Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings
Résumé
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 recent work (collaboration with H. Gloeckner and K.-H. Nee), 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.
Mots clés
exponential map
Lie groups over topological fields
infinite-dimensional Lie groups
diffeomorphism groups
formal group
general and special multilinear group
symmetric space
differentiable manifold
tangent functor
linear and multilinear bundle
jet
linear connection
multilinear connection
curvature
exponential jet
Campbell-Hausdorff formula
multilinear algebra
scalar extension
dual numbers
synthetic differential geometry
Weil functor