Non-commutative differentials and differential or difference Galois theory

We show how the Galois-Picard-Vessiot theory of differential equations and difference equations. and the theory of holonomy groups in differential geometry, are different aspects of a unique Galois theory. The latter is based upon the construction and study of the tensor product of non commutative connections over a commutative base (semi-classical situation), without any curvature assumption. This theory provides for instance an algebraic frame for the study of the confluence arising when the increment of a difference equation tends to 0. (C) 2001 Editions scientifiques et medicales Elsevier SAS.