Torse-forming η-Ricci solitons in almost paracontact η-Einstein geometry

Abstract

Torse-forming ?-Ricci solitons are studied in the framework of almost paracontact metric ?-Einstein manifolds. By adding a technical condition, called regularity and concerning with the scalars provided by the two ?-conditions, is obtained a reduction result for the parallel symmetric covariant tensor fields of order two.

η-Ricci solitons in (ε)-almost paracontact metric manifolds

“…It is easy to verify that (3) and one of (4), (5) and (6) imply the other two equations. If g is a pseudo-Riemannian metric such that…”

“…where £ V denotes the Lie-derivative in the direction V, S stands for the Ricci tensor field, λ and µ are constants and X, Y are arbitrary vector fields on M. In [7] the authors studied η-Ricci solitons on Hopf hypersurfaces in complex space forms. In the context of paracontact geometry η-Ricci solitons were investigated in [4,5,3].…”

Harmonic Aspects in an $\eta$-Ricci Soliton

“…We say that a 1-form γ is a solution of the Schrödinger-Ricci equation if Remark 2.2. Under the hypotheses of Theorem 2.1, if the potential vector field is of constant length k, then from (7) we deduce that the scalar curvature is constant if either the soliton is a Ricci soliton or, (scal +λn+µk 2 )η = ∇ ξ η which implies scal = −λn−µk 2 . Corollary 2.3.…”

Certain Results on η-RICCI Solitions in α-Sasakian Manifolds

“…for any X,Y∈T(M). An important geometrical object in studying Ricci solitons is well-known to be a symmetric (0,2)-tensor field which is parallel with respect to the Levi-Civita connection, some of its geometrical properties being described in [3], [4], [18] etc. In the same manner as in [5] we shall state the existence of η-Ricci solitons in our settings.…”

“…where S is the Ricci tensor associated to g. In this connection we mention the works of Blaga [1,2,3] with η-Ricci solitons. In particular, if μ = 0 then the notion η-Ricci soliton (g,V,λ,μ) reduces to the notion of Ricci soliton (g, V, λ).…”