Elements for a metric tangential calculus

TL;DR

This paper introduces metric jets as a generalization of classical jets in differential geometry, defining tangential maps in metric spaces and exploring their representations and properties.

Contribution

It extends the concept of jets to metric spaces, defining metric tangential calculus and providing examples of their representations.

Findings

Metric jets generalize classical jets to metric spaces.

Defined tangential maps at a point in metric spaces.

Provided examples of metric jet representations.

Abstract

The metric jets, introduced in the first chapter, generalize the jets (at order one) of Charles Ehresmann. In short, for a "good" map $f$ (said to be "tangentiable" at $a$), we define its metric jet tangent at $a$ (composed of all the maps which are locally lipschitzian at $a$ and tangent to $f$ at $a$) called the "tangential" of $f$ at $a$, and denoted T$f_a$ (the domain and codomain of $f$ being metric spaces). Furthermore, guided by the heuristic example of the metric jet T$f_a$, tangent to a map $f$ differentiable at $a$, which can be canonically represented by the unique continuous affine map it contains, we will extend, in the second chapter, into a specific metric context, this property of representation of a metric jet.This yields a lot of relevant examples of such representations.