The aim of this work is to lay the foundations of differential geometry and Lie theory over the general class of topological base fields and -rings for which a differential calculus has been developed in [BGN04], without any restriction on the dimension or on the characteristic. Two basic features distinguish our approach from the classical real (finite or infinite dimensional) theory, namely the interpretation of tangent-and jet functors as functors of scalar extensions and the introduction of multilinear bundles and multilinear connections which generalize the concept of vector bundles and linear connections.
Abstract. We define symmetric spaces in arbitrary dimension and over arbitrary non-discrete topological fields K , and we construct manifolds and symmetric spaces associated to topological continuous quasi-inverse Jordan pairs and -triple systems. This class of spaces, called smooth generalized projective geometries, generalizes the well-known (finite or infinite-dimensional) bounded symmetric domains as well as their "compact-like" duals. An interpretation of such geometries as models of Quantum Mechanics is proposed, and particular attention is paid to geometries that might be considered as "standard models" -they are associated to associative continuous inverse algebras and to Jordan algebras of hermitian elements in such an algebra.Contents.
We define a simplicial differential calculus by generalizing divided differences from the case of curves to the case of general maps, defined on general topological vector spaces, or even on modules over a topological ring K. This calculus has the advantage that the number of evaluation points grows linearly with the degree, and not exponentially as in the classical, "cubic" approach. In particular, it is better adapted to the case of positive characteristic, where it permits to define Weil functors corresponding to scalar extension from K to truncated polynomial rings K[X]/(X k+1 ).2000 Mathematics Subject Classification. 13B02, 13F20, 13N99, 14B10, 39A12, 58A20, 58A32.
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