Construction of connection inducing maps between principal bundles. I (Q1099812)

Let \(X\to V\) and \(Y\to W\) be two \(C^{\infty}\)-smooth principal bundles with the same structure group G and with \(C^{\infty}\)-connection \(\Gamma\) on X and \(\Delta\) on Y, respectively. The problem which the author deals with, is to find a \(C^{\infty}\)-map \(f: V\to K\) such that the induced bundle \(f^*(Y)\) over V with the induced connection \(f^*(\Delta)\) is isomorphic to (X,\(\Gamma)\). She gives some general criteria for the existence of connection inducing maps, and she states the following main result. Let \(\Gamma\) be an arbitrary connection on a trivial 0(p)-bundle over a stably parallelizable n-dimensional manifold V. If \(q\geq p(n+3)/2\), then there exists a connection inducing map \(f: V\to Gr_ p({\mathbb{R}}^ q).\) From a technical point of view, the construction of a connection inducing map amounts to solving a system of partial differential equations. If \(q\geq p(n+1)\), then the PDE for connection inducing maps \(f: V\to Gr_ p({\mathbb{R}}^ q)\) can be reduced to an algebraic system. Moreover, a suitable regularity condition on f makes the linearized PDE algebraically solvable. Then, Nash's implicit function theorem can be applied to the local study and the theory of topological sheaves gives global results.