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

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.Comment: 39 page

The Cartan-Weyl Conformal Geometry of a Pair of Second-OrderPartial-Differential Equations

Abstract: We explore the conformal geometric structures of a pair of second-order partial-differential equations.In particular, we investigate the conditions under which this geometry is conformal to the vacuum Einsteinequations of general relativity. Furthermore, we introduce a new version of the conformal Einstein equations,which are used in the analysis of the conformal geometry of the PDE's