Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings

Wolfgang Bertram (1)

1 IECN - Institut Élie Cartan de Nancy

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. Neeb), 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: 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 map; exponential jet; Campbell-Hausdorff formula; multilinear algebra; scalar extension; dual numbers; synthetic differential geometry; Weil functor

Domaines: Géométrie différentielle [math.DG]

Wolfgang Bertram. Differential Geometry, Lie Groups and Symmetric Spaces over General Base Fields and Rings. Memoirs of the American Mathematical Society, 2008, 00 (00), pp.00. ⟨hal-00004190v2⟩