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The classification of 4-dimensional homogeneous D’Atri spaces revisited. (English) Zbl 1121.53026
A D’Atri space is defined as a Riemannian manifold $$(M,g)$$ whose local geodesic symmetries are volume preserving. See [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 $$F^{*}R_q=R_p.$$
F. Podesta and 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 [P. Bueken, L. Vanhecke, Geom. Dedicata 75, 123–136 (1999; Zbl 0944.53026)] concerning homogeneous spaces is improved.

##### MSC:
 53C21 Methods of global Riemannian geometry, including PDE methods; curvature restrictions 53B20 Local Riemannian geometry 53C25 Special Riemannian manifolds (Einstein, Sasakian, etc.) 53C30 Differential geometry of homogeneous manifolds
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