AI 中文总结
研究可微栈上黎曼度量与微分几何轨道空间上黎曼度量数据的等价性，使用特定框架，证明经典黎曼orbifold与黎曼微分几何orbifold概念等价，还探讨李群胚上黎曼度量诱导条件及恰当性条件。
AI 中文摘要
我们证明，由orbifold群胚给出的可微栈上的黎曼度量数据，等同于其微分几何轨道空间上的黎曼度量数据。因此，我们得出结论，经典的黎曼orbifold概念等同于黎曼微分几何orbifold概念。我们使用了由栗林、酒井和塩原引入的黎曼微分几何框架，我们的结果肯定地回答了他们提出的一个问题。更一般地，我们表明李群胚上的黎曼度量（即del Hoyo和Fernandes意义下的2 - 度量），仅当李群胚是正则时，才会在其微分几何轨道空间上诱导出黎曼度量，并且恰当性是一个充分但非必要条件。
英文摘要
We show that the data of a Riemannian metric on a differentiable stack presented by an orbifold groupoid is equivalent to the data of a Riemannian metric on its diffeological orbit space. As a consequence, we conclude that the classical notion of Riemannian orbifold is equivalent to that of a Riemannian diffeological orbifold. We use the framework for Riemannian diffeology introduced by Kuribayashi, Sakai, and Shiobara, and our result answers a problem they posed in the affirmative. More generally, we show that a Riemannian metric on a Lie groupoid, namely a 2-metric in the sense of del Hoyo and Fernandes, induces a Riemannian metric on its diffeological orbit space only if the Lie groupoid is regular, and that properness is a sufficient but not necessary condition.
Comments25 pages
