Curvature form

Summary

The curvature form of a connection on a principal bundle is a $2$-form on the total space. Its values lie in the Lie algebra of the structure group. It measures the failure of the horizontal distribution of the connection to be integrable, and it vanishes exactly when the connection is flat. For connections on vector bundles it becomes a $2$-form with values in endomorphisms, and for the Levi-Civita connection it recovers the Riemann curvature tensor. Its invariant polynomials give the characteristic forms of Chern–Weil theory.