论文标题
规定的Riemannian对称性
Prescribed Riemannian Symmetries
论文作者
论文摘要
考虑到平滑紧凑的歧管$ m $的紧凑型谎言组$ g $的平滑免费动作，我们表明，$ g $ invariant的riemannian riemannian riemannian指标的空间在$ m $上的自动形态群体正是$ g $在所有$ g $ g $ invariant级别的空间中都是开放的，前提是$ m $ m $ $ $ $ g $ g'是$ $ m $。结果，因此，每个紧凑的谎言组都可以实现为某些紧凑的riemannian歧管的自动形态群体。这在尺寸不太严格的条件下恢复了Bedford-Dadok和Saerens-Zame的先前工作。在此过程中，我们还表明，在维度和动作的限制性较小的情况下，$ g $ invariant的指标的空间在所有$ g $ g $ -Invariant Metrics的空间中都保留了$ G $ -Orbits的$ G $ -ORBITS。
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.
