Random Manifolds have no Totally Geodesic Submanifolds

Abstract

For $n\ge 4$ we show that generic closed Riemannian $n$-manifolds have no nontrivial totally geodesic submanifolds, answering a question of Spivak. An immediate consequence is a severe restriction on the isometry group of a generic Riemannian metric. Both results are widely believed to be true, but we are not aware of any proofs in the literature.

V2: minor changes to address referees' reports, calculations in the local construction have been made more explicit. To appear in Michigan Math. J