Geometry of Vector Sheaves. A-connections

Abstract

As already remarked below the heading, the present chapter marks, in effect, the beginning of our abstract, (viz. axiomatic) approach to the standard Differential Geometry of smooth (i.e. C^∞-) manifolds. A crucial aspect hereby is that we do not employ for this any differential analysis; that is, no calculus is involved at all(!), our tools being just sheaf theory, in particular, vector sheaves and their algebra (linear and multilinear) and/or sheaf cohomology, to the extent that these issues have been developed in the preceding chapters. Nontheless, as we shall realize through the succeeding discussion, several fundamental results of the classical theory, global, as well as, local ones, can still be derived, within the present abstract framework. We start thus right away by displaying the necessary preliminaries, in order next to define (cf. Section 2 below) the important (differential geometric) notion of a connection, or even, in our terminology, of an A-connection, which essentially lies in the core of all the subsequent material; this, with the exception of Chapter X, which, however, again is just an auxiliary one of the next Chapter XI, where anew A-connections of a particular type are further examined.