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

TL;DR

This paper introduces a new geometric framework to characterize unparametrised geodesics across various geometries, leading to explicit formulas for conserved quantities and extending classical concepts like Killing tensors.

Contribution

It provides a unified theory for distinguished curves and their conserved quantities, generalizing classical results and including solutions from a broader class of equations.

Findings

Explicit formulae for curve first integrals derived

Extension of conserved quantities beyond classical Killing tensors

Introduction of a tractor-valued invariant for conformal circles

Abstract

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