On representations of zero curvature, generated by geometry of a second order partial differential equation (Q1896719)

The author studies the geometry of the second-order equation \[ \varphi \left(x^1, \dots,x^n, u, {\partial u \over \partial x^j},\;{\partial^2 u\over \partial x^k \partial x^\ell}\right) = 0. \] The author treats the variables \(x^1,\dots, x^n\), \(u\) has adapted coordinates in an \((n + 1)\)-dimensional fibre bundle \(E\) whose base is an \(n\)-dimensional manifold \(M\) with local coordinates \(x^1,\dots, x^n\). Thus the differential equation can be rewritten in the form \[ \varphi (x^1, \dots, x^n,u, \lambda_j, \lambda_{kl}) = 0 \] where \(x^i\), \(u\), \(\lambda_j\), \(\lambda_{kl}\), \(i,j,k = 1,\dots,n\), are the adapted local coordinates on the manifold \(J^2 E\) of holonomic 2-jets of local sections of the bundle \(E\). Using the invariant analytical method of Laptev and Vasil'ev, the author derives a geometric object with components \(F^{jk}\), \(F^j\), \(F\), \(F_i\). Having examined the differential equations for this object the author concludes that the fundamental object is being associated with some Lie group \(G\) whose structural forms are derived, too. With a geometrical subobject \(F^{ij}\), \(F^i\) he associates a Lie group \(G^1\). In this article a connection in the principal bundle \(H(J^5E, G^1)\) enveloped by the prolonged fundamental object is to be constructed. He reveals conditions under which such a connection can be prolonged to a connection in the bundle \(H(J^5 E, G)\). In invariant manner he points out the case when the curvature forms of an induced connection vanish on the lifts \(\sigma^5 \subset J^5 E\) of local sections \(\sigma \subset E\). Further he finds conditions under which the connection forms vanish on lifts of sections iff the sections are solutions, or in other words, there are found conditions under which connection forms are being pseudopotentials.