  A equivalence relation, preserving the Chern-Weil form, is defined between
connections on a complex vector bundle. Bundles equipped with such an
equivalence class are called Structured Bundles, and their isomorphism classes
form an abelian semi-ring. By applying the Grothedieck construction one obtains
the ring K, elements of which, modulo a complex torus of dimension the sum of
the odd Betti numbers of the base, are uniquely determined by the corresponding
element of ordinary K and the Chern-Weil form. This construction provides a
simple model of differential K-theory, c.f.Hopkins-Singer (2005), as well as a
useful codification of vector bundles with connection.
