N(k)-paracontact three metric as a Eta-Ricci soliton

In this paper, we study Eta-Ricci soliton (η-Ricci soliton) on three dimensional N(k)-paracontact metric manifolds. We prove that the scalar curvature of an N(k)-paracontact metric manifold admitting η-Ricci solitons is constant and the manifold is of constant curvature k. Also, we prove that such manifolds are Einstein. Moreover, we show the condition of that the η-Ricci soliton to be expanding, steady or shrinking. In such a case we prove that the potential vector field is Killing vector field. Also, we show that the potential vector field is an infinitesimal automorphism or it leaves the structure tensor in the direction perpendicular to the Reeb vector field ξ. Finally, we illustrate an example of a three dimensional N(k)-paracontact metric manifold admitting an η-Ricci soliton

A 3-DIMENSIONAL SASAKIAN METRIC AS A YAMABE SOLITON

If a 3-dimensional Sasakian metric on a complete manifold (M, g) is a Yamabe soliton, then we show that g has constant scalar curvature, and the flow vector field V is Killing. We further show that, either M has constant curvature 1, or V is an infinitesimal automorphism of the contact metric structure on M.

Skew-symmetric vector fields on aCR-submanifold of a para-Kählerian manifold

We deal with aCR-submanifoldMof a para-Kählerian manifoldM˜, which carries aJ-skew-symmetric vector fieldX. It is shown thatXdefines a global Hamiltonian of the symplectic formΩonM⊤andJXis a relative infinitesimal automorphism ofΩ. Other geometric properties are given.