It has been known for at least 2,500 years that different Riemannian metrics can have the same geodesics (as unparameterised curves). However, this phenomenon is quite rare: the generic metric has little "mobility." More generally, one could try to fit a metric to a system of curves that one would like to be its geodesics but generically this is hopeless: no "mobility" at all! I'll explain some of this classical area (Beltrami 1865, Liouville 1889) from a modern perspective (tractor bundles and the like).

My notes for the talk

Projective symmetries A follow-up exposition of the PDE controlling the infinitesimal symmetries of a projective structure.