Aspects of overdetermined systems of partial differential equations in projective and conformal geometry

Abstract

This thesis discusses aspects of overdetermined systems of partial differential equations (PDEs) in projective and conformal geometry. The first part deals with projective differential geometry. A projective surface is a 2-dimensional smooth manifold equipped with a projective structure i.e. a class of torsion-free affine connections that have the same geodesics as unparameterised curves. Given any projective surface we can ask whether it admits a torsion-free affine connection (in its projective class) that has skew-symmetric Ricci tensor. This is equivalent to solving a particular overdetermined system of semi-linear partial differential equations. It turns out that there are local obstructions to solving the system of PDEs in two dimensions. These obstructions are constructed out of local invariants of the projective structure. We give examples of projective surfaces that admit skew-symmetric Ricci tensor and examples that do not because of nonvanishing obstructions. We relate projective surfaces admitting skew-symmetric Ricci tensor to 3-webs in 2 dimensions. We also give examples of projective structures in higher dimensions that admit skew-symmetric Ricci tensor. The second part of the thesis deals with conformal differential geometry. On Mobius surfaces introduced in [5], we can define an analogous overdetermined system of semi-linear PDEs as in the projective case. This is called the scalar-flat Mobius Einstein-Weyl equation and is conformally invariant. We derive local algebraic constraints for Mobius surfaces to admit a solution to this equation and give local obstructions. These obstructions are similarly constructed out of local conformal invariants of the Mobius structure. Again we provide examples of Mobius surfaces that admit a solution and examples that do not because of non-vanishing obstructions. Finally, we also investigate the conformally Einstein equation on Mobius surfaces and derive obstructions. In contrast to the previous two equations, the conformally Einstein equation is linear.