  This is the translation to appear in the "SUPERSYMMETRY 2000 - Encyclopaedic
Dictionary" of the original paper, published in March 1980, (C.R. Acad. Sci.
Paris, Ser. A-B, 290, 1980) in which basic notions of noncommutative geometry
were introduced and applied to noncommutative tori. These include connections
on finite projective modules, their curvature, and the Chern character. Finite
projective modules on the noncommutative two-torus $ \Tb^2_{\theta}$ were
realized as Schwartz spaces of vector valued functions on $\Rb$. Explicit
constant curvature connections were constructed and a basic integrality
phenomenon of the total curvature was displayed. The pseudo-differential
calculus and the Atiyah-Singer index theorems were extended to Lie group
actions on $C^*$ algebras and used to explain the above integrality of the
total curvature by an index formula for finite difference-differential
operators on the line. Recent interest in the hep-th literature for basic
notions of noncommutative geometry in the case of noncommutative tori (cf for
instance hep-th/0012145 for an excellent review) prompted us to make the
English translation of the original paper available.
