Comparison theorems for compact symmetric spaces

Min-Oo; Ernst A. Ruh

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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...