Y. Euh, J. Park and K. Sekigawa were the first authors who defined the concept of a weakly Einstein Riemannian manifold as a modification of that of an Einstein Riemannian manifold. The defining formula is expressed in terms of the Riemannian scalar invariants of degree two. This concept was inspired by that of a super-Einstein manifold introduced earlier by A. Gray and T. J. Willmore in the context of mean-value theorems in Riemannian geometry. The dimension $4$ is the most interesting case, where each Einstein space is weakly Einstein. The original authors gave two examples of homogeneous weakly Einstein manifolds (depending on one, or two parameters, respectively) which are not Einstein. The goal of this paper is to prove that these examples are the only existing examples. We use, for this purpose, the classification of $4$-dimensional homogeneous Riemannian manifolds given by L. Bérard Bergery and, also, the basic method and many explicit formulas from our previous article with different topic published in Czechoslovak Math. J. in 2008. We also use Mathematica 7.0 to organize better the tedious routine calculations. The problem of existence of non-homogeneous weakly Einstein spaces in dimension $4$ which are not Einstein remains still unsolved.
The property of being a D’Atri space (i.e., a space with volume-preserving symmetries) is equivalent to the infinite number of curvature identities called the odd Ledger conditions. In particular, a Riemannian manifold $(M,g)$ satisfying the first odd Ledger condition is said to be of type $\mathcal {A}$. The classification of all 3-dimensional D’Atri spaces is well-known. All of them are locally naturally reductive. The first attempts to classify all 4-dimensional homogeneous D’Atri spaces were done in the papers by Podesta-Spiro and Bueken-Vanhecke (which are mutually complementary). The authors started with the corresponding classification of all spaces of type $\mathcal {A}$, but this classification was incomplete. Here we present the complete classification of all homogeneous spaces of type $\mathcal {A}$ in a simple and explicit form and, as a consequence, we prove correctly that all homogeneous 4-dimensional D’Atri spaces are locally naturally reductive.