338 Jin Suk Pak and Yang Jae Shin 2. Preliminaries Let M be a( + )-dimensional almost contact metric manifold with an almost contact metric structure

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1 Comm. Korean Math. Soc. 3(998), No. 2, pp A NOTE ON CONTACT CONFORMAL CURVATURE TENSOR Jin Suk Pak* and Yang Jae Shin Abstract. In this paper we show that every contact metric manifold with vanishing contact conformal curvature tensor is a Sasakian space form.. Introduction The contact conformal curvature tensor ([3]) is a curvature-like tensor dened on a contact metric manifold (M; '; ; ; g) which is constructed from the conformal curvature tensor ([6]) by using the Boothby-Wang's bration ([3]). It seems to play an important role to studying the spectral geometry of compact Sasakian manifolds (cf. [5], [7]). On the other hand Tanno([0, ]) proved that every conformally at K-contact manifold is a space form, and Blair and Koufogiorgos ([2]) improved Tanno's result as follow: Every conformally at contact metric manifold with R' = 'R is a space form, where R denotes the Ricci operator. In this paper we shall prove the following theorem which gives a geometric characterization of a contact metric manifold with vanishing contact conformal curvature tensor: Theorem. Every ( + )(n > 2)-dimensional contact metric manifold with vanishing contact conformal curvature tensor is a Sasakian space form. Received January 4, 998. Revised March 9, Mathematics Subject Classication: 53C5. Key words and phrases: contact metric manifold, contact conformal curvature tensor, Sasakian space form. * This paper is partially supported by Research Fund of Korean Council for University Education, 997 and BSRI

2 338 Jin Suk Pak and Yang Jae Shin 2. Preliminaries Let M be a( + )-dimensional almost contact metric manifold with an almost contact metric structure ('; ; ; g). Then we have by denition '2 =,I + ; ' =0; ' =0 (2.) g('x; 'Y )=g(x; Y ), (X)(Y ) for any vector elds X; Y tangent to M, where I denotes the identity transformation(cf, [], [9]). Denoting by r the Riemannian connection and by the fundamental 2-form dened by (2.2) (X; Y )=g('x; Y ) the almost contact metric structure ('; ; ; g) is called a contact metric structure if satises (2.3) =d: A manifold with a contact metric structure is called a contact metric manifold. (cf. [], [9]). A contact metric manifold for which is Killing is called a K-contact manifold. Also, a manifold with a normal contact metric structure is called a Sasakian manifold. Thus a Sasakian manifold is K-contact but the converse is not true except in dimension 3. (cf. see []). Moreover, the following lemmas are well known (cf. see []), which give inclusion relations among those manifolds. Lemma 2.. On a contact metric manifold, the followings are equivalent to each other: () The manifold is a K-contact manifold. (2) The sectional curvature of plane section containing is equal to. (3) The Ricci curvature in the direction of is. Lemma 2.2. Let M be a ( +)-dimensional Riemannian manifold admitting a unit Killing vector eld such that K(X; Y ) = (Y )X, (X)Y;

3 A note on contact conformal curvature tensor 339 where K denotes the curvature tensor. Then M is a Sasakian manifold. Finally we recall the denition and fundamental properties of D- homothetic deformation due to Tanno ([]), where D means the distribution dened by. D-homothetic deformation g 7! g is dened by g = g + (, ) for a positive constant. The following identities for D-homothetic deformation are well known ([]): (2.4) 8 >< >: K(X; Y )Z = K(X; Y )Z +(, ) fg('x; Z)'Y, g('y; Z)'X +2g('X; Y )'Zg +(, ) 2 f(y )X,(X)Y g(z)+(, )[(X)fg(Y;Z),(Z)Y g, (Y )fg(x; Z), (Z)Xg +(Z)f(Y )X, (X)Y g]; g( RX; Y )=g(rx; Y ), 2(, )g(x; Y ) +2(, )(n + n +)(X)(Y ); s =, s,, (, ): Moreover, if ('; ; ; g) is a Sasakian structure, then ( '; ; ; g) is also a Sasakian structure, where (2.5) ' = '; =, ; = ; g = g + (, ) for a positive constant. In this case it is said that (M; '; ; ; g) is D-homothetic to (M; '; ; ; g)([]). 3. Proof of the main theorem Let (M; '; ; ; g) bea( + )-dimensional contact metric manifold. Then we can consider the following contact conformal curvature tensor

4 340 Jin Suk Pak and Yang Jae Shin C 0 of type (,3) on M, which is dened by ([4]) (3.) C 0 (X; Y )Z = K(X; Y )Z + fr 0(Y;Z)X, R 0 (X; Z)Y + g(y;z)rx, g(x; Z)RY + (Y )R 0 (X; Z), (X)R 0 (Y;Z) + (X)(Z)RY, (Y )(Z)RX + S 0 (X; Z)'Y, S 0 (Y;Z)'X +(X; Z)SY, (Y;Z)SX + 2(X; Y )SZ +2S 0 (X; Y )'Zg + (n +) f2, n, 2+ (n +2)s g f(y;z)'x, (X; Z)'Y, 2(X; Y )'Zg (3n +2)s + fn +2, g (n +) fg(y;z)x, g(x; Z)Y g, (n +) f4n2 +5n +2, f(y )(Z)X, (X)(Z)Y + (X)g(Y;Z), (Y )g(x; Z)g; (3n +2)s g where R 0 and s denote the Ricci tensor and the scalar curvature, respectively, i.e., R 0 (X; Y ):=g(rx; Y ); s := tracer, and (3.2) SX := R('X); S 0 (X; Y ):=g(sx;y ): Using (2.4), we can easily verify that the tensor C 0 is invariant under the D-homothetic deformation dened by (2.5)([4]). From now onwe assume that the contact conformal curvature tensor C 0 vanishes identically on M, i.e. C 0 0. Then from (3.) with C 0 =0 we can easily see that (3.3) RX = f,3'r'x, 3(RX), 2(X)Rg 4n, f(n, 2)s, (n, 2)gX n(4n, 5) 2 + n(4n, 5) f(2 + n, 2), (n, 2)sg(X):

5 A note on contact conformal curvature tensor 34 Putting X = in (3.3) and using (2.), we have (3.4) R =; which and Lemma 2. give Lemma 3.. Every contact metric manifold with C 0 0 is a K- contact manifold. Substituting (3.4) into (3.3), we obtain (3.5) RX =,3 4n, 5 + 2(n, 2) 'R'X + (s, )X n(4n, 5) 2 n(4n, 5) fn(4n2, 3n, 4), (n, 2)sg(X): Applying the operator ' to the both side of (3.6) and using (2.) and (3.4), it can be easily veried that (3.6) g('rx; Y )= 3 2(n, 2)(s, ) g(r'x; Y )+ g('x; Y ) 4n, 5 n(4n, 5) Since ' is a skew- because of R being a symmetric endomorphism. symmetric endomorphism, (3.6) implies g(r'x; Y )= 3 2(n, 2)(s, ) g('rx; Y )+ g('x; Y ); 4n, 5 n(4n, 5) from which together with (3.6), we have g('rx; Y )=g(r'x; Y ) that is, (3.7) 'R = R': Hence it follows from (3.5) and (3.7) that (3.8) g(rx; Y )=( s s, )g(x; Y )+( +, )(X)(Y );

6 342 Jin Suk Pak and Yang Jae Shin provided n>2. Substituting (3.8) into (3.) with C 0 =0,we can easily see that (3.9) K(X; Y )Z = k +3 fg(y;z)x, g(x; Z)Y g 4 + k, fg('y; Z)'X, g('x; Z)'Y 4, 2g('X; Y )'Z, (Y )(Z)X + (X)(Z)Y, (X)g(Y;Z) + (Y )g(x; Z)g; where k = fs, n(3n +)g. Moreover (3.9) yields n(n+) K(X; Y ) = (Y )X, (X)Y; which together with Lemma 2.2 and Lemma 3. implies our main theorem stated in section because s is constant, provided n>2. Combining the theorem with Proposition 4. ([3], p.504) and Corollary ([8], p.282), we have Corollary 3.2. Let M be a ( + )(n > 2)-dimensional complete, simply connected contact metric manifold with vanishing contact conformal curvature tensor. () If s>,, it is D-homothetic to the unit sphere S + ; (2) If s =,, it is isometric to E + (,3) ; (3) If s <,, it is D-homothetic to the universal pseudo-riemannian covering manifold of S +, which is dieomorphic to E S. References [] D. E. Blair, Contact manifolds in Riemannian Geometry, Lecture notes in Math., vol. 509, Springer-Verlag Berlin, 976. [2] D. E. Blair and T. Koufogiorgos, Conformally at contact metric manifolds, preprint. [3] W. M. Boothby and H. C. Wang, On contact manifolds, Ann of Math 68 (958), [4] J. C. Jeong, J. D. Lee, G. H. Oh and J. S. Pak, On the contact conformal curvature tensor, Bull. Korean Math. Soc. 27 (990),

7 A note on contact conformal curvature tensor 343 [5] T. H. Kang and J. S. Pak, Some remarks for the spectrum of the p-laplacian on Sasakian manifolds, J. Korean Math. Soc 32 (995), [6] H. Kitahara, K. Matsuo and J. S. Pak, A conformal curvature tensor eld on hermitian manifolds; Appendium, J. Korean Math. Soc. 27 (990), 7-7; Bull. Korean Math. Soc. 27 (990), [7] J. S. Pak, J. C. Jeong and W. T. Kim, The contact conformal curvature tensor eld and the spectrum of the Laplacian, J. Korean Math. Soc. 28 (99), [8] T. Takahashi, Sasakian manifold with pseudo-riemannian metric, T^ohoku Math J 2 (969), [9] S. Sasaki and Y. Hatakeyama, On dierentiable manifolds with contact metric structures, J. Math. Soc. Japan 4 (962), [0] S. Tanno, Some transformations on manifolds with almost contact and contact metric structure II, T^ohoku Math. J. 5 (963), [], Locally symmetric K-contact Riemannian manifolds, Proc. Japan Acad. 43 (967), [2], The topology of contact Riemannian manifolds, Illinoi J. Math. 2 (968), [3], Sasakian manifolds with constant '-holomorphic sectional curvature, T^ohoku Math. J. 2 (969), Jin Suk Pak Department of Mathematics Education Kyungnam University Masan 63-70, Korea Department of Mathematic Education Kyungpook University Tague , Korea Yang Jae Shin Department of Mathematics Education Kyungnam University Masan 63-70, Korea