A generalization of a 4-dimensional Einstein manifold

Abstract

ABSTRACT. A weakly Einstein manifold is a natural generalization of a 4-dimensional Einstein manifold. In this paper, we shall give a characterization of a weakly Einstein manifold in terms of so-called generalized Singer-Thorpe bases.As an application, we prove a generalization of the Hitchin inequality for compact weakly Einstein 4-manifolds. Examples are provided to illustrate the theorems.

Weakly Einstein critical metrics of the volume functional on compact manifolds with boundary

“…Remark 1 in Section 2). We refer the readers to [1,15,16,17,18,19] for more results on this subject. Here, we shall replace the assumption of Einstein in the Miao Tam result (see Theorem 1) by the weakly Einstein condition, which is weaker that the former one in low dimension.…”

Classification of 4-dimensional homogeneous weakly einstein manifolds

“…The first definition of a weakly Einstein manifold in general dimension appeared in [5] and a more detailed study for dimension 4 continued in [6] and [7]. This definition was inspired by that of a super-Einstein manifold as defined in [8].…”

“…The converse does not hold. In [6], the authors present two different examples of homogeneous weakly Einstein spaces which are not Einstein. where α = 0, β are constants.…”

“…Here, g is the left-invariant Riemannian metric on G determined by the inner product , on g α,β defined by e i , e j = δ ij . In the sequel, we will use for the Riemannian group spaces (G, g) α,β the name EPS spaces giving hereby the credit to the authors of the papers [5], [6], [7].…”

Singer-Thorpe bases for special Einstein curvature tensors in dimension 4

“…Motivated by this result and also by the research in the so-called weakly Einstein spaces (see [3], [4]), Sekigawa put the following, more general question: Let (M, g) be a 4-dimensional Einstein manifold, not necessarily 2-stein, and let {e 1 , . .…”