We analyse the structure of the first order operators in bimodules introduced by A. Connes. We apply this analysis to the theory of connections on bimodules generalizing thereby several proposals.

A general definition of a bimodule connection in noncommutative geometry has been recently proposed. For a given algebra this definition is compared with the ordinary definition of a connection on a left module over the associated enveloping algebra. The corresponding curvatures are also compared. LPTHE Orsay 95/63October, 1995 * Laboratoire associé au CNRS. Introduction and motivationRecently a general definition has been given (Mourad 1995 of a linear connection in the context of noncommutative geometry which makes essential use of the full bimodule structure of the differential forms. A preliminary version of the curvature of the connection was given which had the drawback of not being in general a linear map with respect to the right-module structure. It is in fact analogous to the curvature which is implicitly used by those authors (Chamseddine et al. 1993, Sitarz 1994, Klimčík et al. 1994, Landi et al. 1994 who define a linear connection using the formula for a covariant derivative on an arbitrary left (or right) module (Karoubi 1981, Connes 1986. Our purpose here is to present a modified definition of curvature which is bilinear. Let A be a general associative algebra (with unit element). This is what replaces in noncommutative geometry the algebra of smooth functions on a smooth (compact) manifold which is used in ordinary differential geometry. By 'bilinear' we mean, here and in what follows, bilinear with respect to A. In fact we shall present two definitions of curvature. The first is valid in all generality and reduces to the ordinary definition of curvature in the commutative case. The second definition seems to be better adapted to 'extreme' noncommutative cases, such as the one considered in Section 5.The definition of a connection as a covariant derivative was given an algebraic form in the Tata lectures by Koszul (1960) and generalized to noncommutative geometry by Karoubi (1981) and Connes (1986Connes ( , 1994. We shall often use here the expressions 'connection' and 'covariant derivative' synonymously. In fact we shall distinguish three different types of connections. A 'left A-connection' is a connection on a left A-module; it satisfies a left Leibniz rule. A 'bimodule A-connection' is a connection on a general bimodule M which satisfies a left and right Leibniz rule. In the particular case where M is the module of 1-forms we shall speak of a 'linear connection'. The precise definitions are given below. A bimodule over an algebra A is also a left module over the tensor product A e = A ⊗ C A op of the algebra with its 'opposite'. So a bimodule can have a bimodule A-connection as well as a left A e -connection. These two definitions are compared in Section 2. In Section 3 we discuss the curvature of a bimodule connection. In Section 4 we consider an algebra of forms based on derivations and we compare the left connections with the linear connections. We show that in a sense to be made precise the two definitions yield the same bilinear curvature. That is, the extra restriction which ...