Canonical Curves and First Integrals

Abstract

All parabolic geometries have canonical (unparametrised) curves however, ordinary differential equations governing canonical curves are only known in projective and conformal geometry. In this thesis, we use tractor calculus to systematically find the canonical curves in projective and conformal geometry. For a projectively compact connection, we find projectively invariant first integrals that extend to the boundary. In conformal geometry, we find new first integrals for conformal circles. The first integrals we find are determined by first BGG operators and, in projective geometry, we calculate a detailed collection of first BGG operators and splittings.