# 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.

## Key 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.

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Source: https://tomesphere.com/paper/1703.09240