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

TL;DR

This paper investigates the geometric structures derivable from specific pairs of second-order PDEs with vanishing Wünschmann invariant, revealing their connection to conformal Lorentzian metrics and Cartan normal conformal connections.

Contribution

It demonstrates how to construct all 4D conformal Lorentzian metrics and Cartan normal conformal O(4,2) connections from these PDE pairs, and discusses imposing Einstein equations through additional conditions.

Findings

Derived all 4D conformal Lorentzian metrics from PDE pairs.

Constructed Cartan normal conformal O(4,2) connections from PDE pairs.

Outlined how to impose Einstein equations via PDE restrictions.

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.