We prove that a compact Riemannian 3-symmetric space is globally symmetric if every Jacobi field along a geodesic vanishing at two points is the restriction to that geodesic of a Killing field induced by the isotropy action or, in particular, if the isotropy action is variationally complete.
J. Differential Geom.
83(2):
273-288
(October 2009).
DOI: 10.4310/jdg/1261495332