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The exterior derivative is a flat connection on = (the trivial line bundle over M). More generally, there is a canonical flat connection on any flat vector bundle (i.e. a vector bundle whose transition functions are all constant) which is given by the exterior derivative in any trivialization.
Trivial line bundles can have several flat bundle structures. An example is the trivial bundle over {}, with the connection forms 0 and .The parallel vector fields are constant in the first case, and proportional to local determinations of the square root in the second.
Suppose that X is a smooth complex algebraic variety.. Riemann–Hilbert correspondence (for regular singular connections): there is a functor Sol called the local solutions functor, that is an equivalence from the category of flat connections on algebraic vector bundles on X with regular singularities to the category of local systems of finite-dimensional complex vector spaces on X.
In particular, on any associated vector bundle the principal connection induces a covariant derivative, an operator that can differentiate sections of that bundle along tangent directions in the base manifold. Principal connections generalize to arbitrary principal bundles the concept of a linear connection on the frame bundle of a smooth manifold.
An Ehresmann connection is a connection in a fibre bundle or a principal bundle by specifying the allowed directions of motion of the field. A Koszul connection is a connection which defines directional derivative for sections of a vector bundle more general than the tangent bundle.
If is the canonical vector-valued 1-form on the frame bundle, the torsion of the connection form is the vector-valued 2-form defined by the structure equation = + =, where as above D denotes the exterior covariant derivative.
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. [7]
Similarly, a fibre bundle with discrete fibre is a local system, because each path lifts uniquely to a given lift of its basepoint. (The definition adjusts to include set-valued local systems in the obvious way). A local system of k-vector spaces on X is equivalent to a k-linear representation of (,).