Differential Geometric Aspects of Rigidity

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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.