Local and Global Hartogs-Bochner Phenomenon in Tubes

A generalization of the Hartogs theorem is proved for a class of Tubes structures. We assume that the intervening commutative Lie algebra admits at least a number of globally solvable generators greater or equal to the structure codimension. We give necessary and sufficient conditions for the first cohomological group with compact support to be trivial. A such global result was previously obtained only when the supporting manifold was the Euclidean space. Further but canonical work is need to recover the Bochner extension theorem.