Another class of warped product CR-submanifolds in Kenmotsu manifolds

Abstract

Recently, Arslan et al. [K. Arslan, R. Ezentas, I. Mihai, C. Murathan, J. Korean Math. Soc., 42 (2005), 1101-1110] studied contact CR-warped product submanifolds of the form M T × f M ⊥ of a Kenmotsu manifold M, where M T and M ⊥ are invariant and anti-invariant submanifolds of M, respectively. In this paper, we study the warped product submanifolds by reversing these two factors, i.e., the warped products of the form M ⊥ × f M T which have not been considered in earlier studies. On the existence of such warped products, a characterization is given. A sharp estimation for the squared norm of the second fundamental form is obtained, and in the statement of inequality, the equality case is considered. Finally, we provide two examples of non-trivial warped product submanifolds.

Another inequality for skew CR-warped products in Kenmotsu manifolds

“…1. If dim(M θ ) = 0 in a contact skew CR-warped product, then it reduces to contact CR-warped products of the form M = M ⊥ × f M T studied in [32]. In this case, the statement of Theorem 3.5 will be: where ∇ ⊥ (ln f ) is the gradient of ln f along M ⊥ .…”

Warped product submanifolds of Kenmotsu manifolds with slant fiber

“…If we assume θ = 0 in Theorem 3.5, then the warped product becomes M = M ⊥ × f M T of a Kenmotsu manifoldM, where M T and M ⊥ are invariant and anti-invariant submanifolds ofM, respectively, which is a case of warped product contact CR-submanifolds which have been studied in [31]. Thus, Theorem 3.1 of [31] is a special case of Theorem 3.5.…”

“…Also, if we consider θ = 0 in Theorem 4.1, then the warped product is of the form M = M ⊥ × f M T of a KenmotsuM, where M T and M ⊥ are invariant and anti-invariant submanifolds ofM, respectively and hence the inequality (23) will be h 2 ≥ 2p ∇(ln f ) 2 − 1 . Thus, Theorem 3.2 of[31] is again a special case of Theorem 4.1.…”

A class of warped product submanifolds of Kenmotsu manifolds

“…Then many authors studied warped product submanifolds of different ambient manifolds, see [16,17,19]. Warped product submanifolds of Kenmotsu manifolds are studied in ( [1][2][3], [21,22,25,26], [32][33][34][35]).…”

“…Theorem 3.3 of[35]) Let M be a contact CR-submanifold ofM such that ξ is orthogonal to D T . Then M is locally a warped product submanifold of the form M ⊥ × f M T if and only ifA φZ X = −{(Zµ) − η(Z)}φX for any X ∈ Γ(D T ) and Z ∈ Γ(D ⊥ ⊕ {ξ}) ,where µ is any smooth function on M such that Y (µ) = 0 , Y ∈ Γ(D T ).…”