Prescribed Riemannian Symmetries

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 and 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 𝐺δ in the space of all 𝐺-invariant metrics.