Remove real-analyticity and codimension-one assumptions in arithmeticity rigidity

Establish that, for closed negatively curved Riemannian manifolds of dimension at least three containing infinitely many properly immersed totally geodesic submanifolds, the real-analyticity assumption can be removed and the codimension-one hypothesis can be replaced by maximality in dimension at least two, while retaining the conclusion that the manifold is, up to rescaling, an arithmetic locally symmetric space.

Background

The paper situates its arithmeticity theorem within broader rigidity questions for negatively curved manifolds. Existing results require a real-analytic metric and infinitely many totally geodesic hypersurfaces. The stated conjecture proposes substantially weakening both assumptions: smooth negative curvature should suffice, and hypersurfaces should be replaced by maximal totally geodesic submanifolds of dimension at least two.

This problem is distinct from the paper’s main theorem, which concerns infinite-volume hyperbolic manifolds with Fuchsian ends and proves arithmeticity under those hypotheses. The conjecture concerns a broader negatively curved, not necessarily locally symmetric, setting and is explicitly left unresolved.