19 June 2019 Mon 13:15-14:45 REALF A010

Rod Gover (University of Auckland)

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

A new characterisation is described for the unparametrised geodesics, or distinguished curves, for affine, pseudo-Riemannian, conformal, and projective geometry. The characterisation is a type of moving incidence relation and most importantly it leads naturally to a very general theory and construction of quantities that are necessarily conserved along the curves. In this the usual role of Killing tensors and conformal Killing tensors is recovered, but the construction shows that a significantly larger class of equation solutions can also yield curve first integrals.

In particular any normal solution to an equation from the class of first BGG equations can potentially 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 and these are via a fundamental tractor-valued invariant of such curves. This quantity is parallel if and only if the curve is an unparametrised conformal circle.

This talk is based on joint work with Daniel Snell and Arman Taghavi-Chabert.