Almost contact structures and curvature tensors (original) (raw)

Manifolds with almost contact 3-structure and metrics of Hermitian-Norden type

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.

A new curvaturelike tensor field in an almost contact Riemannian manifold II

Publications de l'Institut Math?matique (Belgrade), 2018

In the last paper, we introduced a new curvaturlike tensor field in an almost contact Riemannian manifold and we showed some geometrical properties of this tensor field in a Kenmotsu and a Sasakian manifold. In this paper, we define another new curvaturelike tensor field, named (CHR)3-curvature tensor in an almost contact Riemannian manifold which is called a contact holomorphic Riemannian curvature tensor of the second type. Then, using this tensor, we mainly research (CHR)3-curvature tensor in a Sasakian manifold. Then we define the notion of the flatness of a (CHR)3-curvature tensor and we show that a Sasakian manifold with a flat (CHR)3-curvature tensor is flat. Next, we introduce the notion of (CHR)3-?-Einstein in an almost contact Riemannian manifold. In particular, we show that Sasakian (CHR)3- ?-Einstein manifold is ?-Einstain. Moreover, we define the notion of (CHR)3- space form and consider this in a Sasakian manifold. Finally, we consider a conformal transformation of an ...

Almost Hermitian structures on the products of two almost contact metric manifolds

In this paper, we study the almost Hermitian structure on the product of two almost contact metric manifolds. We give some properties that each factor should satisfy to make the almost Hermitian structure on the product manifold in a certain class of almost Hermitian manifolds. In addition, opposite to Chinea-Gonzales, we show that semi-cosymplectic manifolds do not contain the class C 12. M.S.C. 2010: 53D10, 57R15.

Structures on the Product of Two Almost Hermitian Almost Contact Manifolds

International electronic journal of geometry, 2016

The purpose of this paper is to define some classes of almost contact metric 3-structures manifolds and almost quaternionic metric with an almost Hermitian almost contact metric structure. Next, we construct an almost quaternionic Hermitian structure on the product of two almost Hermitian almost contact metric structures. This gives a new positive answer to a question raised by T. Tshikuna-Matamba [7].

ALMOST CONTACT B-METRIC MANIFOLDS WITH CURVATURE TENSORS OF KÄHLER TYPE

On 5-dimensional almost contact B-metric manifolds, the form of any ϕ-Kähler-type tensor (i. e. a tensor satisfying the properties of the curvature tensor of the Levi-Civita connection in the special class of the parallel structures on the manifold) is determined. The associated 1-forms are derived by the scalar curvatures of the ϕ-Kähler-type tensor for the ϕ-canonical connection on the manifolds from the main classes with closed associated 1-forms.

Curvature Properties on Some Classes of Almost Contact Manifolds with B-Metric

arXiv: Differential Geometry, 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 In this work 1 we continue the investigations on a manifold M with an almost contact structure (',�,�) which is equipped with a B-metric g, i.e. a pseudo-Riemannian metric of signature (n,n+1) with the opposite compat- ibility of the metric with the structure in comparison with the known almost contact metric structure. Moreover, the B-metric is an odd-dimensional ana- logue of the Norden metric on almost complex manifolds. Recently, manifolds with neutral metrics and various tensor structures have been object of interest in theoretical physics. The classes of the almost contact B-metric manifolds from the so-called vertical component is an object of special interest in this paper. The goal of the present work is the investigation of some p...

Pseudohermitian geometry on contact Riemannian manifolds

2002

Starting from work by S. Tanno, [39], and E. Barletta et al., [3], we study the geometry of (possibly non integrable) almost CR structures on contact Riemannian manifolds. We characterize CR-pluriharmonic functions in terms of differential operators naturally attached to the given contact Riemannian structure. We show that the almost CR structure of a contact Riemannian manifold (M, η) admitting global 276 DAVID E. BLAIR – SORIN DRAGOMIR [2] nonzero closed sections (with respect to which η is volume normalized) in the canonical bundle is integrable and η is a pseudo-Einstein contact form. The pseudohermitian holonomy of a Sasakian manifold M is shown to be contained in SU(n)× 1 if and only if the Tanaka-Webster connection is Ricci flat. Also, for any quaternionic Sasakian manifold (M, (F, T, θ, g)) either the Tanaka-Webster connection of (M, θ) is Ricci flat or m = 1 and then (M, θ) is pseudo-Einstein if and only if 4p + ρ∗ θ is closed, where p is a local 1-form on M such that ∇G = ...