Linear connections in non-commutative geometry

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Abstract
A construction is proposed for linear connections on non-commutative algebras. The construction relies on a generalization of the Leibniz rules of commutative geometry and uses the bimodule structure of . A special role is played by the extension to the framework of non-commutative geometry of the permutation of two copies of . 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 general of the Dirac operator based differential calculus of Connes and other differential calculuses is given. The covariant derivative obtained admits an extension to the tensor product of several copies of . These constructions are illustrated with the example of the algebra of matrices.
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