The classification of 4-dimensional homogeneous D'Atri spaces revisited (Q884011)

A D'Atri space is defined as a Riemannian manifold \((M,g)\) whose local geodesic symmetries are volume preserving. See [\textit{O. Kowalski, F. Prüfer, L. Vankecke}, Topics in geometry. In memory of Joseph D'Atri. Boston, MA: Birkhäuser. Prog. Nonlinear Differ. Equ. Appl. 20 241--284 (1996; Zbl 0862.53039)] for a survey about this topic. A Riemannian manifold \(M\) is said to belong to the class \({\mathcal A}\) (the first approximation of D'Atri's property) or to be of type \({\mathcal A}\), if its Ricci curvature tensor \(\rho \) is cyclic-parallel, that is, if \((\nabla _X\rho )(X,X)=0,\) for all vector fields \(X\) tangent to \(M.\) A smooth Riemannian manifold \(M\) is called curvature homogeneous if, for any two points \(p, q \in M,\) there exists a linear isometry \(F: T_pM \mapsto T_qM\) such that \(F^{*}R_q=R_p.\) \textit{F. Podesta} and \textit{A. Spiro} [Geom. Dedicata 54, 225--243 (1995; Zbl 0835.53056)] published a classification theorem for 4-dimensional homogeneous Riemannian manifolds of type \({\mathcal A},\) non Einstein, with at most three distinct Ricci principal curvatures. In the present paper it is proved that the classification is incomplete by providing a new family of examples. The underlying manifold considered is the direct product SU\((2)\times {\mathbb R}.\) It is shown that there is a one dimensional family of left invariant metrics on it (in fact a homothety class) which is of type \({\mathcal A},\) and which is irreducible, not locally symmetric and have exactly three distinct Ricci eigenvalues. Also a result from [\textit{P. Bueken, L. Vanhecke}, Geom. Dedicata 75, 123--136 (1999; Zbl 0944.53026)] concerning homogeneous spaces is improved.