Prescribed Riemannian Symmetries

TL;DR

This paper proves that for large enough manifolds, most G-invariant Riemannian metrics have automorphism groups exactly G, enabling realization of any compact Lie group as an automorphism group of some Riemannian manifold.

Contribution

It establishes the genericity of G-invariant metrics with automorphism group exactly G and shows every compact Lie group can be realized as such automorphism groups.

Findings

Most G-invariant metrics have automorphism group exactly G.

Every compact connected Lie group can be realized as an automorphism group.

The space of G-invariant metrics with automorphism groups preserving G-orbits is dense G_delta.

Abstract

Given a smooth free action of a compact connected Lie group $G$ on a smooth compact manifold $M$, we show that the space of $G$-invariant Riemannian metrics on $M$ whose automorphism group is precisely $G$ is open dense in the space of all $G$-invariant metrics, provided the dimension of $M$ is "sufficiently large" compared to that of $G$. As a consequence, it follows that every compact connected Lie group can be realized as the automorphism group of some compact connected Riemannian manifold; this recovers prior work by Bedford-Dadok and Saerens-Zame under less stringent dimension conditions. Along the way we also show, under less restrictive conditions on both dimensions and actions, that the space of $G$-invariant metrics whose automorphism groups preserve the $G$-orbits is dense $G_{\delta}$ in the space of all $G$-invariant metrics.