The Ledger curvature conditions and D'Atri geometry
The main result of this paper is the following Theorem. Let \((M,g)\) be a Riemannian 3-manifold which is ``cyclic Ricci-parallel'', i.e., it satisfies the identity \((\nabla_X\text{Ric})(X,X)=0\). Then \((M,g)\) is a D'Atri space (i.e., a space with volume-preserving local geodesic symmetries) and, in particular, it is locally homogeneous. The reviewer [Rend. Semin. Mat., Torino, Fasc. Spec., 131-158 (1983; Zbl 0631.53033)] classified all 3-dimensional D'Atri spaces using the above identity (called the 3rd Ledger condition \(L_3)\) and the next natural identity of higher order (the Ledger condition \(L_5)\), which are both necessary conditions for a Riemannian manifold to be a D'Atri space. Hence, the present authors proved that the condition \(L_5\) follows from the condition \(L_3\) in dimension 3. On the other hand, they give a 5-dimensional example satisfying the condition \(L_3\) but not the condition \(L_5\).
scientific article; zbMATH DE number 1367712
scientific article