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

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 Research Article Turk J Math (2013) 37: 340 – 347 ¨ ITAK ˙ c TUB doi: 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: • Accepted: • Published Online: • Printed: 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

• Toán học
• Contact 3 structure
• Warped product
• Contact QR warped product
• Sasakian space form
• Cauchy Riemann manifold