Isotropic Jacobi fields on compact 3-symmetric spaces. (English) Zbl 1186.53064

González-Dávila, José Carmelo

J. Differ. Geom. 83, No. 2, 273-288 (2009).

A Jacobi field along a geodesic \(\gamma\) on a homogeneous Riemannian manifold \((M,g)\) is called isotropic if it is the restriction to \(\gamma\) of a Killing vector field. It is known that any Jacobi field of a symmetric space which vanishes at two points is isotropic. The author proves that this property characterize symmetric spaces inside the class of 3-symmetric spaces:
A compact Riemannian 3-symmetric space is a symmetric space if and only if all Jacobi fields which vanish at two points are isotropic.
In the case of 3-symmetric spaces of inner type, the proof is based on some general arguments which use root systems. In the case of 3-symmetric spaces of outer type, the proof consists of explicit construction of a non isotropic Jacobi field for each non symmetric 3-symmetric manifold.

Reviewer: Dmitri Alekseevsky (Hamburg)

MSC:

53C35	Differential geometry of symmetric spaces
53C30	Differential geometry of homogeneous manifolds

Keywords:

3-symmetric; variationally complete action; Jacobi fields; symmetric spaces; Killing fields