ar X iv : h ep - t h / 94 11 12 7 v 2 1 9 D ec 1 99 4 Linear Connections on Matrix Geometries

A general definition of a linear connection in noncommutative geometry has been recently proposed. Two examples are given of linear connections in noncommutative geometries which are based on matrix algebras. They both possess a unique metric connection. 1 Introduction and Motivation The extension to noncommutative algebras of the notion of a differential calculus has been given both without (Connes 1986) and with (Dubois-Violette 1988) use of the derivations of the algebra. A definition has been given (Chamseddine et al. 1993) of a possible noncommutative generalization of a linear connection which uses the left-module structure of the differential forms. Recently a different definition has been given (Mourad 1994, Dubois-Violette et al. 1994) which makes essential use of the full bimodule structure of the differential forms. We shall use this definition here to consider linear connections on two examples of noncommutative geometries based on matrix algebras. Both have a unique linear connection, which is metric and torsion free. In this respect they are similar to the quantum plane, which is not based on a finite-dimensional algebra. The general definition of a linear connection is given in this section and in Section 2 some basic formulae from matrix geometry are recalled. In Section 3 we consider an algebra of forms based on derivations and we show that there is a unique metric linear connections without torsion. This case is very similar to ordinary differential geometry and the calculations follow closely those of this section. In Section 4 we consider a more abstract differential geometry whose differential calculus is not based on derivations. Here we find that there is a unique 1-parameter family of connections, which is without torsion. The condition that the connection be metric fixes the value of the parameter. We first recall the definition of a linear connection in commutative geometry, in a form (Koszul 1960) which allows for a noncommutative generalization. Let V be a differential manifold and let (Ω * (V), d) be the ordinary differential calculus on V. Let H be a vector bundle over V associated to some principle bundle P. Let C(V) be the algebra of smooth functions on V and H the left C(V)-module of smooth sections of H. A connection on P is equivalent to a covariant derivative on H, which in turn can be characterized as a linear map H