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In this note, the generalized Lorentzian Sasakian-space-form M 1 2 n + 1 (f 1 , f 2 , f 3) satisfying certain constraints on the M -projective curvature tensor W * is considered. Here, we characterize the structure M 1 2 n + 1 (f 1 , f 2 , f 3) when it is, respectively, M -projectively flat, M -projectively semisymmetric, M -projectively pseudosymmetric, and φ − M -projectively semisymmetric. Moreover, M 1 2 n + 1 (f 1 , f 2 , f 3) satisfies the conditions W * (ζ , V 1) · S = 0 , W * (ζ , V 1) · R = 0 and W * (ζ , V 1) · W * = 0 are also examined. Finally, illustrative examples are given for obtained results. [ABSTRACT FROM AUTHOR]

The object of this paper is to study (∈)-Lorentzian para-Sasakian manifolds. Some typical identities for the curvature tensor and the Ricci tensor of (∈)-Lorentzian para-Sasakian manifold are investigated. Further, we study globally φ-Ricci symmetric and weakly φ-Ricci symmetric (∈)-Lorentzian para-Sasakian manifolds and obtain interesting results. [ABSTRACT FROM AUTHOR]

In this paper, we study α-cosymplectic manifold M admitting ∗ -Ricci tensor. First, it is shown that a ∗ -Ricci semisymmetric manifold M is ∗ -Ricci flat and a ϕ -conformally flat manifold M is an η -Einstein manifold. Furthermore, the ∗ -Weyl curvature tensor W ∗ on M has been considered. Particularly, we show that a manifold M with vanishing ∗ -Weyl curvature tensor is a weak ϕ -Einstein and a manifold M fulfilling the condition R E 1 , E 2 ⋅ W ∗ = 0 is η -Einstein manifold. Finally, we give a characterization for α-cosymplectic manifold M admitting ∗ -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. [ABSTRACT FROM AUTHOR]

In this paper we study NMCP manifolds, which are (2p+2q+2)--dimensional differential manifolds with an normal metric contact pair structure. The aim of the study is to examine the Riemannian geometry of normal metric contact pair manifolds under certain conditions related to the *--Ricci tensor. We prove that a *--Ricci-semi-symmetric NMCP manifold is *--Ricci-flat, and that a *--generalized quasi-Einstein NMCP manifold cannot be *--Ricci-semi-symmetric. Finally, we consider the concircular curvature tensor on NMCP manifolds and prove that a concircular flat NMCP manifold is locally isometric to (2p+2q+2)-dimensional hyperbolic space. [ABSTRACT FROM AUTHOR]