Distinguished curves and integrability in Riemannian, conformal, and projective geometry

We give a new characterisation of the unparametrised geodesics, or distinguished curves, for affine, pseudo-Riemannian, conformal, and projective geometry. This is type moving incidence relation. The used to provide very general theory construction quantities that are necessarily conserved along curves. formalism immediately yields explicit formulae these curve first integrals. usual role Killing tensors conformal recovered as special case, but shows significantly larger class equation solutions also yield In particular any normal solution an from BGG equations can such quantity. For some condition normality not required. For nowhere-null curves in pseudo-Riemannian geometry additional results available. fundamental tractor-valued invariant this quantity parallel if only circle.