  We obtain a global version of the Hamiltonian KAM theorem for invariant
Lagrangean tori by glueing together local KAM conjugacies with help of a
partition of unity. In this way we find a global Whitney smooth conjugacy
between a nearly-integrable system and an integrable one. This leads to
preservation of geometry, which allows us to define all the nontrivial
geometric invariants like monodromy or Chern classes of an integrable system
also for near integrable systems.
