Differential Calculus, Manifolds and Lie Groups over Arbitrary Infinite Fields

2003/03/25 by Wolfgang Bertram, Helge Glockner, Bertram, Wolfgang +5
Mathematics · #26E15 #26E20 #26E30 #46T05 (Secondary) #58C20 (Primary) 22E65 #Advanced Algebra and Geometry #Advanced Operator Algebra Research #Advanced Topics in Algebra #Differential Geometry (math.DG) #FOS: Mathematics #General Mathematics (math.GM) #math.DG #math.GM #msc:22E65 #msc:26E15 #msc:26E20 #msc:26E30 #msc:46T05 #msc:58C20

paper · pdf · doi:10.48550/arxiv.math/0303300

70 pages

arxiv created 2003/03/25 · openalex publication_date 2003/03/25 · arxiv updated 2009/11/30 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28

Abstract

We present an axiomatic approach to finite- and infinite-dimensional differential calculus over arbitrary infinite fields (and, more generally, suitable rings). The corresponding basic theory of manifolds and Lie groups is developed. Special attention is paid to the case of mappings between topological vector spaces over non-discrete topological fields, in particular ultrametric fields or the fields of real and complex numbers. In the latter case, a theory of differentiable mappings between general, not necessarily locally convex spaces is obtained, which in the locally convex case is equivalent to Keller's C^k_c-theory.

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• Lie groups over non-discrete topological fields
• Axiomatic Differential Geometry II-1 Vector Fields
• Aspects of differential calculus related to infinite-dimensional vector bundles and Poisson vector spaces
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