**
ACTA MATHEMATICA UNIVERSITATIS COMENIANAE **

Vol. LXXIII, 2 (2004)

p. 235 - 278

Infinitesimal Differential Geometry

P. Giordano

**Abstract**.
Using standard analysis only, we present an extension $\ER$ of the real field containing nilpotent
infinitesimals. On the one hand we want to present a very simple setting to formalize infinitesimal methods in
Diffe\-ren\-tial 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 $\Man$ of finite dimensional $\Cn$ manifolds in a cartesian closed category. In it we have a
functor $\ext(-)$ which extends these spaces adding new infinitesimal points and with values in another full cartesian
closed embedding of $\Man$. We present a first development of Differential Geometry using these infinitesimals.

**Keywords**:
Spaces of mappings, nilpotent infinitesimals, differential manifolds, foundations.

**AMS Subject classification:** 58D15, 58B10, 58A05, 26E15.

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Acta Mathematica Universitatis Comenianae

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