Curvature

In differential geometry, curvature is the infinitesimal obstruction to integrability of the horizontal distribution determined by a connection.

The notion takes different but related forms depending on context:

1. Principal bundles. For a principal connection on a principal $G$-bundle, the curvature is the curvature $2$-form $\Omega \in \Omega^2(P;\mathfrak{g})$, defined by
   $$\Omega = d\omega + \tfrac{1}{2}[\omega \wedge \omega].$$

In a local trivialization with gauge potential $A$, this pulls back to the local curvature $F = dA + \tfrac{1}{2}[A \wedge A]$.

1. Vector bundles. For a connection $\nabla$ on a vector bundle, the curvature is the curvature endomorphism $R^\nabla$, an $\operatorname{End}(E)$-valued $2$-form satisfying
   $$R^\nabla(X,Y)s = \nabla_X\nabla_Ys-\nabla_Y\nabla_Xs-\nabla_{[X,Y]}s.$$

1. Frame bundles. The curvature in a frame relates the principal bundle and vector bundle viewpoints: a connection on a vector bundle induces a principal connection on its frame bundle, and their curvatures correspond.

A connection is flat when its curvature vanishes. Flatness is equivalent to integrability of the horizontal distribution. For a flat connection, parallel transport depends only on the endpoint-preserving homotopy class of a path; it need not be independent of the path when the base has nontrivial fundamental group.

The curvature appears fundamentally in the Chern–Weil theorem, where invariant polynomials applied to the curvature yield characteristic classes.