Abstract

We develop a comprehensive tractor based theory and calculus for conformal submanifolds.This includes a conformally invariant Gauss-Codazzi-Ricci theory, and canbe used for example to proliferate submanifold invariants. The set-up is designed tooptimally capture links to the conformal sphere as model and within that a correspondencespace picture of the distinguished totally geodesic submanifolds.On general conformal manifolds we show that the equations for unparametrisedconformal geodesics are the special case of a uniform notion of distinguished submanifolds.Moreover this class of special submanifolds is exactly the class that is weaklyconformally geodesic, meaning that ambient conformal geodesics remain in the submanifold.For conformal geodesics and and such distinguished submanifolds there isan essentially uniform way to construct first integrals from suitable solutions of firstBGG equations.Moreover the notions of minimal submanifolds, CMC submanifolds, and relatedconcepts are also nicely captured in the tractor theory for submanifolds that we develop,and this also means these notions are generalised in that they are well defined atthe conformal singularities of metrics, as arise in Poincare-Einstein (and more generallyconformally compact metrics). This applies in particular to geodesics which maybe viewed as minimal 1-manifolds. This description of minimal submanifolds also providessimpler proofs of some recent results in the literature, and this will be described.This is joint work with Sean Curry and Daniel Snell.