Existence of non-isotropic conjugate points on rank one normal homogeneous spaces

TL;DR

This paper proves that simply connected rank one normal homogeneous spaces are symmetric if all conjugate points are isotropic, confirming Chavel's conjecture and characterizing spaces with variational complete isotropy actions.

Contribution

It establishes the equivalence between isotropic conjugate points and symmetry in rank one normal homogeneous spaces, confirming a longstanding conjecture.

Findings

All simply connected rank one normal homogeneous spaces with variational complete isotropy are symmetric.

Confirmed Chavel's conjecture regarding conjugate points and symmetry.

Characterized conditions under which these spaces are symmetric.

Abstract

We give a positive answer to the Chavel's conjecture [J. Diff. Geom. 4 (1970), 13-20]: a simply connected rank one normal homogeneous space is symmetric if any pair of conjugate points are isotropic. It implies that all simply connected rank one normal homogeneous space with the property that the isotropy action is variational complete is a rank one symmetric space.