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arXiv · 1705.02437

Hyperbolic rank rigidity for manifolds of $\frac14$-pinched negative curvature

Chris Connell·Thang Nguyen·Ralf Spatzier
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Abstract

A Riemannian manifold $M$ has higher hyperbolic rank if every geodesic has a perpendicular Jacobi field making sectional curvature -1 with the geodesic. If in addition, the sectional curvatures of $M$ lie in the interval $[-1,-\frac14]$, and $M$ is closed, we show that $M$ is a locally symmetric space of rank one. This partially extends work by Constantine using completely different methods. It is also a partial converse to Hamenst\"{a}dt's hyperbolic rank rigidity result for sectional curvatures $\leq -1$, and complements well-known results on Euclidean and spherical rank rigidity.

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Chris Connell, Thang Nguyen, Ralf Spatzier. 2017-05-06. Hyperbolic rank rigidity for manifolds of $\frac14$-pinched negative curvature. https://doi.org/10.1017/etds.2018.113

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