Differential Geometric Aspects of Rigidity

PI建议研究刚性的微分几何方面：哪些群可以作用于紧流形上，保持给定类型的刚性几何结构？当一个给定的群起作用时，底层流形的几何和拓扑性质是什么？PI的工作集中在伪黎曼度量和共形结构的设置这些问题。提出的研究从三个活跃的数学分支：微分几何，动力学和李群。研究的对象，伪黎曼度量，特别是洛伦兹度量，出现在广义相对论中。相对主义者感兴趣的时空是非紧化的，但他们经常考虑共形紧化。等距或共形自同构在寻找时空模型中很重要，因为它们简化了背景偏微分方程，并且它们符合守恒定律。

The PI proposes to study differential-geometric aspects of rigidity: which groups can act on a compact manifold preserving a rigid geometric structure of given type? When a given group acts, what can be the geometric and topological properties of the underlying manifold? The PI's work focuses on these questions in the setting of pseudo-Riemannian metrics and conformal structures.The proposed research draws from three active branches of mathematics: differential geometry, dynamics, and Lie groups. The objects of study, pseudo-Riemannian and, in particular, Lorentzian metrics, arise in general relativity. The spacetimes of interest to relativists are noncompact, but they often consider conformal compactifications. Isometric or conformal automorphisms are important in the search for models of spacetime, because they simplify the background partial differential equations, and they correspond to conservation laws.