Linear Connections in Non - Commutative Geometry

A construction is proposed for linear connections on non-commutative algebras. The construction relies on a generalisation of the Leibnitz rules of commutative geometry and uses the bimodule structure of Ω 1. A special role is played by the extension to the framework of non-commutative geometry of the permutation of two copies of Ω 1. The construction of the linear connection as well as the definition of torsion and curvature is first proposed in the setting of the derivations based differential calculus of Dubois-Violette and then a generalisation to the framework proposed by Connes as well as other non-commutative differential calculi is suggested. The covariant derivative obtained admits an extension to the tensor product of several copies of Ω 1. These constructions are illustrated with the example of the algebra of n × n matrices.