Random 3-manifolds have no totally geodesic submanifolds

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 \(C^{q}\)–topology for any \(q\ge 2.\) In Lytchak and Petrunin’s work, the “generic set” is a dense \(G_{\delta }\) in the \(C^{q}\)–topology for any \(q\ge 2.\) 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 \(C^{q}\)–topology, provided \(q\ge 3.\)