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Contact 3-structure QR-warped product submanifold in Sasakian space form

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In the present paper we obtain sharp estimates for the squared norm of the second fundamental form in terms of the mapping function for contact 3 -structure CR-warped products isometrically immersed in Sasakian space form. | Turkish Journal of Mathematics http://journals.tubitak.gov.tr/math/ Research Article Turk J Math (2013) 37: 340 – 347 ¨ ITAK ˙ c TUB doi:10.3906/mat-1102-23 Contact 3-structure QR-warped product submanifold in Sasakian space form Esmaiel ABEDI∗, Ghorbanali HAGHIGHATDOOST, Muhammad ILMAKCHI, Zahra NAZARI Department of Mathematics, Azarbaijan University of Tarbiat Moallem, Tabriz 53751 71379, Iran Received: 12.02.2011 • Accepted: 08.10.2011 • Published Online: 19.03.2013 • Printed: 22.04.2013 Abstract: In the present paper we obtain sharp estimates for the squared norm of the second fundamental form in terms of the mapping function for contact 3 -structure CR-warped products isometrically immersed in Sasakian space form. Key words: Warped product, contact QR-warped product, Sasakian space form 1. Introduction ˜ be a hermitian manifold and denoted by J the almost complex structure on M ˜ . Yano and Ishihara (see Let M [13]) considered a submanifold M whose tangent bundle T M splits into a complex subbundle D and a totally real subbundle D⊥ . Later, such a submanifold was called a CR-submanifold [4],[3]. Blair and Chen [4] proved that a CR-submanifold of a locally conformal Ka¨ aler manifold is a Cauchy-Riemann manifold in the sense of Greenfield. Recently, Chen [5] introduced the notion of a CR-warped product submanifold in a Ka¨ aler manifold. He established a sharp relationship between the mapping function f of a warped product CR-submanifold M1 ×f M2 ˜ and the squared norm of the second fundamental form h [5]. of a Ka¨ aler manifold M In 1971, Kenmotsu [7] introduced a class of almost contact metric manifolds, called Kenmotsu manifold, ¨ ur [10] which is not Sasakian. Kenmotsu manifolds have been studied by several authors such as Piti¸s [12], Ozg¨ ¨ ur and De [11]. and Ozg¨ ¯ (n+p) 4 be a quaternionic Ka¨ aler manifold with real dimension of n + p. Let M be an n-dimensional Let M ¯ (n+p) 4 QR-submanifold of QR dimension (p − 3) isometrically immersed