Jacobi relations on naturally reductive spaces

Ponente(s): Tillmann Jentsch Jentsch, Gregor Weingart

Naturally reductive spaces in general can be seen as an adequate generalization of Riemannian symmetric spaces. Nevertheless there are some whose geometric properties are closer to symmetric spaces than others. On the one hand there is the series of Hopf fibrations over complex space forms including the Heisenberg groups with their metrics of type H. On the other hand there exist certain naturally reductive spaces in dimensions six and seven whose torsion forms have a distinguished algebraic property. All these spaces generalize geometric or algebraic properties of $3$--dimensional naturally reductive spaces and have the following property in common: along every geodesic the Jacobi operator satisfies an ordinary differential equation with constant coefficients which can be chosen independently of the given geodesic.