Jacobi relations on naturally reductive spaces (Q2227508)

It is well known that naturally reductive spaces generalize Riemannian symmetric spaces. On the other hand, among naturally reductive spaces there are spaces whose properties are even closer to those of Riemannian symmetric spaces. The paper under review is devoted to the study of naturally reductive spaces with the following property: along every geodesic, the Jacobi operator satisfies an ordinary differential equation with constant coefficients which can be chosen independently of the given geodesic (in other words, there exist linear Jacobi relations). The main result of the paper is Theorem 2.8. It states that on the following naturally reductive spaces, there exist linear Jacobi relations: 1) The Berger metric on the \((2n+1)\)-dimensional total space \(\widehat{M}^n(k,r)\) of the Hopf fibration of circles of radius \(r\) over an \(n\)-dimensional complex space form \(M^n(k)\) of holomorphic sectional curvature \(k \neq 0\) and the \((2n+1)\)-dimensional Heisenberg group \(\widehat{M}^n(k,1/c^2)\), equipped with a multiple \(1/c^2\) of the canonical left-invariant metric of type H; 2) a seven-dimensional simply connected standard normal homogeneous space of positive sectional curvature except for \(S^7=\operatorname{SU}(4)\operatorname{SU}(3)\) and \(S^7=\operatorname{Sp}(2)/\operatorname{Sp}(1)\); 3) a six-dimensional homogeneous strict nearly Kähler manifold. The paper under consideration also contains many other remarkable results. Among such results, one should note Theorem 7.17, which gives interesting observations about naturally reductive structures on simply connected homogeneous nearly parallel \(G_2\)-spaces, that are similar in some sense to six-dimensional strict nearly Kähler manifolds.