Abstract
In this paper, we study α-cosymplectic manifoldMadmitting∗-Ricci tensor. First, it is shown that a∗-Ricci semisymmetric manifoldMis∗-Ricci flat and aϕ-conformally flat manifoldMis anη-Einstein manifold. Furthermore, the∗-Weyl curvature tensorW∗onMhas been considered. Particularly, we show that a manifoldMwith vanishing∗-Weyl curvature tensor is a weakϕ-Einstein and a manifoldMfulfilling the conditionRE1,E2⋅W∗=0isη-Einstein manifold. Finally, we give a characterization for α-cosymplectic manifoldMadmitting∗-Ricci soliton given as to be nearly quasi-Einstein. Also, some consequences for three-dimensional cosymplectic manifolds admitting∗-Ricci soliton and almost∗-Ricci soliton are drawn.
Highlights
In the last few years, theory of almost contact geometry and related topics are an active branch of research due to elegant geometry and applications to physics
Some generalizations of Einstein manifolds have been defined in the literature, and there have been obtained some applications of these kinds of manifolds in theoretical physics
Contact manifolds are special Riemann manifolds with almost contact structures
Summary
In the last few years, theory of almost contact geometry and related topics are an active branch of research due to elegant geometry and applications to physics. A simple example of almost cosymplectic manifolds is given by the products of almost Kaehler manifolds and the real line R or the circle S1. Similar to a complex case, the ∗ -Ricci tensor of an almost contact metric manifold has been defined as follows: S∗. E concept of the ∗ -Ricci tensor has been studied in contact case. Venkatesha and his group ([9, 10]) recently studied some of the curvature properties on Sasakian manifold and contact metric generalized (κ, μ)-space form using the ∗ -Ricci tensor. An α-cosymplectic manifold satisfying ∗ -Ricci semisymmetric and φ-conformally flat conditions are studied in Section 3 and shown that a φ-conformably flat α-cosymplectic manifold is η-Einstein. We have proved some important results of α-cosymplectic manifold admitting ∗ -Ricci soliton
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