj^ifc—5'hfc^u)l. (6.6) RHM=J^-^Hjy'-Suyh),. ^*U*tf=^.<W,. (13).
(15) 14 Izumi HASEGAWA, Yukihiko OKUYAMA and Tomoji ABE. 1 16. Rhct U k= Rh*a' V lc*=~ic<> hKi. the other independent components are zero. Now it is checked that the relation (3. 10) holds for c=-3p. We denote Rw+p with this structure by Rsn+p[-3p].. (2) (CD"XRP) [c], c<-3p. Let CD" be a simply connected bounded complex domain in C" with a constant holomorphic sectional curvature k<0. We denote by (J, g) a Kaehlerian structure on CD". Since the fundamental 2-form Q of this Kaehierian structure is closed, Q=da) for some real analytic 1-form <y. Let t=(t\ •••••-, t") denote the coordinate on R" and put T]a =K*wJrdta on a product space CD"XR1', where TT : CD'1XR1' — > C£>" is the projection. We put '?: =(li, ••••••, r)p), then 1 is a connection form on the trivial bundle CDnXRP. Moreover, we put Qa: =a/3ta (a=l, ••••••, p), < , > : =K*g+ S^^®??a, (ilZ=(J(^^))* for any vector field X on CD"XR'', where (J(n-^X))* is a horizontal lift of J(TC^X). We then see that (^S, ^i, ••••••, ^p, ^i, ••••••, '7p, < , » is a p-th Sasakian structure on CD"XRP.. We are able to prove that CD" X R" is a p-th Sasakian manifold with constant <zi -sectional curvature c<—3p. First, we see. (R{X,Y)Z)*=-^\R{X*,Y*)Z*-^p^a([Y*, Z*m^a +-1S^W Z*])F,.^+S^W Y*])^^*I. ^~l. '~. '". a**. for tangent vector fields X, Y, Z on CD", where X is the horizontal lift of X, V a connection on CD" XR", R a Riemannian curuature tensor of CD" and R a Riemannian curvature tensor of. CD"XR1' (cf. [6] ). From this we obtain R(X, y)Z=fet<^, <^Z>^Y-<<f>Y, <f>Z>^X\ +(fe-4p)t<0Y, Z><t>X-<<t>X, Z>-f>Y+2<X, 0Y>^Z!. +4(E^(X))(E^(Z))^Y-(S,a(y))(S^(Z))0^ a=l. „. ^. p. ~.. p. +4<^v, 0z>(E^a))(s^)-4<^, ^zxs^(y))(s^). a==i. '. b=i. By Theorem 3. 2, CD"XRP has the constant 0-sectional curvature c : =k-3p«-3p).. (3) (Sln+iWXRP-')[c], c>-3p (p^l). Let S2"+'(2) be a (2n+l)-dimensional ordinary sphere of radius 2 and M=S2"+1(2) XRtl~t a hypersurface in a (p-l)-th Sasakian space form fi2n+2+lp-"[-3(p-l)]. Let (x\ •••••-, xn+\ y\ •••••-, y"+I, z\ ••••••, ^-') be the cartesian coordinate system and (0, ^i, ••••••, ^p_,, 771, ••••••, ^p-i, < , » a (p-l)-th Sasakian structure. on /i2"+2+lp-"[-3(p-l)] as example (1). For simplicity, we put ?a : =^a and ?;„ : =ria (a=l, ••••••, p—1) on M.. The normalized mean curvature vector s is represented by e=-F,xt3/3xl-^yl3/3y'-{Exiyt)(r,3/3za). (=1. (-1. 1=1. According to Lemma 5. 1 we put. n-l. ^..=-^=-^yt3/3xi+T.xl3/ay'-(r,(ytr)(^3/3za) and. rip(X) : =<X, ^>. (14).
(16) On p-th Sasakian Manifolds 15 for tangent vector X of M . In this case, we have. h(X, Y)=<(7(Y, e> =l-'Z(X{Y'+X"Y't)+^(^X"a-p^Xlyi)('WJxJ-YJyJ)) 4. ^1. '"". ~. °. (I1"1. (==I. ~':=. +-^(SY"a-p'^Ylyt)('^(X'JxJ-Yjy])) 8 'S='i ~ " ^ ~ j-^. =<fX, fY>+{^rja(X))r),(Y)+(^rja(Y)}rj,(X) where X= S Xt3/3xt+ E X't3/9yi+ E Z'/a3/a^° and 7= S y'a/a.K'+S y/ia/3i/'+. p_,. (=l. '-I. *. a=l. ^1. •. T~i. Zl Y 3/3za denote the tangent vector fields on M. Therefore M is a totally ^-contact umbilical hypersurface in Rm+wt'~"[-3(p-l)] with constant mean curvature 2n+l_ . Using Theorem 5. 2, we see that M admits a ^-th Sasakian structure {<{>, ^i, ••••••, ^p, ??i, ••••", ?7p, <. » with constant ^-sectional curvature 4—3p. We now consider the deformed structure on M=Sm+t(2)X R1'-1 (cf. [12]). ^ •• =CtT]a , SS : =a~1^ (a=l, ••••••, P),. <**;=(*, < , >*:==a< , >+a(ff-l)E^®?7a, a=l. where a is a positive constant. Then (</>*, £*, ••••••, ^, T]*, ••••••, rjf, < , >*) is a. p-th Sasakian structure on M with costant 0-sectional curvature c=-^-—3p. For p = 0, it is well-known that the complex projective space CP" with Fubni-Study metric is a complex space form with a positive constant holomorphic sectional curvature (i. e., 0-th Sasakian space form with a positive constant ^-sectional curvature).. References. [1] D. E. Blair ; Contact manifolds in Riemannian geometry, Springer-Verlag, 1976.. [2] I. Hasegawa : Remarks on extrinsic spheres and totally contact umbilical submanifolds in Sasakian manifolds, J. Hokkaiodo Univ. Ed. (Sect. H A), 36 (1986), 39-48. [3] I. Hasagawa : Totally fe-contact umbilical submanifolds in p-tb Sasakian manifolds, to appear. [4] M. Matsumoto and G. Chuman : On the C-Bochner curvature tensor, TRU Math., 5 (1969), 21-30.. [5] K. Ogiue : On almost contact manifolds admitting axiom of planes or axiom of free mobility, Kodai Math. Sem. Rep., 16 (1964), 223-232.. [6] K. Ogiue: On fiberings of almost contact manifolds, Kodai Math. Sem. Rep., 17 (1965), 53-62. [7] M. Okumura : On infmitesimal conformal and projective transformations of normal contact spaces, Tohoku Math. J., 14 (1962), 398-412. [8] M. Okumura ; Certain almost contact hypersurfaces in Kaehlerian manifolds of constant holomorphjc sectional curvatures, Tohoku Math. J., 16 (1964), 270-284.. [9] M. Okumura: Contact hypersurfaces in certain Kaehlerian manifolds, Tohoku Math. J., 18 (1966), 74-102. [10] S.Sasaki: Almost cotact manifolds, Tohoku Univ., 1 (1965), 2 (1967), 3 (1968). [11] S.Tachibana: On the Bochner curvature tensor, Nat. Sci. Rep. Ochanomizu Univ., 18 (1967), 15-19.. [12] S. Tanno : The topology of contact Riemannian manifolds, Illinois J. Math., 12 (1968), 700-717. [13] S. Tanno : Sasakian manifolds with constant <f> -holomorphic sectional curvature, Tohoku Math. J., 21. (1969), 501-507. [14] Y. Tashiro : On contact structure of hypersurfaces in complex manifolds I , Tohoku Math. J., 15 (19Q3),. (15).
(17) ,^ Izumi HASEGAWA, Yukihilm OKUYAMA and Tomoji ABE 62-78.. [15] Y. Tashiro : On contact structure of hypersurfaces in complex manifolds V. , Tohoku Math. J., 15 (1963), 167-175.. [16] K. Yano and M. Kon : Structures on manifolds, World Scientific, 1984.. (16).
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