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Abstract We prove that the existence of one horosphere in the universal cover of a closed Riemannian manifold of dimension n 3 with strongly 1/4 -pinched or relatively 1/2 -pinched sectional curvature, on which the stable holonomy along one horosphere coincides with the Riemannian parallel transport, implies that the manifold is homothetic to a real hyperbolic manifold.
Besson et al. (Wed,) studied this question.
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