• Corpus ID: 7904057

Infinitesimal affine geometry of metric spaces endowed with a dilatation structure

  title={Infinitesimal affine geometry of metric spaces endowed with a dilatation structure},
  author={Marius Buliga},
  journal={arXiv: Metric Geometry},
  • Marius Buliga
  • Published 1 April 2008
  • Mathematics
  • arXiv: Metric Geometry
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… 

Deformations of normed groupoids and differential calculus. First part

Differential calculus on metric spaces is contained in the algebraic study of normed groupoids with $\delta$-structures. Algebraic study of normed groups endowed with dilatation structures is

A characterization of sub-riemannian spaces as length dilatation structures constructed via coherent projections

We introduce length dilatation structures on metric spaces, tempered dilatation structures and coherent projections and explore the relations between these objects and the Radon-Nikodym property and

Sub-riemannian geometry from intrinsic viewpoint

Gromov proposed to extract the (di erential) geometric content of a sub-riemannian space exclusively from its Carnot-Carath eodory distance. One of the most striking features of a regular

Braided spaces with dilations and sub-riemannian symmetric spaces

Braided sets which are also spaces with dilations are presented and explored in this paper, in the general frame of emergent algebras arxiv:0907.1520. Examples of such spaces are the sub-riemannian

Uniform refinements, topological derivative and a differentiation theorem in metric spaces

For the importance of differentiation theorems in metric spaces (starting with Pansu Rademacher type theorem in Carnot groups) and relations with rigidity of embeddings see the section 1.2 in Cheeger

Emergent algebras as generalizations of differentiable algebras, with applications

We propose a generalization of differentiable algebras, where the underlying differential structure is replaced by a uniform idempotent right quasigroup (irq). Algebraically, irqs are related with

Maps of metric spaces

This is a pedagogical introduction covering maps of metric spaces, Gromov-Hausdorff distance and its "physical" meaning, and dilation structures as a convenient simplification of an exhaustive

COLIN implies LIN for emergent algebras

Emergent algebras, first time introduced in arXiv:0907.1520, are families of quasigroup operations indexed by a commutative group, which satisfy some algebraic relations and also topological

Metric Lie groups admitting dilations

We consider left-invariant distances $d$ on a Lie group $G$ with the property that there exists a multiplicative one-parameter group of Lie automorphisms $(0, \infty)\rightarrow\mathtt{Aut}(G)$,

The em-convex rewrite system

We introduce and study em (or "emergent"), a lambda calculus style rewrite system inspired from dilations structures in metric geometry. Then we add a new axiom (convex) and explore its consequences.



The tangent space in sub-riemannian geometry

Tangent spaces of a sub-Riemannian manifold are themselves sub-Riemannian manifolds. They can be defined as metric spaces, using Gromov’s definition of tangent spaces to a metric space, and they turn

Dilatation structures in sub-riemannian geometry

Based on the notion of dilatation structure arXiv:math/0608536, we give an intrinsic treatment to sub-riemannian geometry, started in the paper arXiv:0706.3644 . Here we prove that regular

Metric Structures for Riemannian and Non-Riemannian Spaces

Length Structures: Path Metric Spaces.- Degree and Dilatation.- Metric Structures on Families of Metric Spaces.- Convergence and Concentration of Metrics and Measures.- Loewner Rediscovered.-

Differentiability of mappings in the geometry of Carnot manifolds

We study the differentiability of mappings in the geometry of Carnot-Carathéodory spaces under the condition of minimal smoothness of vector fields. We introduce a new concept of hc-differentiability

The differential of a quasi-conformal mapping of a Carnot-Caratheodory space

The theory of quasi-conformal mappings has been used to prove rigidity theorems on hyperbolic n space over the division algebras ℝ, ℂ, ℍ, and \({\Bbb O}\), by studying quasi-conformal mappings on

Tangent bundles to sub-Riemannian groups

This paper is about non-Euclidean analysis on Lie groups endowed with left invariant distributions, seen as sub-Riemannian manifolds. This is a an updated version, which will be modified according

Dilatation structures I. Fundamentals

A dilatation structure is a concept in between a group and a differential structure. In this article we study fundamental properties of dilatation structures on metric spaces. This is a part of a

Contractible Lie groups over local fields

Let G be a Lie group over a local field of characteristic p > 0 which admits a contractive automorphism α : G → G (i.e., αn(x) → 1 as n → ∞, for each x ∈ G). We show that G is a torsion group of