Editing Connection form (section)

===Curvature===
{{main|Curvature form}}
The '''curvature two-form''' of a connection form in ''E'' is defined by
:<math>\Omega(\mathbf e) = d\omega(\mathbf e) + \omega(\mathbf e)\wedge\omega(\mathbf e).</math>
Unlike the connection form, the curvature behaves tensorially under a change of frame, which can be checked directly by using the [[Poincaré lemma]].  Specifically, if '''e''' → '''e''' ''g'' is a change of frame, then the curvature two-form transforms by
:<math>\Omega(\mathbf e\, g) = g^{-1}\Omega(\mathbf e)g.</math>
One interpretation of this transformation law is as follows.  Let '''e'''<sup>*</sup> be the [[dual basis]] corresponding to the frame ''e''.  Then the 2-form
:<math>\Omega={\mathbf e}\Omega(\mathbf e){\mathbf e}^*</math>
is independent of the choice of frame.  In particular, Ω is a vector-valued two-form on ''M'' with values in the [[endomorphism ring]] Hom(''E'',''E'').  Symbolically,
:<math>\Omega\in \Gamma(\Lambda^2T^*M\otimes \text{Hom}(E,E)).</math>

In terms of the exterior connection ''D'', the curvature endomorphism is given by
:<math>\Omega(v) = D(D v) = D^2v\, </math>
for ''v'' ∈ ''E'' (we can extend ''v'' to a local section to define this expression).  Thus the curvature measures the failure of the sequence
:<math>\Gamma(E)\ \stackrel{D}{\to}\ \Gamma(E\otimes\Lambda^1T^*M)\ \stackrel{D}{\to}\ \Gamma(E\otimes\Lambda^2T^*M)\ \stackrel{D}{\to}\ \dots\ \stackrel{D}{\to}\ \Gamma(E\otimes\Lambda^nT^*(M))</math>
to be a [[chain complex]] (in the sense of [[de Rham cohomology]]).