by differential 2-forms and the connection is expressed by the connection 1-forms, a skew-symmetric matrix of differential 1-forms . The exterior covariant derivative of , as a vector-valued differential 3-form, is expressed relative to the local frame by many differential 3-forms, defined by In the case of the trivial real line bundle with its standard connection, vector-valued differential forms and differential forms can be naturally identified with one another, and each of the above definitions coincides with the standard exterior derivative. Given a principal bundle, any linear representation of the structure group defines an associated bundle, and any connection on the principal bundle induces a connection on the associated vector bundle. Differential forms valued in the vector bundle may be naturally identified with fully anti-symmetric tensorial forms on the total space of the principal bundle. Under this identification, the notions of exterior covariant derivative for the principal bundle and for the vector bundle coincide with one another. The curvature of a connection on a vector bundle may be defined as the composition of the two exterior covariant derivatives and , so that it is defined as a real-linear map . It is a fundamental but not immediately apparent fact that only depends on , and does so linearly. As such, the curvature may be regarded as an element of . Depending on how the exterior covariant derivative is formulated, various alternative but equivalent definitions of curvature (some without the language of exterior differentiation) can be obtained. It is a well-known fact that the composition of the standard exterior derivative with itself is zero: . In the present context, this can be regarded as saying that the standard connection on the trivial line bundle has zero curvature. Example Bianchi's second identity, which says that the exterior covariant derivative of Ω is zero (that is, ) can be stated as: . Notes References Connection (mathematics) Differential geometry Fiber bundles