Infinitesimal affine geometry of metric spaces endowed with a dilatation structure

TL;DR

This paper explores the algebraic and geometric properties of metric spaces with dilatation structures, leading to a generalized affine geometry characterized by noncommutative addition and ratio concepts, encompassing various group-based geometries.

Contribution

It introduces a new framework of metric affine geometry derived from dilatation structures, generalizing classical affine geometry to include noncommutative operations and normed affine group spaces.

Findings

Develops a generalized affine geometry from dilatation structures

Defines noncommutative vector addition in metric spaces

Includes a broad class of groups like Carnot and homogeneous groups

Abstract

We study algebraic and geometric properties of metric spaces endowed with dilatation structures, which are emergent during the passage through smaller and smaller scales. In the limit we obtain a generalization of metric affine geometry, endowed with a noncommutative vector addition operation and with a modified version of ratio of three collinear points. This is the geometry of normed affine group spaces, a category which contains the ones of homogeneous groups, Carnot groups or contractible groups. In this category group operations are not fundamental, but derived objects, and the generalization of affine geometry is not based on incidence relations.