Editing Connection (principal bundle) (section)

=== Flat connections and characterization of bundles with flat connections ===
We say that a connection <math>\omega</math> is '''flat''' if its curvature form <math>\Omega = 0</math>. There is a useful characterization of principal bundles with flat connections; that is, a principal <math>G</math>-bundle <math>\pi: E \to X</math> has a flat connection<ref name=":0" /><sup>pg 68</sup> if and only if there exists an open covering <math>\{U_a\}_{a\in I}</math> with trivializations <math>\left\{ \phi_a \right\}_{a \in I}</math> such that all transition functions<blockquote><math>g_{ab}: U_a\cap U_b \to G</math></blockquote>are constant. This is useful because it gives a recipe for constructing flat principal <math>G</math>-bundles over smooth manifolds; namely taking an open cover and defining trivializations with constant transition functions.