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.
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Besson et al. (2024) studied this question.