LINEAR CONNECTIONS IN NONCOMMUTATIVE GEOMETRY

A construction is proposed for linear connections on non-commutative a lgebras. The construction relies on a generalization of the Leibniz ru les of commutative geometry and uses the bimodule structure of Omega(1 ). A special role is played by the extension to the framework of non-c ommutative geometry of the permutation of two copies of Omega(1). The construction of the linear connection as well as the definition of tor sion and curvature is first proposed in the setting of the derivations based differential calculus of Dubois-Violette and then a generalizat ion to the framework of the Dirac operator based differential calculus of Connes and other differential calculuses is given. The covariant d erivative obtained admits an extension to the tenser product of severa l copies of Omega(1). These constructions are illustrated with the exa mple of the algebra of n x n matrices.