Transformation Groups in Conformal and Projective Geometry

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The Principal Investigator will study differential-geometric aspects of rigidity: what can be the symmetries of a geometric structure on a compact space? Assuming a high degree of symmetry, what can be the geometric and topological properties of the space? The PI's work focuses on these questions in the setting of pseudo-Riemannian, conformal, and projective geometry. Pseudo-Riemannian and, in particular, Lorentzian metrics, arise in General Relativity; here compact spacetimes are not realistic, but conformal or projective compactifications of noncompact spacetimes are quite important. In physics, symmetries correspond to conservation laws. In geometry, they provide a framework for classification. The PI is mentoring two graduate students in research on aspects of this program, who are expected to be supported by the award.Building on her recent advance, the proof of the Lorentzian Lichnerowicz Conjecture for real-analytic manifolds with finite fundamental group, the PI will investigate the conjecture for manifolds with infinite fundamental group but low dimension, and deformation rigidity of flat conformal structures with essential conformal group. The PI will work to apply her techniques in the projective setting, where an analogous conjecture for projective structures arising from pseudo-Riemannian metrics remains unsolved in higher signature. The PI will continue her investigation of relations between transformation groups and tractor solutions. In the context of smooth hyperbolic dynamics, the PI will build on the construction of invariant differential-geometric structures for nonuniformly contracting cocycles, to find more refined invariant structures under suitable hypotheses, and applications of these to rigidity results.This award reflects NSF's statutory mission and has been deemed worthy of support through evaluation using the Foundation's intellectual merit and broader impacts review criteria.