  The notions of a cleft extension and a cross product with a Hopf algebroid
are introduced and studied. In particular it is shown that an extension (with a
Hopf algebroid $H= (H_L,H_R)$) is cleft if and only if it is $H_R$-Galois and
has a normal basis property relative to the base ring $L$ of $H_L$. Cleft
extensions are identified as crossed products with invertible cocycles. The
relationship between the equivalence classes of crossed products and gauge
transformations is established. Strong connections in cleft extensions are
classified and sufficient conditions are derived for the Chern-Galois
characters to be independent on the choice of strong connections. The results
concerning cleft extensions and crossed product are then extended to the case
of weak cleft extensions of Hopf algebroids hereby defined.
