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 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 δ in the space of all G-invariant metrics.