Almost contact metric manifolds with certain condition

Ukrainian Mathematical Journal, 2013

The object of the present paper is to study a transformation called the D-homothetic deformation of normal almost contact metric manifolds. In particular, it is shown that, in a (2n + 1)-dimensional normal almost contact metric manifold, the Ricci operator Q commutes with the structure tensor φ under certain conditions, and the operator Qφ − φQ is invariant under a D-homothetic deformation. We also discuss the invariance of η-Einstein manifolds, φ-sectional curvature, and the local φ-Ricci symmetry under a D-homothetic deformation. Finally, we prove the existence of such manifolds by a concrete example. Метою цiєї статтi є вивчення перетворення, що називається D-гомотетичною деформацiєю нормальних майже контактних многовидiв. Зокрема, показано, що у (2n + 1)-вимiрному нормальному майже контактному многовидi оператор Рiччi Q комутує за певних умов iз структурним тензором φ, а оператор Qφ − φQ є iнварiантним щодо Dгомотетичної деформацiї. Також розглянуто питання про iнварiантнiсть η-ейнштейнiвських многовидiв, φ-секцiйну кривину та локальну φ-симетрiю Рiччi при D-гомотетичнiй деформацiї. Iснування таких многовидiв доведено на конкретному прикладi.

Communications of the Korean Mathematical Society, 2009

The object of the present paper is to study 3-dimensional normal almost contact metric manifolds satisfying certain curvature conditions. Among others it is proved that a parallel symmetric (0, 2) tensor field in a 3-dimensional non-cosympletic normal almost contact metric manifold is a constant multiple of the associated metric tensor and there does not exist a non-zero parallel 2-form. Also we obtain some equivalent conditions on a 3-dimensional normal almost contact metric manifold and we prove that if a 3-dimensional normal almost contact metric manifold which is not a β-Sasakian manifold satisfies cyclic parallel Ricci tensor, then the manifold is a manifold of constant curvature. Finally we prove the existence of such a manifold by a concrete example.

2011

Certain curvature properties and scalar invariants of the manifolds belonging to one of the main classes almost contact manifolds with Norden metric are considered. An example illustrating the obtained results is given and studied.

2021

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.

Trends in Differential Geometry, Complex Analysis and Mathematical Physics - Proceedings of 9th International Workshop on Complex Structures, Integrability and Vector Fields, 2009

In this paper we study submanifolds of almost contact manifolds with Norden metric of codimension two with totally real normal spaces. Examples of such submanifolds as a Lie subgroups are constructed.

It is introduced a differentiable manifold with almost contact 3-structure which consists of an almost contact metric structure and two almost contact B-metric structures. The corresponding classifications are discussed. The product of this manifold and a real line is an almost hypercomplex manifold with Hermitian-Norden metrics. The vanishing of the Nijenhuis tensors and their associated tensors is considered. It is proven that the introduced manifold of cosymplectic type is flat. Some examples of the studied manifolds are given.

Tamkang Journal of Mathematics, 2021

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.

Comptes Rendus De L Academie Bulgare Des Sciences Sciences Mathematiques Et Naturelles, 2011

Almost contact manifolds with B-metric are considered. Of special interest are the so-called vertical classes of the almost contact B-metric manifolds. Curvature properties of these manifolds are studied. An example of 5-dimensional manifolds is constructed and characterized.

2011

Almost contact manifolds with B-metric are considered. A special linear connection is introduced, which preserves the almost contact B-metric structure on these manifolds. This connection is investigated on some classes of the considered manifolds.

Georgian Mathematical Journal

We study the conformal curvature tensor and the contact conformal curvature tensor in Sasakian and/or K-contact manifolds. We find a necessary and sufficient condition for a Sasakian manifold to be ϕ-conformally flat. We also find some necessary conditions for a K-contact manifold to be ϕ-contact conformally flat. Then we give a structure theorem for ϕ-contact conformally flat Sasakian manifolds. It is also proved that a Sasakian manifold cannot be ξ-contact conformally flat.