Infinitesimal Differential Geometry

Details Infinitesimal Differential Geometry Infinitesimal Differential Geometry

2003-08-13

arxiv.org/abs/math/0308119v2

Mathematics

Differential Geometry

Submitted to: AMUC, December 2003. We added a sheaf property to the definition of $C^n$ space so that now they generalize diff

Scientific paper

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 Man${}^n$ of finite dimensional $\mathcal{C}^n$ 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 Man${}^n$. We present a first development of Differential Geometry using these infinitesimals.

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