Jacobi relations on naturally reductive homogeneous spaces

Abstract

Naturally reductive spaces in general can be seen as an adequate generalization of symmetric spaces. Nevertheless there are non-symmetric naturally reductive spaces whose geometric properties come closer to symmetric spaces than others. We consider a distinguished class of non-symmetric naturally reductive spaces intimately related to special geometries in dimension six and seven. For this class we establish the following property: along every geodesic the Jacobi operator satisfies an ordinary differential equation with constant coefficients which can be choosen independent of the given geodesic.