Prescribed Riemannian Symmetries

• Published in: Symmetry, Integrability and Geometry: Methods and Applications
• Date: 2021
• Main Author: Chirvasitu, Alexandru
• Format: Article
• Language: English
• Published: Інститут математики НАН України 2021
• Online Access: https://nasplib.isofts.kiev.ua/handle/123456789/211319
• Journal Title: Digital Library of Periodicals of National Academy of Sciences of Ukraine
• Cite this: Prescribed Riemannian Symmetries. Alexandru Chirvasitu. SIGMA 17 (2021), 030, 17 pages

Given a smooth free action of a compact connected Lie group on a smooth compact manifold , we show that the space of -invariant Riemannian metrics on whose automorphism group is precisely is open and dense in the space of all -invariant metrics, provided the dimension of is ''sufficiently large'' compared to that of . 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 -invariant metrics whose automorphism groups preserve the -orbits is dense δ in the space of all -invariant metrics.