Random 3-manifolds have no totally geodesic submanifolds
Hasan M. El-Hasan, Frederick Wilhelm · Annals of Global Analysis and Geometry · 2025
Abstract Murphy and the second author showed that a generic closed Riemannian manifold has no totally geodesic submanifolds, provided the ambient space is at least four dimensional. Lytchak and Petrunin established a similar result in dimension 3. For the higher dimensional result, the “generic set” is open and dense in the $$C^{q}$$ C q –topology for any $$q\ge 2.$$ q ≥ 2 . In Lytchak and Petrunin’s work, the “generic set” is a dense $$G_{\delta }$$ G δ in the $$C^{q}$$ C q –topology for any $$q\ge 2.$$ q ≥ 2 . Here we show that the set of such metrics on a compact 3–manifold actually contains a set that is that is open and dense set in the $$C^{q}$$ C q –topology, provided $$q\ge 3.$$ q ≥ 3 .