Connection (vector Bundle) - Curvature

The curvature of a connection ∇ on E → M is a 2-form F∇ on M with values in the endomorphism bundle End(E) = E⊗E*. That is,

It is defined by the expression

where X and Y are tangent vector fields on M and s is a section of E. One must check that F∇ is C∞-linear in both X and Y and that it does in fact define a bundle endomorphism of E.

As mentioned above, the covariant exterior derivative d∇ need not square to zero when acting on E-valued forms. The operator (d∇)2 is, however, strictly tensorial (i.e. C∞-linear). This implies that it is induced from a 2-form with values in End(E). This 2-form is precisely the curvature form given above. For an E-valued form σ we have

A flat connection is one whose curvature form vanishes identically.