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Hyperbolic rank rigidity for manifolds of $\frac{1}{4}$-pinched negative curvature

Part of: Dynamical systems with hyperbolic behavior Global differential geometry

Published online by Cambridge University Press:  08 October 2018

CHRIS CONNELL ,
THANG NGUYEN  and
RALF SPATZIER
CHRIS CONNELL
Affiliation:
Department of Mathematics, Indiana University, Bloomington, IN 47405, USA email connell@indiana.edu
THANG NGUYEN
Affiliation:
Courant Institute of Mathematical Sciences, New York University, New York, NY 10012, USA email tnguyen@nyu.edu
RALF SPATZIER
Affiliation:
Department of Mathematics, University of Michigan, Ann Arbor, MI 48109, USA email spatzier@umich.edu
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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 $[-1,-\frac{1}{4}]$ 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 counterpart to Hamenstädt’s hyperbolic rank rigidity result for sectional curvatures $\leq -1$, and complements well-known results on Euclidean and spherical rank rigidity.

Keywords

smooth dynamicsHamiltonian dynamicsgeodesic flowsrigidity

MSC classification

Primary: 53C24: Rigidity results
Secondary: 53C20: Global Riemannian geometry, including pinching 37D40: Dynamical systems of geometric origin and hyperbolicity (geodesic and horocycle flows, etc.)
Type
Original Article
Information
Ergodic Theory and Dynamical Systems , Volume 40 , Issue 5 , May 2020 , pp. 1194 - 1216
Copyright
© Cambridge University Press, 2018

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