Jacobi relations on naturally reductive homogeneous spaces

TL;DR

This paper explores Jacobi relations on naturally reductive homogeneous spaces, revealing that along every geodesic, the Jacobi operator satisfies a constant coefficient ODE, generalizing properties of symmetric spaces.

Contribution

It demonstrates that certain naturally reductive spaces have Jacobi operators satisfying geodesic-independent constant coefficient differential equations.

Findings

Jacobi operators satisfy ODEs with constant coefficients along geodesics.

Spaces include Hopf fibrations and specific 6- and 7-dimensional examples.

Generalizes properties of symmetric spaces to broader classes.

Abstract

Naturally reductive spaces, in general, can be seen as an adequate generalization of Riemannian symmetric spaces. Nevertheless, there are some that 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 point 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.