Spherical rank rigidity and Blaschke manifolds. (English) Zbl 1082.53051

Shankar, K.; Spatzier, R.; Wilking, B.

Duke Math. J. 128, No. 1, 65-81 (2005).

In this paper the authors 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, the authors show that a complete Riemannian manifold \(M\) is locally isometric to a compact, rank one symmetric space if \(M\) has sectional curvature at most 1 and each normal geodesic in \(M\) has a conjugate point at \(\pi\).

Reviewer: Antonio Masiello (Bari)

MSC:

53C35	Differential geometry of symmetric spaces
53C20	Global Riemannian geometry, including pinching

Keywords:

Riemannian manifolds; symmetric spaces; geodesics; conjugate points

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