A note on quasi-bi-slant submanifolds of Sasakian manifolds

The object of the present paper is to study the notion of quasi-bi-slant submanifolds of almost contact metric manifolds as a generalization of slant, semi-slant, hemi-slant, bi-slant, and quasi-hemi-slant submanifolds. We study and characterize quasi-bi-slant submanifolds of Sasakian manifolds and provide non-trivial examples to signify that the structure presented in this paper is valid. Furthermore, the integrability of distributions and geometry of foliations are researched. Moreover, we characterize quasi-bi-slant submanifolds with parallel canonical structures.

New types of almost contact metric submersions

We introduce the concept of conjugaison in contact geometry. This concept allows to define new structures which are used as base space of a Riemannian submersion. With these new structures, we study new three types of almost contact metric submersions.

ON A CERTAIN TRANSFORMATION IN ALMOST CONTACT METRIC MANIFOLDS

In this work, we investigate a new deformations of almost contact metric manifolds. New relations between classes of 3-dimensional almost contact metric have been discovered. Several concrete examples are discussed.

Real hypersurfaces of non-flat complex space forms with two generalized conditions on the Jacobi structure operator

We aim to classify the real hypersurfaces M in a Kaehler complex space form Mn (c) satisfying the two conditions φ l = l φ , $\varphi l=l\varphi ,$ where l = R ( ⋅ , ξ ) ξ and φ $l=R(\cdot ,\xi )\xi \text{ and }\varphi $ is the almost contact metric structure of M, and ( ∇ ξ l ) X = $\left( {{\nabla }_{\xi }}l \right)X=$ ω(X)ξ, where where ω(X) is a 1-form and X is a vector field on M. These two conditions imply that M is a Hopf hypersurface and ω = 0.

Optimal inequalities for contact CR-submanifolds in almost contact metric manifolds

In [An optimal inequality for CR-warped products in complex space forms involving CR δ-invariant, Internat. J. Math. 23(3) (2012)], B.-Y. Chen introduced the CR δ-invariant for CR-submanifolds. Then, in [Two optimal inequalities for anti-holomorphic submanifolds and their applications, Taiwan. J. Math. 18 (2014), 199–217], F. R. Al-Solamy, B.-Y. Chen and S. Deshmukh proved two optimal inequalities for anti-holomorphic submanifolds in complex space forms involving the CR δ-invariant. In this paper, we obtain optimal inequalities for this invariant for contact CR-submanifolds in almost contact metric manifolds.

A Study of New Class of Almost Contact Metric Manifolds of Kenmotsu Type

In this paper, we characterized a new class of almost contact metric manifolds and established the equivalent conditions of the characterization identity in term of Kirichenko’s tensors. We demonstrated that the Kenmotsu manifold provides the mentioned class; i.e., the new class can be decomposed into a direct sum of the Kenmotsu and other classes. We proved that the manifold of dimension 3 coincided with the Kenmotsu manifold and provided an example of the new manifold of dimension 5, which is not the Kenmotsu manifold. Moreover, we established the Cartan’s structure equations, the components of Riemannian curvature tensor and the Ricci tensor of the class under consideration. Further,the conditions required for this to be an Einstein manifold have been determined.

∇N-EINSTEIN ALMOST CONTACT METRIC MANIFOLDS

On an almost contact metric manifold M, an N-connection ∇N defined by the pair (∇,N), where ∇ is the interior metric connection and N: TМ → TM is an endomorphism of the tangent bundle of the manifold M such that Nξ = 0, 􀁇 􀁇 N (D) ⊂ D , is considered. Special attention is paid to the case of a skew-symmetric N-connection ∇N, which means that the torsion of an N-connection considered as a trivalent covariant tensor is skew-symmetric. Such a connection is uniquely defined and corresponds to the endomorphism N = 2ψ, where the endomorphism ψ is defined by the equality ω( X ,Y ) = g (ψX ,Y ) and is called in this work the second structure endomorphism of an almost contact metric manifold. The notion of a ∇N-Einstein almost contact metric manifold is introduced. For the case N = 2ψ, conditions under which almost contact manifolds are ∇N-Einstein manifolds are found.

B.-Y. Chen’s inequality for pointwise CR-slant warped products in cosymplectic manifolds

Recently, pointwise CR-slant warped products introduced by Chen and Uddin in [14] for Kaehler manifolds. In the context of almost contact metric manifolds, in this paper, we study these submanifolds in cosymplectic manifolds. We investigate the geometry of such warped product and prove establish a lower bound relation between the second fundamental form and warping function. The equality case is also investigated.