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SOOCHOW JOURNAL OF MATHEMATICS Volume 31, No. 4, pp. 611-616, October 2005 SYMMETRY PROPERTIES OF SASAKIAN SPACE FORMS BY MOHAMED BELKHELFA, RYSZARD DESZCZ AND LEOPOLD VERSTRAELEN Dedicated to Professor Dr. Zbigniew Olszak on his 60th birthday Abstract. The only pseudo-symmetric Kählerian manifolds of dimension n ≥ 6 are the semi-symmetric ones ([5], [6]). A similar result was obtained for Ricci pseudosymmetry ([10]). Therefore Olszak introduced a variant of pseudo-symmetric Kählerian manifolds [10]. For dimension 4 Olszak gave an example of non semisymmetric pseudo-symmetric Kählerian manifold [11]. In contrast however there do exist pseudo-symmetric Sasakian manifolds which are not semi-symmetric. In the present paper, we investigate the pseudo-symmetry of Sasakian space forms. 1. Sasakian Manifold Let M = (M 2n+1 , g) be a (2n + 1)-dimensional Riemannian manifold and let (φ, ξ, η) be tensor fields of type (1, 1), (1, 0) and (0, 1) respectively on M , such that: φ2 (X) = −X + η(X)ξ, η ◦ φ = 0, η(ξ) = 1, g(φX, φY ) = g(X, Y ) − η(X)η(Y ), for all vector field X, Y of M . If in addition, dη(X, Y ) = g(X, φY ), then M is called contact Riemannian manifold. If, moreover M is normal, i.e. if φ2 [X, Y ]+[φX, φY ]−φ[φX, Y ]−φ[X, φY ]+2dη⊗ξ = 0, then M is called Sasakian manifold, for more details we refer to [3], [4], [17]. The sectional curvature of the plane section spanned by the unit tangent vector field X orthogonal to ξ and φX is called a φ-sectional curvature. If M has a constant φ-sectional curvature c, then M is called a Sasakian space forms and denoted by M 2n+1 (c) ([3], [4], [17]). The Riemannian curvature tensor of Sasakian space forms is given by the following formula R(X, Y )Z = c+3 (g(Y, Z)X − g(X, Z)Y ) 4 Received May 3, 2004; revised January 4, 2005. AMS Subject Classification. 53B20, 53C25. Key words. semi-symmetry, Ricci-symmetry, pseudo-symmetry, Sasakian space forms. 611 612 MOHAMED BELKHELFA, RYSZARD DESZCZ AND LEOPOLD VERSTRAELEN c−1 (η(X)η(Z)Y − η(Y )η(Z)X) 4 c−1 {g(X, Z)η(Y )ξ − g(Y, Z)η(X)ξ + 4 +g(Z, φY )φX − g(Z, φX)φY + +2g(X, φY )φZ}. (1.1) The Ricci curvature S and the scalar curvature κ are given by S(X, Y ) = n(c + 3) + c − 1 (n + 1)(c − 1) g(X, Y ) − η(X)η(Y ) 2 2 and 1 (n(2n + 1)(c + 3) + n(c − 1)). 2 Example 1.1. We consider R2n+1 with coordinates (xi , y i , z), i = 1, . . . , n P and its usual contact form 12 (dz − ni=1 y i dxi ). The characteristic vector field ξ is 0 δij 0 ∂ given by ξ = 2 ∂z , the tensor field φ is given by the matrix −δij 0 0 and the κ= 0 2 yj 0 2 Riemannian metric g = η ⊗ η + 41 ni=1 ((dxi ) + (dy i ) ) is an associated metric for η, with this metric R2n+1 is a Sasakian form with φ-sectional curvature c = −3 denoted by R2n+1 (−3). P For more details, see [3], [4]. 2. Pseudo-Symmetry In the following, by R we will also denote the curvature operator, such that R(X, Y ) = ∇X ∇Y − ∇Y ∇X − ∇[X,Y ] . As is well known, every (1, 1)-tensor field A on a differential manifold determines a derivation A· of the tensor algebra on this manifold, which commutes with contractions. In particular, the anti-symmetric (1, 1)-tensor field R(X, Y ) induce derivations R(X, Y )·, thus associating with any (0, k)-tensor fields T, the (0, 2+k)tensor field R · T defined by (R · T )(X1 , X2 , . . . , Xk ; X, Y ) = R(X, Y ) · (T (X1 , X2 , . . . , Xk )) = −T (R(X, Y )X1 , X2 , . . . , Xk ) − · · · − T (X1 , X2 , . . . , R(X, Y )Xk ) (2.1) SYMMETRY PROPERTIES OF SASAKIAN SPACE FORMS 613 one has R(X, Y ) · T = ∇X (∇Y T ) − ∇Y (∇X T ) − ∇[X,Y ] T. By R · R we denote in this way the (0, 6)-tensor field obtained by the derivation of the second R, the (0, 4)-curvature tensor, using the curvature operator, R·. When R · R = 0, (2.2) then M is called a semi-symmetric space. Clearly all locally symmetric spaces and all 2-dimensional Riemannian space are semi-symmetric. Through studies of semi-symmetric Riemmannian manifolds were done by Z.I. Szabó [12], [13], [14]. To state the definition of the pseudo-symmetry, we first determine a specific field (0, k + 2)-tensor field Q(g, T ) associated with any (0, k)-tensor field T on Riemannian manifold M by Q(g, T )(X1 , X2 , . . . , Xk ; X, Y ) = (X ∧g Y ) · (T (X1 , X2 , . . . , Xk )) (2.3) = −T ((X ∧g Y )X1 , X2 , . . . , Xk ) − · · · − T (X1 , X2 , . . . , Xk−1 , (X ∧g Y )Xk ), where X ∧g Y is the endomorphism given by (X ∧g Y )Z = g(Y, Z)X − g(X, Z)Y. A Riemannian manifolds M is said to be pseudo-symmetric (in the sense of R. Deszcz) if the (0, 6)-tensor fields R · R and Q(g, R) on M are linearly dependent, i.e. if there exists a function LR : M −→ R such that R · R = LR Q(g, R) holds on UR = {x ∈ M | R − κ n(n−1) G (2.4) 6= 0 at x} where G is the (0, 4) tensor field of M defined by G(X1 , X2 , X3 , X4 ) = g((X1 ∧g X2 )X3 , X4 ) ([1], [2], [7], [16]). Clearly, all semi-symmetric manifolds are pseudo-symmetric. The converse is not true: there are many proper pseudo-symmetric manifolds, i.e. pseudosymmetric manifolds which are not semi-symmetric. In this paper we give other examples. 614 MOHAMED BELKHELFA, RYSZARD DESZCZ AND LEOPOLD VERSTRAELEN Theorem 2.1.([9]) If a normal contact space is a symmetric one, it is a space of constant curvature one. Theorem 2.2.([15]) A Sasakian manifold is semi-symmetric if and only if it is a space of constant curvature one. Theorem 2.3. Every Sasakian space forms M 2n+1 (c) is pseudo-symmetric, more precisely for every Sasakian space forms: R · R = Q(g, R). Proof. As is well known that the Riemannian curvature tensor of every Sasakian manifold satisfies the following property R(X, ξ)Z = (X ∧ ξ)Z. Therefore R(X, ξ) · R = (X ∧ ξ) · R. (2.5) We remark that in a Sasakian space forms M which is not a real space form, both the left and right hand side of (2.5) are different from zero, so that if M is pseudosymmetric the coefficient function necessarily has to be 1. Then to proof that M is effectively pseudo-symmetric it remains to prove that R(X, Y ) · R = (X ∧ Y ) · R, for all X, Y orthogonal to ξ. First we remark that the Riemannian curvature tensor of M can be written as follows: R(X, Y ) = X ∧ Y + c−1 2 {φ X ∧ φ2 Y + φX ∧ φY + 2g(X, φY )φ}. 4 (2.6) We observe that in any Sasakian manifold: φ · R = 0, as can be verified using the property that R(X, Y ) − X ∧ Y = R(φX, φY ) − φX ∧ φY ([17]) or can be obtained from [3, p.93]. From (2.6) we have thus R(X, Y ) · R = (X ∧ Y ) · R + c−1 {(φ2 X ∧ φ2 Y ) · R + (φX ∧ φY ) · R}. (2.7) 4 Therefore it is needed to prove that for any X, Y perpendicular to ξ, the following holds (φX ∧ φY ) · R = −(X ∧ Y ) · R, which can be verified by a long but straight forward computations. Corollary 2.4. The Sasakian space forms R 2n+1 (−3) is pseudo-symmetric but not semi-symmetric. SYMMETRY PROPERTIES OF SASAKIAN SPACE FORMS 615 Denoting X ∧±φ Y = X ∧ Y ± φX ∧ φY, respectively, we obtain the following characterization of Sasakian space forms. Proposition 2.5. Let M 2n+1 , n ≥ 2, be a Sasakian manifold. Then, (X ∧ +φ Y ) · R = 0 for all X, Y orthogonal to ξ if and only if M 2n+1 is a Sasakian space form. Proof. We already proved that in a Sasakian space forms we have ((X ∧ +φ Y )·R) = 0. Next we assume that (X ∧+φ Y )·R = 0, for all X, Y orthogonal to ξ in M 2n+1 . Assuming that X, Y are orthonormal, by straight-forward computations, we find that ((X ∧+φ Y ) · R)(Y, φX, X, φX) (2.8) = R(X, φX, φX, X) + R(Y, φX, Y, φX) + R(Y, φY, X, φX) + R(Y, φX, X, φY ) ((X ∧+φ Y ) · R)(X, φY, Y, φY ) (2.9) = −R(Y, φY, φY, Y ) − R(X, φY, X, φY ) − R(X, φX, Y, φY ) − R(X, φY, Y, φX). By (2.8), (2.9) and the following property of the Riemannian curvature of any Sasakian manifold: R(φX, Y ) + R(X, φY ) = φX ∧ Y − φY ∧ X. We obtain that: R(X, φX, φX, X) = R(Y, φY, φY, Y ). This implies that M 2n+1 has constant φsectional curvature. Remark 2.6. As it is well-known that, if a Sasakian manifold M 2n+1 , n > 3 is conformally flat or semi-symmetric it is necessary a space of constant curvature, ([9], [15]). It is known (see e.g. [5]) that on every Riemannian manifold of dimension ≥ 5 the conditions R · R = 0 and R · C = 0 are equivalent on the set UC = {x ∈ M | C 6= 0 at x} , 1 κ where C(X, Y ) = R(X, Y ) − n−2 (X ∧g SY + SX ∧g Y − n−1 X ∧g Y ) is the Weyl tensor. Proposition 2.7. Every (2n + 1)-dimensional Sasakian manifold M 2n+1 , n > 1 is Weyl semi-symmetric if and only if it is a space of constant curvature one. 616 MOHAMED BELKHELFA, RYSZARD DESZCZ AND LEOPOLD VERSTRAELEN References [1] M. Belkhelfa, Differential geometry of semi-Riemannian manifolds and submanifolds, Ph.D. thesis, K. U. Leuven, 2001. [2] M. Belkhelfa, R. Deszcz, M. Glogowska, M. Hotloś, D. Kowalczyk and L. Verstraelen, On some type of curvature conditions, Publ. Banach Center, 57(2002), 179-194. [3] D. E. Blair, Contact Manifolds in Riemannian Geometry, Lecture Notes in Math, Springer Verlag, Vol.509, 1976. [4] D. E. Blair, Riemannian Geometry of Contact Manifolds and Symplectic Manifolds, Progress in Mathematics (Birkhäuser Boston), 2002. [5] F. Defever, R. Deszcz and L. Verstraelen, On pseudo-symmetric para-Kähler manifolds, Colloq. Math, 74(1997), 253-260. [6] J. Deprez, R. Deszcz and L. Verstraelen, Pseudo-symmetric curvature conditions on hypersurfaces of Euclidean spaces and on Kählerian manifolds, Ann. Fac. Sci. Univ. Toulouse, 9(1988), 183-192. [7] R. Deszcz, On pseudo-symmetric spaces, Bull. Soc. Math. Belg., Ser. A, 44(1992), 1-34. [8] R. Deszcz and W. Grycak, On manifolds satisfying some curvature conditions, Colloq. Math., 57(1989), 89-92. [9] M. Okumura, Some remarks on space with a certain contact structure, Tohoku Math. J., 14(1962), 135-145. [10] Z. Olszak, Bochner flat Kählerian manifolds with certain condition on the Ricci tensor, Simon Stevin, 63(1989), 295-303. [11] Z. Olszak, On the existence of pseudo-symmetric Kählerian manifolds, Colloq. Math, 95 (2003), 185-189. [12] Z. I. Szabó, Structure theorems on Riemannian spaces satisfying R(X, Y ) · R = 0, J. Differential Geom, 17: I, The local version (1982), 531-582 [13] Z. I. Szabó, Classification and construction of complete hypersurfaces satisfying R(X, Y ) · R = 0, Acta Sci. Math, 47(1984), 321-348. [14] Z. I. Szabó, Structure theorems on Riemannian spaces satisfying R(X, Y ) · R = 0, Geom. Dedicata, 19: II, Global version (1985), 65-108. [15] T. Takahashi, Sasakian φ symmetric spaces, Tohoko Math. J., 29(1977), 91-113. [16] L. Verstraelen, Comments on pseudo-symmetry in sense of R. Deszcz, in: Geometry and Topology of Submanifolds, World Sci. Singapore, 6(1994), 199-209. [17] K. Yano and M. Kon, Structures on Manifolds, In Pure Mathematics, Vol. 3, World Scientific, P.O. Box 128, Farrer Road, Singapore 9128, 1984. Centre Universitaire Mustapha Stombouli, Laboratoire de Physique Quantique de la Matiere et Modelisations Mathmatiques - L.P.Q. 3M - Route de Mamounia, Mascara 29000 (Algerie). E-mail: [email protected] Department of Mathematics, Agricultural University of Wroclaw, Grunwaldzka 53, PL - 50-357 Wroclaw, Poland. E-mail: [email protected] Department of Mathematics, K.U. Leuven, Celestijnenlaan 200B Heverlee Belgium. E-mail: [email protected]