Random Manifolds have no Totally Geodesic Submanifolds

TL;DR

This paper proves that generic closed Riemannian manifolds of dimension four or higher do not contain nontrivial totally geodesic submanifolds, significantly restricting their symmetry groups and addressing a longstanding question.

Contribution

It provides the first proof that generic high-dimensional Riemannian manifolds lack nontrivial totally geodesic submanifolds, confirming a widely held belief.

Findings

Generic closed Riemannian n-manifolds (n≥4) have no nontrivial totally geodesic submanifolds.

Severe restrictions on the isometry groups of generic Riemannian metrics.

Addresses and answers a question posed by Spivak.

Abstract

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.