Cartan Normal Conformal Connections from Pairs of 2nd Order PDE's

Abstract

We explore the different geometric structures that can be constructed from the class of pairs of 2nd order PDE's that satisfy the condition of a vanishing generalized W\"{u}nschmann invariant. This condition arises naturally from the requirement of a vanishing torsion tensor. In particular, we find that from this class of PDE's we can obtain all four-dimensional conformal Lorentzian metrics as well as all Cartan normal conformal O(4,2) connections. To conclude, we briefly discuss how the conformal Einstein equations can be imposed by further restricting our class of PDE's to those satisfying additional differential conditions.