Metrics without isometries are generic

TL;DR

This paper proves that, for any compact manifold of dimension greater than one, most pseudo-Riemannian metrics have no nontrivial symmetries, meaning the set of metrics with only trivial isometries is both large and typical.

Contribution

It establishes that the generic pseudo-Riemannian metric on such manifolds has a trivial isometry group, showing this property is open and dense in the space of all metrics.

Findings

Metrics with trivial isometry groups form an open and dense subset.

Most metrics on higher-dimensional compact manifolds have no symmetries.

Symmetry-free metrics are typical in the space of all pseudo-Riemannian metrics.

Abstract

We prove that for any compact manifold of dimension greater than $1$, the set of pseudo-Riemannian metrics having a trivial isometry group contains an open and dense subset of the space of metrics.