Connections with prescribed curvature
We discuss the problem of prescribing the curvature of a connection on a principal bundle whose base manifold is three-dimensional. In particular, we consider the local question: Given a curvature form F, when does there exist locally a connection A such that F is the curvature of A ? When the structure group of the bundle is semisimple, a finite number of nonlinear identities arise as necessary conditions for local solvability of the curvature equation. We conjecture that these conditions are also generically sufficient, and we prove this for bundles whose structure group is of low rank. Nilpotent structure groups are also discussed.
scientific article; zbMATH DE number 4019925
scientific article