Symmetry, Integrability and Geometry: Methods and Applications (SIGMA)

SIGMA 17 (2021), 030, 17 pages      arXiv:2008.10072      https://doi.org/10.3842/SIGMA.2021.030

Prescribed Riemannian Symmetries

Alexandru Chirvasitu
Department of Mathematics, University at Buffalo, Buffalo, NY 14260-2900, USA

Received September 27, 2020, in final form March 10, 2021; Published online March 25, 2021

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.

Key words: compact Lie group; Riemannian manifold; isometry group; isometric action; principal action; principal orbit; scalar curvature; Ricci curvature.

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