Connell–Nguyen–Spatzier higher hyperbolic rank conjecture

Let $(M^n,g)$ be a closed negatively curved manifold with higher hyperbolic rank, meaning that for every geodesic there is a nonvanishing Jacobi field whose span with the geodesic vector has sectional curvature $-1$. Assume that the sectional curvature satisfies $K\geq -1$. Connell–Nguyen–Spatzier conjecture. The manifold $(M^n,g)$ is locally symmetric.

Higher hyperbolic rank is conjectured to characterize locally symmetric spaces among closed negatively curved manifolds. The source notes results under additional curvature assumptions, including $K\leq -1$ and $1/4$-pinching, while this formulation remains open.