Infinitesimal Differential Geometry

TL;DR

This paper introduces a simple, analysis-based extension of the real numbers with nilpotent infinitesimals to formalize infinitesimal methods in differential geometry, including infinite-dimensional cases, within a cartesian closed category.

Contribution

It presents a new infinitesimal extension of the real numbers and embeds finite-dimensional manifolds into a cartesian closed category to facilitate differential geometry.

Findings

Defined a nilpotent infinitesimal extension of real numbers.

Embedded finite-dimensional manifolds into a cartesian closed category.

Developed initial differential geometric methods using these infinitesimals.

Abstract

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.