Characterizations and integral formulae for generalized m-quasi-Einstein metrics

The aim of this paper is to present some structural equations for generalized m-quasi-Einstein metrics (M ⁿ , g, ? f, λ), which was defined recently by Catino in 11]. In addition, supposing that M ⁿ is an Einstein manifold we shall show that it is a space form with a well defined potential f. Finally, we shall derive a formula for the Laplacian of its scalar curvature which will give some integral formulae for such a class of compact manifolds that permit to obtain some rigidity results.