  For $n\geq 4$ 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.
