Natural generalizations of locally symmetric spaces (Q1801380)

Let \((M,g)\) be a Riemannian manifold, \(\nabla\) its Levi-Civita connection and \(R\) the Riemannian curvature tensor. \((M,g)\) is locally symmetric if \(\nabla R = 0\). Such spaces form a subclass of that of the semi-symmetric spaces, i.e., spaces such that \(R_{XY}\cdot R = 0\) for all tangent vectors \(X,Y\). (Here \(R_{XY}\) acts as a derivation.) Further, let \(\gamma\) be a geodesic and consider the field \(R_ \gamma\) of symmetric Jacobi operators \(R(\cdot ,\gamma')\gamma'\) along \(\gamma\). As is well- known, locally symmetric spaces satisfy the following conditions: (C) the eigenvalues of \(R_ \gamma\) are constant along \(\gamma\) for each \(\gamma\); (P) \(R_ \gamma\) can be diagonalized by a parallel orthonormal frame along \(\gamma\) for each \(\gamma\). Manifolds satisfying (C) (resp. (P)) are called \(\mathcal C\)-spaces (resp. \(\mathcal P\)-spaces). The have been introduced and studied by \textit{J. Berndt} and the reviewer [see for example, Differ. Geom. Appl. 2, No. 1, 57-80 (1992; Zbl 0747.53013)]. In this paper the author uses Szabó's classification of semi-symmetric spaces to prove the following results: (i) semi-symmetric \(\mathcal C\)-spaces are locally symmetric; (ii) complete semi-symmetric \(\mathcal P\)-spaces are local products of symmetric spaces and spaces of type \(LP(M^ 2 \times \mathbb{R}^ k)\), i.e. local products of surfaces and Euclidean space \(\mathbb{R}^ k\). He also proves that the semi-symmetric spaces of cone type (real and Kählerian) are \(\mathcal P\)-spaces.