Differential geometry, Lie groups and symmetric spaces over general base fields and rings

Abstract

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.

Symmetric bundles and representations of Lie triple systems

“…This work can be read on two different levels: the reader may take K = R to be the real base field and understand by "manifold" finite-dimensional real manifolds in the usual sense; then our symmetric spaces and Lie groups are the same as in [16] or [13], or one may consider a commutative topological field or ring K, having dense unit group K × and such that 2 is invertible in K; then we refer to [5] for the definition of manifolds and Lie groups over K. Readers interested in the general case should just keep in mind that, in general,…”

“…Clearly, the concept of symmetric bundle could be adapted to other classes of bundles whenever the fibers belong to a category that admits direct products (e.g., multilinear bundles in the sense of [5]): it suffices to replace (SB2) by the requirement that the map F q × F p → F µ(p,q) be a morphism in that category. Also, it is clear that such concepts exist for any category of manifolds equipped with binary, ternary or other "multiplication maps", such as generalized projective geometries (cf.…”

Inner ideals and intrinsic subspaces of linear pair geometries

“…, equipped with a multilinear product which is constructed by using the third order tangent bundle (essentially, this is an algebraic version of the construction of the associated Lie triple system of a symmetric space, see [Be06,Chapter 27] (2) Assume now that a friendly sub-pair (W + , W − ) of (V + , V − ) is given, and define W ± = G W .o ± as in the claim. The main point of the proof of Part (2) is to show that…”

Some aspects of differential theories Respectfully dedicated to the memory of Serge Lang

“…As a consequence, there are several theories of infinite-dimensional manifolds, Lie groups and differential geometric structures. Changing the real or complex ground field to a more general topological field or ring, even more general differential calculus, Lie theory and differential geometry may be constructed [3,4]. In this subsection we briefly explain the approach to differential calculus originated by A. D. Michel [25] and A. Bastiani [2], and popularized by J. Milnor [27] and R. Hamilton [13].…”