Cubics, curvature and asymptotics

Geodesics are a generalisation of straight lines to Riemannian manifolds and other spaces equipped with an a ne connection. Interpolation and approximation problems motivate analogous generalisations of cubic polynomials. There are several approaches. Cubic polynomials in Euclidean space are critical points of the mean norm-squared acceleration, motivating Riemannian cubics which are critical points of the mean normsquared covariant acceleration. Cubic polynomials are also curves of constant jerk, and this motivates Jupp and Kent's cubics or JK-cubics which are curves of covariantly constant jerk. This thesis contains results on Riemannian and JK-cubics. The primary concerns are asymptotic properties of these curves for large time. There are results on integration of the ordinary di erential equations in special cases. JK-cubics and a special family of Riemannian cubics known as null are studied in matrix groups. Asymptotics are described in generic cases. We also consider these curves in Riemannian manifolds of strictly negative curvature. In such manifolds we show the existence of an open family of Riemannian cubics which cannot be extended for all time. We prove that all JK-cubics in Riemannian manifolds of strictly negative curvature are asymptotically approximated by reparametrised geodesics.