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 K¨ ahlerian manifolds of dimension n ≥ 6 are the semi-symmetric ones ([5], [6]). A similar result was obtained for Ricci pseudo- symmetry ([10]). Therefore Olszak introduced a variant of pseudo-symmetric K¨ ahlerian manifolds [10]. For dimension 4 Olszak gave an example of non semi- symmetric pseudo-symmetric K¨ ahlerian 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: φ 2 (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 φ 2 [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 4 (g(Y,Z )X − g(X, Z )Y ) 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