Author:

Keywords:

einstein-like manifold, curvature homogeneous manifold, homogeneous manifold, cyclic-parallel ricci tensor, d'atri space, naturally reductive space, principal ricci curvatures, spaces, rho(1)=rho(2)not-equal-rho(3), Science & Technology, Physical Sciences, Mathematics, Einstein-like manifold, cyclic-parallel Ricci tensor, D'Atri space, PRINCIPAL RICCI CURVATURES, SPACES, RHO(1)=RHO(2)NOT-EQUAL-RHO(3), 0101 Pure Mathematics, General Mathematics, 4904 Pure mathematics

Abstract:

The aim of this paper is to study three- and four-dimensional Einstein-like Riemannian manifolds which are Ricci-curvature homogeneous, that is, have constant Ricci eigenvalues. In the three- dimensional case, we present the complete classification of these spaces while, in the four-dimensional case, this classification is obtained in the special case where the manifold is locally homogeneous. We also present explicit examples of four-dimensional locally homogeneous Riemannian manifolds whose Ricci tensor is cyclic-parallel (that is, are of type A) and has distinct eigenvalues. These examples are invalidating an expectation stated by F. Podesta and A. Spiro, and illustrating a striking contrast with the three- dimensional case (where this situation cannot occur). Finally, we also investigate the relation between three- and four-dimensional Einstein-like manifolds of type A and D'Atri spaces, that is, Riemannian manifolds whose geodesic symmetries are volume-preserving (up to sign).