Auteur(s):
Le document est une prépublicationCode(s) de Classification MSC:
Résumé: The aim of this work is to lay the foundations of differential geometry over the general class of topological base fields and rings for which a differential calculus has been developed in [BGN93], without any restriction on the dimension or on the characteristic. Our point of view is classical in the sense that we work in the context of manifolds which are defined in the usual way using charts and atlasses, but we believe that the main ingredients carry over to more general and more formal concepts (cf. Appendix G). Two basic features distinguish our approach from the classical real (finite or infinite dimensional) theory (as laid e.g. in [La99]): the use of scalar extensions and the introduction of a linear algebra concept for non-linear bundles.
The use of scalar extensions by dual numbers in order to modelize tangent objects is known from algebraic geometry and from "synthetic differential geometry " (cf., e.g., [MR91]). We establish a natural version of this procedure in the context of general differential calculus: the tangent functor is the functor of scalar extension by dual numbers; iterating this, we get as powerful tool the interpretation of jet functors as functors of scalar extensions by "jet rings".
The main topic of the present work is the definition and study of linear connections on vector bundles, of their curvature- and torsion forms and the application of the theory to symmetric spaces and Lie groups. The basic idea is very simple: as is well-known, the tangent bundle $T F$ of a linear (i.e., vector) bundle $p : F \longrightarrow M$, seen as a bundle over $M$, is no longer a linear bundle. We show that it is a bilinear bundle, i.e. a bundle modelled on a bilinear space - the concept of a bilinear space, only based on elementary (bi)-linear algebra, is developed in the Appendix BA of this work. A linear connection on $F$ then is simply a linear structure on $T F$ over $M$ that are compatible with the structure if bilinear bundle. We construct canonical linear connections on the tangent bundle of a Lie group, resp. of a symmetric space.
Third and higher order differential geometry will be developed in Part II of this work. This concerns in particular the discussion of curvature which is related to a trilinear bundle, namely to the iterated tangent bundle $TTF$ , seen as a bundle over $M$.
Mots Clés: bundle ; differentiable manifold ; jet ; Lie group ; linear connection ; multilinear algebra ; scalar extension ; symmetric space ; synthetic differential geometry
Date: 2003-11-07