The objective of the present paper is to study weakly Ricci symmetric spacetimes. Among others, we prove that a weakly Ricci symmetric spacetime obeying Einstein’s field equation without cosmological constant represents stiff matter. Moreover, it is shown that the local cosmological structure of a weakly Ricci symmetric perfect fluid spacetime can be identified as Petrov type [Formula: see text], [Formula: see text] or [Formula: see text]. Next, we prove that a dust and dark fluid weakly Ricci symmetric spacetime satisfying Einstein’s field equation without cosmological constant is vacuum. Finally, we show the non-existence of radiation era in such a spacetime.
The object of the present paper is to characterize two classes of almost Kenmotsu manifolds admitting Ricci-Yamabe soliton. It is shown that a (k, µ) ′ -almost Kenmotsu manifold admitting a Ricci-Yamabe soliton or gradient Ricci-Yamabe soliton is locally isometric to the Riemannian product H n+1 (−4) × R n . For the later case, the potential vector field is pointwise collinear with the Reeb vector field. Also, a (k, µ)-almost Kenmotsu manifold admitting certain Ricci-Yamabe soliton with the curvature property Q · P = 0 is locally isometric to the hyperbolic space H 2n+1 (−1) and the non-existense of the curvature property Q · R = 0 is proved.Mathematics Subject Classification 2010: 53D15, 35Q51.
We consider η-Ricci solitons on Lorentzian para-Sasakian manifolds with Codazzi type of the Ricci tensor. Then we study η-Ricci solitons on ϕ-conformally semi-symmetric, ϕ-Ricci symmetric, and conformally Ricci semi-symmetric Lorentzian para-Sasakian manifolds. Finally, we construct an example of a three dimensional Lorentzian para-Sasakian manifold which admits η-Ricci solitons with non-constant scalar curvature. 2010 Mathematics Subject Classification. 53C15, 53C25. Key words and phrases. Ricci solitons; η-Ricci solitons; Lorentzian para-Sasakian manifolds; Codazzi type of the Ricci tensor. Debabrata Kar is supported by the Council of Scientific and Industrial Research, India (File no. 09/028(1007)/2017-EMR-1).
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