The extended quasi-Einstein manifolds with generalised Ricci solitons

As a generalization of Einstein manifolds, the nearly quasi-Einstein manifolds and pseudo quasi-Einstein manifolds are both interesting and useful in studying the general relativity. In this paper, we study the extended quasi-Einstein manifolds which derive from pseudo quasi-Einstein manifolds. After showing the existence theorem of extended quasi-Einstein manifold, we give some special geometric properties of such manifolds. At the same time, we also discuss the extended quasi-Einstein manifolds with certain soliton like generalised Ricci soliton or Riemann soliton. Furthermore, we construct some nontrivial example to illustrate these extended quasi-Einstein manifolds.