Infinitesimal Variations of Hypersurfaces of a Kaehlerian Manifold

J. Math. Soc. Japan Vol. 29, No. 2, 1977 Infinitesimal variations of hypersurfaces of a Kaehlerian manifold Dedicated to Prof. Shokichi Iyanaga on his seventieth birthday By Kentaro YANG (Received Feb. 10, 1976) (Revised May 27, 1976) § 0. Introduction. B. Y. Chen [1, 2, 3], C. S. Houh [2], S. Tachibana [10] and the present author [3, 9, 10] have recently studied infinitesimal variations of submanif olds of Riemannian and Kaehlerian manifolds. (See also [7].) The main purpose of the present paper is to study infinitesimal variations of hypersurfaces of a Kaehlerian manifold, to obtain variations of structure tensors of the almost contact metric structure induced on the hypersurf aces from the Kaehlerian structure of the ambient manifold and to prove theorems on variations of Sasakian hypersurf aces with f -sectional curvature a2. (For the definition of a Sasakian manifold with f -sectional curvature a2, see § 1.) In § 1 we state some preliminaries on almost contact metric structure induced on a hypersurface from the Kaehlerian structure of the ambient manifold. In § 2 we consider infinitesimal variation of a hypersurface of a Kaehlerian manifold and obtain variations of structure tensors of almost contact metric structure induced on the hypersurf ace. § 3 is devoted to the study of what the author calls parallel variation and § 4 to that of variation of the second fundamental form of the hypersurface. In the last § 5, we consider variations of a Sasakian hypersurf ace with f sectional curvature a2 and obtain the condition that an infinitesimal variation carries a Sasakian hypersurf ace with f -sectional curvature a2 into a hypersurface of the same kind. § 1. Preliminaries. Let Men+2 (n>1) be a real (2n+2)-dimensional almost Hermitian manifold covered by a system of coordinate neighborhoods {U; xh} and Fih be its almost complex structure tensor and g;1 its almost Hermitian metric tensor, where 288 K. YANo and in the sequel the indices h, i, j, k, run over the range {1', 2', , (2n+2)'}. Then we have (1.1) FitFth==-az and (1.2) FjtFisgts=gji Let M2n+1be a (2nd 1)-dimensional orientable Riemannian manifold covered by a system of coordinate neighborhoods { V; ya} and g,b be its fundamental metric tensor, where and in the sequel the indices a, b, c, run over the range {1, 2, , 2n+1}. We assume that M2n+1is isometrically immersed into M2n+2 by the immersion i ; M2 1-M22 and identify i(M21) with M2n+1. We represent the immersion by (1.3) xh=xh(ya) and put (1.4) Bbh= abxh , (ab= aI a?) . Then Bbh are 2n+1 linearly independent vectors of M2n+2tangent to M2n+1 Since the immersion is isometric we have (1.5) gcb=gjiBciBb2 We represent the unit normal to M2n+1by Ch. We then have (1.6) gj1BbiCi=O and (1.7) gjiCjC1 =1 . Now the transform FitBbz of Bbi by Fih can be written as (1.8) Fit BbI =f baBah+fbCh, where f ba is a tensor field of type (1, 1) and f b a 1-form of M2n+1and the transform FihC1 of C by Fih, being orthogonal to Ch, can be written as (1.9) FihCI = - f aBah , where f a=f bgba is a vector field of M2n+1,gba being contravariant components of gcb. Applying F to the both sides of (1.8) and using (1.1), (1.8) and (1.9), we find (1.10) fbefea-~Ub~fbfa, fbefe-O. Applying F to the both sides of (1.9) and using (1.1), (1.8) and (1.9), we Infinitesimal variations of hypersurfaces 289 find (1.11) feafe=0, fefe`=1. From (1.2), (1.5), (1.6), (1.7) and (1.8), we have (1.12) f cefbdged=gcbrfcfb Equations (1.10), (1.11) and (1.12;) show that the set (f, gebff b) defines the so-called almost contact metric structure on Men+1 (Tashiro [6]). We denote by T the Christoffel symbols formed with g;1 and by F b those formed with geb. Then it is well known that F and T b are related by (1.13) r ~= (aCBbhI 1 ~iBcb)Bah, where Bc(=BC'Bbti and Bah=BbigbagZh We define the van; der, Waerden-Bor- tolotti covariant derivative of Bb' along Men+1 by (1.14) VcBbh= a~Bbh+r, Br bBah and that of C h by (1.15) V~Ch=dCCh+r BcjCi. Then equations of Gauss can be written as (1.16) VCBbh=hcbCh, where hCbis the second fundamental tensor of Men+1 and equations of Wein- garten as (1.17) VcCh=-hcaBah, where hca= hcbgba Equations of Gauss and Codazzii are respectively written as (1.18) Kdcba= KkjihBdcbh+hdahcb-hcahdb and (1.19) KkjihBdcbCh= V dhcb_V chdb , Kdcbaand Kkjih being curvature tensors of Men+1and Men+2 respectively, where Bdcbh=BdkBCJBb2Bah,Bdcb=BdkBciBb2 and Ch= C1gih. We now assume that the ambient manifold Men+2 is Kaehlerian, that is, (1.20) V,F~h=0, where V; denotes the operator of covariant differentiation with respect to F. Differentiating (1.8) covariantly along Men+1 and using (1.16), (1.17) and 290 K. YANo (1.20), we find (1.21) Qcfba= -hcbf a+hcafb and (1.22) Qcf b = -hcef be. Differentiating (1.9) covariantly along Men+1 and using (1.16), (1.17) and (1.20), we find (1.23) Qcf a= hcefea . Equations (1.22)`and (1.23) are equivalent because of the relation f ba-fbehea =-fab~ We now consider the tensor field Scba defined by (1.24) Scba= Ncba+ (Qcfb-Qbf c)f a , where Ncba is the Ni jenhuis tensor formed with f ba: (1.25) Ncba_'feeQefba-J i, efea-(Qefbe-Qbfce)fea . When the tensor field Scba vanishes identically, the almost contact metric structure (f b', gcb, fb) is said to be normal. Substituting (1.21) and (1.22) into (1.24), we find (1.26) Scba'_ (fcehea`-heefea)fb-(fbehea^hbefea)fc Thus if the almost contact metric structure is normal, we have (f eehea-hcefea)f b (fbehea-hbefea)f c - , from which, using (1.10) and (1.11), (1.27) f eehea-hcefea = 0 and (1.28) heafe - hfa , where h=hc6 fc fb. When the almost contact metric structure (f a, gcb,f b) satisfies (1.29) Qcfb-Qbfc=2fcb, the structure is said to be contact. Substituting (1.22) into (1.29), we obtain (1.30) fcehea+"hcef ea = 2f ca . When the almost contact metric structure is normal and contact, we call the structure a Sasakian structure [5]. For a Sasakian structure, we have Infinitesimal variations of hypersurfaces 291 (1.27) and (1.30) and consequently (1.31) f cehea-f ca , from which, using (1.10), (1.11) and (1.28), (1.32) hcb=geb+(h-1)fcfb For a Sasakian structure, (1.21), (1.22) and (1.23) reduce respectively to (1.33) Qcfba--gcbf a+~afb (1.34) Qcfb=fcb and (1.35) Qbf = f ba . But if, instead of (1.29), we assume (1.36) Qcf b-abf c= 2af eb, a being a function, we obtain, substituting (1.22) into (1.36), (1.37) fcehea+heef ea= 2afca. Thus, from (1.27) and (1.37), we find (L38) f cehea= of ca from which, using (1.10), (1.11) and (1.28), (1.39) heb=agcb+(h-a)fcfb Thus (1.21), (1.22) and (1.23) reduce respectively to (1.40) Qcfba= a(--gcbf a+ ~cfb) , (1.41) Qcf b = of cb and (1.42) Qbfa = a fba . From (1.36) and (1.40), we have Qd(Qcfb-Qbfc) = 2(Qda)f cb+ 2a2(-gdcfb~gdbfc) , from which, using the Ricci formula and the Bianchi identity, we obtain (Qda)f cb+(Qba)f dcT (Qca)fbd =0. Thus if n> 1, we have (1.43) a = constant. 292 K. YANo For a Sasakian structure, we have, from (1.33) and (1.35), (1.44) Kcbeaf e= oaf oafc which shows that the sectional curvature with respect to a plane section con- taining f a is always 1. But for the structure under consideration, we have, from (1.40) and (1.42), (1.45) Kcbeaf e _- a2(Oafb Ubfc) , which shows that the sectional curvature with respect to a plane section con- taining f a is always a2. So we call an almost metric structure satisfying (1.40), (1.41) and (1.42) a Sasakian structure with f -sectional curvature a2 (Okumura [4]). § 2. Infinitesimal variation of a hypersurf ace of a Kaehlerian manifold. We now consider an infinitesimal variation of the hypersurface M2n+1 in n f2n+2 given by (2.1)h=xh~hCY)~ h being a vector field of M2n+2 defined along M2n+1 , where s is an infinitesimal. We then have (2.2) Bbh = Bbh+ (ab h)~ , where Bbh=abxh are 2n+1 linearly independent vectors tangent to the varied hypersurface at the varied point (xh). We displace vectors Bbh parallelly from the varied point (.Ch) to the original point (xh) and obtain Bbh ^ Bbh-+ ji(x+~)E'Bb~E , or N (2.3) Bbh_ Bbh+(Qbh)~ , neglecting the terms of order higher than one with respect to ~, where (2.4) Qbh=~bh-~ I';2Bb~~. In the sequel we always neglect terms of order higher than one with respect to e. Putting N (2.5) Bbh=Bbh-Bbh , we have N Infinitesimal variations of hypersurfaces 293 (2.6) ~Bbh== (Qbh)~ . If we put (2.7)h =aBah+ ~Ch , ea and A being respectively a vector field and a scalar function on Men+1,we have (2.8) Qb h=(Qbr_Ahba)Bah+(QbA+hbe e)Ch We denote by Ch the unit normal to the varied hypersurf ace.

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