The closure conjecture for nonclosed maximal totally geodesic submanifolds

From papers

Let (M,g)(M,g) be a closed Riemannian manifold with negative sectional curvature, of dimension n3n\geq 3. An immersed totally geodesic submanifold NN in MM is maximal if it is not contained in another proper closed immersed totally geodesic submanifold. The closure conjecture. If MM contains a maximal totally geodesic immersed submanifold NN which is not closed, then the closure of NN is an immersion of a totally geodesic locally symmetric space of rank 11 in MM. This is presented as a generalization of the corresponding theorem for nonclosed totally geodesic submanifolds; the source does not state that the conjecture has been resolved.

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Sources & referencesView supporting material

Primary source

Simion Filip, David Fisher and Ben Lowe, “Finiteness of totally geodesic hypersurfaces”, arXiv:2408.03430 (2025).

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