The closure conjecture for nonclosed maximal totally geodesic submanifolds
The closure conjecture for nonclosed maximal totally geodesic submanifolds
Let be a closed Riemannian manifold with negative sectional curvature, of dimension . An immersed totally geodesic submanifold in is maximal if it is not contained in another proper closed immersed totally geodesic submanifold. The closure conjecture. If contains a maximal totally geodesic immersed submanifold which is not closed, then the closure of is an immersion of a totally geodesic locally symmetric space of rank in . 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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