We consider η-Ricci solitons on Lorentzian para-Sasakian manifolds satisfying certain curvature conditions: R(ξ,X)•S = 0 and S•R(ξ,X)=0. We prove that on a Lorentzian para-Sasakian manifold (M,φ,ξ,η,g), if the Ricci curvature satisfies one of the previous conditions, the existence of η-Ricci solitons implies that (M,g) is Einstein manifold. We also conclude that in these cases there is no Ricci soliton on M with the potential vector field ξ. On the other way, if M is of constant curvature, then (M,g) is elliptic manifold. Cases when the Ricci tensor satisfies different other conditions are also discussed.

If the potential vector field of an ?-Ricci soliton is of gradient type, using Bochner formula, we derive from the soliton equation a nonlinear second order PDE. In a particular case of irrotational potential vector field we prove that the soliton is completely determined by f . We give a way to construct a gradient ?-Ricci soliton on a warped product manifold and show that if the base manifold is oriented, compact and of constant scalar curvature, the soliton on the product manifold gives a lower bound for its scalar curvature.

We provide properties of almost η-Ricci and almost η-Yamabe solitons on submanifolds isometrically immersed into a Riemannian manifold whose potential vector field is the tangential component of a ...

We consider almost η-Ricci solitons in (ε)-para Sasakian manifolds satisfying certain curvature conditions. In the gradient case we give an estimation for the norm of the Ricci curvature tensor and express the scalar curvature of the manifold in terms of the two functions that define the soliton. We also prove that if the Ricci operator is of Codazzi type, then the gradient η-Ricci soliton is expanding if M is spacelike or shrinking if M is timelike.