Riemannian manifolds in which certain curvature operator has constant eigenvalues along each circle
J. Berndt and the reviewer introduced the notions of a \(C\)-space and a \(P\)-space, respectively, as Riemannian manifolds \((M,g)\) such that the Jacobi operator has constant eigenvalues or parallel eigenspaces, respectively, along geodesics. These spaces are natural extensions of locally symmetric spaces. A classification for \(\dim M=2\) and 3 and many geometric properties are given in [\textit{J. Berndt} and \textit{L. Vanhecke}, Differ. Geom. Appl. 2, 57-80 (1992; Zbl 0747.53013)]. In this paper, the authors follow a similar procedure to introduce and study two other classes of extensions. Let \(c:(-\delta,\delta)\to (M,g)\) be a smooth curve, parametrized by arc length, such that \(\nabla_{\dot c}\dot c=n\), \(\nabla_{\dot c}n=-\dot c\). Such a curve is a unit circle on \((M,g)\). Let \(\kappa_c=R(\dot c,n)\) be the skew-symmetric endomorphism field associated to the two-plane \((\dot c,n)\), where \(R\) denotes the Riemann curvature tensor. Then \((M,g)\) is said to be an \(O\)-space if \(\kappa_c\) has constant eigenvalues along \(c\) for any unit circle and it is called a \(T\)-space if \(\kappa_c\) has a parallel Jordan basis along \(c\) for any unit circle. \((M,g)\) is locally symmetric if and only if it is an \(O\)- and \(T\)-space. In this paper, the authors derive several criterions for an \((M,g)\) to be an \(O\)-space or a \(T\)-space, respectively. Furthermore, a local classification of two- and three-dimensional \(O\)-spaces is given, providing in this way and for \(\dim M=3\) locally homogeneous as well as curvature homogeneous, but not locally homogeneous, examples. The existence of these last ones contrasts to the case of three-dimensional \(C\)-spaces where only locally homogeneous spaces can exist. \(T\)-spaces will be considered in a forthcoming paper by the same authors.
scientific article; zbMATH DE number 1014200