  We define a new algebra of noncommutative differential forms for any Hopf
algebra with an invertible antipode. We prove that there is a one to one
correspondence between anti-Yetter-Drinfeld modules, which serve as
coefficients for the Hopf cyclic (co)homology, and modules which admit a flat
connection with respect to our differential calculus. Thus we show that these
coefficient modules can be regarded as ``flat bundles'' in the sense of Connes'
noncommutative differential geometry.
