First order partial differential equations on the curvature of 3-dimensional Heisenberg bundles (Q1370801)

Let \(P \to M \) be a principal bundle with structure group \(G\). If \(\omega\) is a connection 1-form (with values in the Lie algebra \(\mathfrak g\) of \(G\)), then its curvature 2-form \(\Omega\) is defined by the structure equation \[ \Omega = d\omega + \frac12 [\omega, \omega]. \] The author studies the problem of characterisation of \(\mathfrak g\)-valued 2-forms \(\Omega\) on \(P\) which are the curvature form of some connection \(\omega\). The Bianchi identity \[ d\Omega = [\Omega, \omega ] \] shows that the differential \(d\Omega\) of a such form has to belong to the image of the following Bianchi map \[ B_{\Omega} : {\mathfrak g}\otimes \Lambda^1(P) \to {\mathfrak g}\otimes \Lambda^3(P),\qquad \alpha \mapsto [\Omega , \alpha] . \] These conditions may be considered as a system of first order partial differential equations on \(\Omega\). The author shows that generically these equations exhaust all differential identities of first order for the curvature \(\Omega\). In the second part of the paper the author describes all such first order differential equations on \(\Omega\) in an explicit form in the case that \(G\) is the 3-dimensional Heisenberg group. The case when the manifold \(M\) has dimension 5 plays a very special role.