2003/08/13 by Paolo Giordano, Giordano, Paolo
Mathematics · Physics and Astronomy · #26E15 #58A05 #58B10 #58D15 #Algebraic and Geometric Analysis #Differential Geometry (math.DG) #FOS: Mathematics #FOS: Physical sciences #Mathematical Physics (math-ph) #Mathematics and Applications #math-ph #math.DG #math.MP #msc:26E15 #msc:58A05 #msc:58B10 #msc:58D15
paper · pdf · doi:10.48550/arxiv.math/0308119
Submitted to: AMUC, December 2003. We added a sheaf property to the definition of $C^n$ space so that now they generalize diffeological spaces and every extended space has now a topology. We also added a final section which compares our construction with other theories of infinitesimals like NSA, SDG and Weil functors
openalex publication_date 2003/08/13 · arxiv created 2003/12/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
Using standard analysis only, we present an extension ^\bullet\R of the real field containing nilpotent infinitesimals. On the one hand we want to present a very simple setting to formalize infinitesimal methods in Differential Geometry, Analysis and Physics. On the other hand we want to show that these infinitesimals may be also useful in infinite dimensional Differential Geometry, e.g. to study spaces of mappings. We define a full embedding of the category Mann of finite dimensional Cn manifolds in a cartesian closed category. In it we have a functor ^\bullet (-) which extends these spaces adding new infinitesimal points and with values in another full cartesian closed embedding of Mann. We present a first development of Differential Geometry using these infinitesimals.