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A condition for warped product semi-invariant submanifolds to be Riemannian product semi-invariant submanifolds in locally Riemannian product manifolds

In this article, we give a necessary and sufficient condition for warped product semi-invariant submanifolds to be Riemannian product semi-invariant submanifolds in a locally Riemannian product manifold whose factor manifolds are real space form. | Turk J Math 32 (2008) , 349 – 362. ¨ ITAK ˙ c TUB A Condition for Warped Product Semi-Invariant Submanifolds to be Riemannian Product Semi-Invariant Submanifolds in Locally Riemannian Product Manifolds Mehmet At¸ceken Abstract In this article, we give a necessary and sufficient condition for warped product semi-invariant submanifolds to be Riemannian product semi-invariant submanifolds in a locally Riemannian product manifold whose factor manifolds are real space form. Key Words: Real space form, Riemannian product and Warped product. 1. Introduction It is well-known that the notion of warped products plays some important role in dif- ferential geometry as well as physics. The geometry of warped product was introduced by B.Y. Chen and it has been studied in the different manifold types by many geometers[see references]. Recently, B.Y. Chen have introduced the notion of CR-warped product in Kaehlerian manifolds and showed that there exist no warped product CR-submanifolds in the form M = M⊥ ×f MT in Kaehlerian manifolds. Therefore, he considered warped product CRsubmanifolds in the form M = MT ×f M⊥ , which is called CR-warped product, where 2000 AMS Mathematics Subject Classification: 53C15, 53C42 349 ATC ¸ EKEN MT is an invariant submanifold and M⊥ is an anti-invariant submanifold of Kaehlerian ¯ [5]. manifold M We note that CR-warped products in Kaehlerian manifold correspond to semi-invariant warped products in Riemannian product manifolds. Recently, we showed that there exist no warped product semi-invariant submanifolds in the form M = MT ×f M⊥ in contrast to Kaehlerian manifolds [1, 5]. So, in the remainder of this paper we consider warped product semi-invariant submanifolds in the form M = M⊥ ×f MT , where M⊥ is an anti-invariant submanifold and MT is an invariant submanifold of Riemannian product manifold M and it is called warped product semi-invariant submanifold in the rest of this paper. 2. Preliminaries ¯ be a Riemannian manifold and M be an