Warped product submanifolds of Kaehler manifolds with pointwise slant fiber

Abstract

It was shown in [15,16] that there does not exist any warped product submanifold of a Kaehler manifold such that the spherical manifold of the warped product is proper slant. In this paper, we introduce the notion of warped product submanifolds with a slant function. We show that there exists a class of nontrivial warped product submanifolds of a Kaehler manifold such that the spherical manifold is pointwise slant by giving an example and a characterization theorem. We also prove that if the warped product is mixed totally geodesic then the warping function is constant.

Characterization theorems on warped product semi-slant submanifolds in Kenmotsu manifolds

“…Thus, by Lemma 4.4 (i)-(ii) and the relation (23) (ii), we get the desired result (26) with λ = ln f . Conversely, if M is a semi-slant submanifold of a Kenmotsu manifoldM such that the given condition (26) holds, then we have…”

Bi-warped product submanifolds of Kenmotsu manifolds and their applications

“…Using Lemma 4(i) and relations (18)- (24) in first two terms in the right-hand side of above inequality and using Lemma 4(ii) and (25) in the last two terms, we derive…”

Chen Inequalities for Warped Product Pointwise Bi-Slant Submanifolds of Complex Space Forms and Its Applications

“…If θ 1 = 0 or θ 2 = 0, in this case, M n is a coinciding pointwise semi-slant submanifold (see [14,34]).…”

“…From the motivation studied in [14,34], we present the following consequence of Theorem 2 by using the Remark 2 for a nontrivial warped product pointwise pseudo-slant submanifold of a complex space, such that:…”

“…The role of immersibility and non-immersibility in studying the submanifolds geometry of a Riemannian manifold was affected by the pioneering work of the Nash embedding theorem [1], where every Riemannian manifold realizes an isometric immersion into a Euclidean space of sufficiently high codimension. This becomes a very useful object for the submanifolds theory, and was taken up by several authors (for instance, see [2][3][4][5][6][7][8][9][10][11][12][13][14][15]). Its main purpose was considered to be how Riemannian manifolds could always be treated as Riemannian submanifolds of Euclidean spaces.…”