Conformal submanifolds, distinguished submanifolds, and integrability

Abstract

For conformal geometries of Riemannian signature, we provide a comprehensive and explicit treatment of the core local theory for embedded submanifolds of arbitrary dimension. This is based in the conformal tractor calculus and includes a conformally invariant Gauss formula leading to conformal versions of the Gauss, Codazzi, and Ricci equations. It provides the tools for proliferating submanifold conformal invariants, as well for extending to conformally singular Riemannian manifolds the notions of mean curvature and of minimal and CMC submanifolds. A notion of distinguished submanifold is defined by asking the tractor second fundamental form to vanish. We show that for the case of curves this exactly characterises conformal geodesics (a.k.a. conformal circles) while for hypersurfaces it is the totally umbilic condition. So, for other codimensions, this unifying notion interpolates between these extremes, and we prove that in all dimensions this coincides with the submanifold being weakly conformally circular, meaning that ambient conformal circles remain in the submanifold. Stronger notions of conformal circularity are then characterised similarly and an extensive collection of examples is provided. Next we provide a very general theory and construction of quantities that are necessarily conserved along distinguished submanifolds. This first integral theory vastly generalises the results available for conformal circles in [56]. We prove that any normal solution to an equation from the class of first BGG equations yields such conserved quantities, and show that it is easy to provide explicit formulae for these. Finally we prove that the property of being distinguished is also captured by a type of moving incidence relation and use this to show that, for suitable solutions of conformal Killing-Yano equations, the zero locus of the solution is necessarily a distinguished submanifold.