On contact metric manifolds

Kodai Mathematical Journal, 1990

2012

The object of the present paper is to study three-dimensional general- ized (k; ) contact metric manifolds with recurrent Ricci tensor and harmonic curvature tensor. Ricci symmetric generalized (k; ) contact metric manifolds of dimension three are also considered. Each sections are followed by examples to illustrate the obtained results. 2000 Mathematics Subject Classication : 53C15, 53C40.

2007

In the present study, we considered 3-dimensional generalized (•;")-contact metric manifolds. We proved that a 3-dimensional gener- alized (•;")-contact metric manifold is not locally `-symmetric in the sense of Takahashi. However such a manifold is locally `-symmetric pro- vided thatand " are constants. Also it is shown that if a 3-dimensional generalized (•;") -contact metric manifold is Ricci-symmetric, then it is a (•;")-contact metric manifold. Further we investigated certain condi- tions under which a generalized (•;")-contact metric manifold reduces to a (•;")-contact metric manifold. Then we obtain several necessary and su-cient conditions for the Ricci tensor of a generalized (•;")-contact metric manifold to be ·-parallel. Finally, we studied Ricci-semisymmetric generalized (•;")-contact metric manifolds and it is proved that such a manifold is either ∞at or a Sasakian manifold.

2011

We study the Riemann curvature tensor of (\kappa,\mu,\nu)-contact metric manifolds, which we prove to be completely determined in dimension 3, and we observe how it is affected by D_a-homothetic deformations. This prompts the definition and study of generalized (\kappa,\mu,\nu)-space forms and of the necessary and sufficient conditions for them to be conformally flat.

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.

2015

We study the Riemann curvature tensor of (κ, µ, ν)-contact metric manifolds, which we prove to be completely determined in dimension 3, and we observe how it is affected by Da-homothetic deformations. This prompts the definition and study of generalized (κ, µ, ν)-space forms and of the necessary and sufficient conditions for them to be conformally flat.Ministerio de Educación y CienciaFondo Europeo de Desarrollo RegionalPlan Andaluz de Investigación (Junta de Andalucía

In the present study, we considered 3-dimensional generalized (κ, µ)-contact metric manifolds. We proved that a 3-dimensional generalized (κ, µ)-contact metric manifold is not locally φ-symmetric in the sense of Takahashi. However such a manifold is locally φ-symmetric provided that κ and µ are constants. Also it is shown that if a 3-dimensional generalized (κ, µ)-contact metric manifold is Ricci-symmetric, then it is a (κ, µ)-contact metric manifold. Further we investigated certain conditions under which a generalized (κ, µ)-contact metric manifold reduces to a (κ, µ)-contact metric manifold. Then we obtain several necessary and sufficient conditions for the Ricci tensor of a generalized (κ, µ)-contact metric manifold to be η-parallel. Finally, we studied Ricci-semisymmetric generalized (κ, µ)-contact metric manifolds and it is proved that such a manifold is either flat or a Sasakian manifold. M.S.C. 2000: 53C15, 53C05, 53C25.

Tamkang Journal of …, 2008

Abstract. We prove that a (k,µ)-manifold with vanishing Endo curvature tensor is a Sasakian manifold. We find a necessary and sufficient condition for a non-Sasakian (k,µ)-manifold whose Endo curvature tensor Bes satisfies Bes (ξ, X ) · S = 0, where S is the Ricci tensor. Using ...

Journal of Geometry, 2013

Monatshefte für Mathematik, 2015

We study the Riemann curvature tensor of (κ, µ, ν)-contact metric manifolds, which we prove to be completely determined in dimension 3, and we observe how it is affected by Da-homothetic deformations. This prompts the definition and study of generalized (κ, µ, ν)-space forms and of the necessary and sufficient conditions for them to be conformally flat.