tailieunhanh - Slant submanifolds of Kaehler product manifolds

In this paper, we study slant submanifolds of a Kaehler product manifold. We show that an F-invariant slant submanifold of Kaehler product manifold is a product manifold. We also obtain some curvature inequalities in terms of scalar curvature and Ricci tensor. | Turk J Math 31 (2007) , 65 – 77. ¨ ITAK ˙ c TUB Slant submanifolds of Kaehler Product Manifolds Bayram S ¸ ahin and Sadık Kele¸s Abstract In this paper, we study slant submanifolds of a Kaehler product manifold. We show that an F -invariant slant submanifold of Kaehler product manifold is a product manifold. We also obtain some curvature inequalities in terms of scalar curvature and Ricci tensor. Key Words: Slant Submanifold, Kaehler Manifold, Ricci tensor. 1. Introduction Submanifolds of a Kaehler manifold are defined with respect to the behaviour of complex structure J. More precisely, a real submanifold M of a Kaehler manifold is called invariant if J(T M ) = T M , where T M denotes the tangent bundle of M . M is called totally real if J(T M ) ⊂ T M ⊥ and M is called CR-submanifold [1] if there are orthogonal complement two distributions D⊥ , D such that D is invariant and D⊥ is totally real. Recently, B. Y. Chen introduced slant submanifolds as follows: Let M be a ¯ for each non zero vector X ∈ Tp M , we denote the submanifold of a Kaehler manifold M, angle between JX and Tp M by θ(X). Then M is said to be slant ([2]) if the angle θ(X) is constant, ., it is independent of the choice of p ∈ M and X ∈ Tp M. The angle θ of a slant immersion is called the slant angle of the immersion. Invariant and anti-invariant immersions are slant immersions with slant angle θ = 0 and θ = π2 , respectively. A proper slant immersion is neither invariant nor anti-invariant. 2000 AMS Mathematics Subject Classification: 53C15, 53C42 ˙ on¨ This study was supported by In¨ u University, Project No. 2002/8. 65 ˙ KELES S¸AHIN, ¸ The geometry of submanifolds of a Kaehler manifold has been investigated by many authors. In [6], and M. Kon studied the geometry of F -invariant and F -antiinvariant submanifolds of Kaehler product manifolds and showed that an F invariant, invariant submanifold of a Kaehler product manifold is also a product manifold. Same result was obtained