发表机构
Texas A&M University（德州农工大学）
机构由 AI 辅助整理，请以论文原文为准。AI 中文总结
本文把 Molino 理论从紧流形上的黎曼叶状结构推广到正则黎曼群胚，证明其正则表示作用于轨道叶状结构的 Molino 结构，并给出基本李代数胚的同构条件。
AI 中文摘要
黎曼群胚描述了黎曼叶状结构及其对称性。在本文中，我们将经典的 Molino 理论（该理论涉及紧流形上黎曼叶状结构）推广到具有紧连通对象流形的正则黎曼群胚的设定中。我们观察到，与正则黎曼群胚相关联的轨道叶状结构在对象流形上定义了一个黎曼叶状结构。主要结果表明，黎曼群胚的正则表示自然扩展为对轨道叶状结构的 Molino 结构的作用，从而给出了正则黎曼群胚的基本结构描述。除主要结果外，我们还阐明了基本李代数胚在 Molino 理论中所起的关键作用，并将正则表示等同于伴随表示（直至同伦）的一次上同调表示。我们还在黎曼 Morita 等价下建立了比较结果。对于正则黎曼群胚，我们给出了充分条件，使得相关联的基本李代数胚在拉回到一个公共加细后成为同构的。
英文摘要
Riemannian groupoids describe Riemannian foliations together with their symmetries. In this article, we extend classical Molino's theory, which concerns the structure of Riemannian foliations on compact manifolds, to the setting of regular Riemannian groupoids with compact connected object manifolds. We observe that the orbit foliation associated with a regular Riemannian groupoid defines a Riemannian foliation on the object manifold. The main result shows that the normal representation of a Riemannian groupoid extends naturally to an action on Molino's structures of the orbit foliation, thereby yielding the fundamental structural description of a regular Riemannian groupoid. In addition to the main result, we clarify the essential role played by basic Lie algebroids in Molino's theory and identify the normal representation with the degree-one cohomology representation of the adjoint representation up to homotopy. We also establish comparison results under Riemannian Morita equivalence. For regular Riemannian groupoids, we give sufficient conditions for the associated basic Lie algebroids to become isomorphic after pullback to a common refinement.
