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Warped product semi-slant submanifolds in Kenmotsu manifolds

In this paper, we research the existence or non-existence of warped product semi-slant submanifolds in Kenmotsu manifolds. Consequently, we see that there are no proper warped product semi-slant submanifolds in Kenmotsu manifolds such that totally geodesic and totally umbilical submanifolds of warped product are proper semi-slant and invariant (or anti-invariant), respectively. | Turk J Math 34 (2010) , 425 – 432. ¨ ITAK ˙ c TUB doi:10.3906/mat-0901-6 Warped product semi-slant submanifolds in Kenmotsu manifolds Mehmet At¸ceken Abstract In this paper, we research the existence or non-existence of warped product semi-slant submanifolds in Kenmotsu manifolds. Consequently, we see that there are no proper warped product semi-slant submanifolds in Kenmotsu manifolds such that totally geodesic and totally umbilical submanifolds of warped product are proper semi-slant and invariant (or anti-invariant), respectively. Key Words: Warped product, Slant submanifold and Kenmotsu manifold. 1. Introduction In [2], the notion of warped product manifolds was introduced by Bishop and O’Neill in 1969 and it was studied by many mathematicians and physicists. These manifolds are generalization of Riemannian product manifolds. Also, the notion of slant submanifolds in a complex manifold was defined and studied by B-Y. Chen as a natural generalization of both invariant and anti-invariant submanifolds. Examples of slant submanifolds of complex Euclidean space C 2 and C 4 were given by B-Y. Chen[7]. Moreover, A. Lotta has defined and studied of slant immersions of a Riemannian manifold into an almost contact metric manifold and proved some properties of such immersions [12]. In [3, 4], Authors studied slant immersions in K-contact and Sasakian manifolds. They introduced many interesting examples of slant submanifolds in almost contact metric manifolds and Sasakian manifolds. They characterized slant submanifolds by means of the covariant derivative of the square of the tangent projection T over the submanifold of almost contact structure of a K-contact manifold. In [9], Authors studied slant submanifolds of a Kenmotsu manifold and gave a necessary and sufficient condition for a 3-dimensional submanifold of a 5-dimensional Kenmotsu manifold to be minimal proper slant submanifold. Recently, we have studied warped product semi-slant submanifolds in Locally .