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 6 and 7 whose torsion forms have a distinguished algebraic property. All these spaces generalize geometric or algebraic properties of three-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.

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自然还原空间上的雅可比关系

一般而言，自然还原空间可以看作是黎曼对称空间的充分推广。尽管如此，还是有一些比其他的更接近对称空间。一方面，在复杂的空间形式上有一系列的霍普夫纤维化，包括海森堡群的 H 型度量。另一方面，在第 6 维和第 7 维存在某些自然还原空间，它们的扭转形式具有显着的特点： ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ? ?代数性质。所有这些空间概括了三维自然还原空间的几何或代数性质，并具有以下共同点：沿着每个测地线，雅可比算子满足常系数的常微分方程，该方程可以独立于给定的测地线进行选择。