Abstract:AbstractWe define and study the theory of derivation-based connectionson a recently introduced class of bimodules over an algebra whichreduces to the category of modules whenever the algebra is commu-t...AbstractWe define and study the theory of derivation-based connectionson a recently introduced class of bimodules over an algebra whichreduces to the category of modules whenever the algebra is commu-tative. This theory contains, in particular, a noncommutative gener-alization of linear connections. We also discuss the different noncom-mutative versions of differential forms based on derivations. Then weinvestigate reality conditions and a noncommutative generalization ofpseudo-riemannian structures. 1 Introduction and notations There are several noncommutative generalizations of the calculus of differen-tial forms and, more generally, of the differential calculus of classical differ-ential geometry, e.g. [2 to 10]. As stressed in [3], the extension of classicaltools to the noncommutative setting is never straightforward. This meansthat, in order to produce relevant objects, one must have in mind a lot ofexamples coming both from mathematics and from physics. In this paper,we concentrate on the differential calculus based on derivations as generaliza-tion of vector fields, [4]. It was shown in [5] that this differential calculus isnatural for quantum mechanics in the sense that with it, quantum mechanicshas the same relation to noncommutative symplectic geometry as classicalmechanics to classical symplectic geometry. For finite quantum spin systemsthis was already pointed out in [6].In this paper, Ais an associative algebra over K = R or C with a unit 1l. Thealgebra Ais to be considered as the generalization of the algebra of smoothfunctions and the Lie algebra Der(A) of all derivations of Aas the generaliza-tion of the Lie algebra of smooth vector fields. The Lie algebra Der(A) is alsoa module over the center Z(A) of Aand furthermore Z(A) is stable by theaction of Der(A). The corresponding Lie algebra homomorphism of Der(A)2Read More