Curvature 2-form in a frame

The matrix of 2-forms computed from a local connection 1-form by dA plus A wedge A.

Curvature 2-form in a frame

Let $E\to M$ be a vector bundle with connection $\nabla$ , and let $U\subset M$ be an open set with a chosen local frame. Let $A$ denote the local connection 1-form (connection matrix) on $U$ in that frame.

Definition. The curvature 2-form of $\nabla$ in the chosen frame is the matrix $F$ of differential 2-forms on $U$ defined by

F := dA + A\wedge A,

where $d$ is the exterior derivative applied entrywise and $A\wedge A$ uses matrix multiplication together with the wedge product of forms.

The matrix $F$ represents the curvature operator in the sense that, writing the frame vectors as $e_j$ ,

R^\nabla(X,Y)e_j = \sum_i F^i{}_j(X,Y)\,e_i,

so $F$ encodes the same information as the curvature tensor .

Under a change of frame by a gauge matrix $g:U\to \mathrm{GL}(r)$ , the curvature transforms tensorially:

F' = g^{-1} F g.

Examples

Line bundle. For rank $1$ , the wedge term vanishes (there is no matrix noncommutativity), so $F=dA$ is just the exterior derivative of the connection 1-form.
Trivial connection. If $A=0$ in some frame, then $F=0$ in that frame and the connection is flat on $U$ .
Constant coefficient connection. If $A=\sum_k A_k\,dx^k$ with constant matrices $A_k$ , then $dA=0$ and $F=\sum_{k<\ell}[A_k,A_\ell]\,dx^k\wedge dx^\ell$ .