In this paper we give a characterization of locally compact rank one symmetric spaces, which can be seen as an analogue of Ballmann's and Burns and Spatzier's characterizations of nonpositively curved symmetric spaces of higher rank, as well as of Hamenstädt's characterization of negatively curved symmetric spaces. Namely, we show that a complete Riemannian manifold is locally isometric to a compact, rank one symmetric space if has sectional curvature at most and each normal geodesic in has a conjugate point at .
K. Shankar. R. Spatzier. B. Wilking. "Spherical rank rigidity and Blaschke manifolds." Duke Math. J. 128 (1) 65 - 81, 15 May 2005. https://doi.org/10.1215/S0012-7094-04-12813-1