 Subriemannian geometry from intrinsic viewpoint. These are the notes prepared for the course "Metric spaces with dilations and subriemannian geometry from intrinsic point of view", CIMPA research
school on subriemannian geometry (2012).
 Gromov proposed to extract the (differential) geometric content of a subriemannian space exclusively from its CarnotCarath\'eodory distance. One of the most striking features of a regular subriemannian space is that it has at any point a metric tangent space with the algebraic structure of a Carnot group, hence a homogeneous Lie group. Siebert characterizes homogeneous Lie groups as locally compact groups admitting a contracting and continuous oneparameter group of automorphisms. Siebert result has not a metric character.
In these notes I show that subriemannian geometry may be described by about 12 axioms, without using any a priori given differential structure, but using dilation structures instead. Dilation structures bring forth the other intrinsic ingredient, namely the dilations, thus blending Gromov metric point of view with Siebert algebraic one.
 LambdaScale, a lambda calculus for spaces with dilations (2012).
 LambdaScale is an enrichment of lambda calculus which is adapted to emergent algebras. It can be used therefore in metric spaces with dilations.
 A characterization of subriemannian spaces as
length dilation structures constructed via coherent projections
Commun. Math. Anal. 11 (2011), No. 2, pp. 70111, (pdf)
 We introduce length dilation structures on metric spaces,
tempered dilation structures and coherent projections and
explore the relations between these objects and the RadonNikodym
property and Gammaconvergence of length functionals.
Then we show that the main properties of subriemannian spaces can
be obtained from pairs of length dilation structures, the
first being a tempered one and the second obtained via a
coherent projection.
 Infinitesimal affine geometry of metric spaces
endowed with a dilatation structure (2008), Houston Journal of
Mathematics, 36, 1 (2010), 91136. Here is a link to the
journal paper.
 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.
 Braided spaces with dilations and
subriemannian symmetric spaces. in: Geometry. Exploratory Workshop on Differential Geometry and its
Applications, eds. D. Andrica, S. Moroianu, ClujNapoca 2011, 2135 (book chapter)
 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 subriemannian symmetric spaces.
Keywords: braided sets, quandles; emergent algebras;
dilatation structures (spaces with dilations); subriemannian
symmetric spaces.
 Introduction to metric spaces with dilations
(2010)
 This paper gives a short introduction into the metric theory
of spaces with dilations.
 Uniform refinements, topological derivative and a
differentiation theorem in metric spaces (2009)
 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
and Kleiner paper
arXiv:math/0611954
and its bibliographic references. Here we propose another type of
differentiation theorem, which does not involve measures.
It is therefore different from Rademacher type theorems.
Instead, this differentiation theorem (and the concept of
uniformly topological derivable function) is formulated in terms
of filters in topological spaces.
 Deformations of normed groupoids and
differential calculus. First part (2009)
 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 contained in the differential calculus on metric spaces.
Thus all algebraic properties of the small world of normed groups with dilatation structures have equivalent formulations (of comparable complexity) in the big world of metric spaces admitting a differential calculus.
Moreover these results non trivially extend beyond metric spaces, by using the language of groupoids.
 Emergent algebras as generalizations of
differentiable algebras, with applications (2009)
 We propose a generalization of differentiable algebras, where
the underlying differential structure is replaced by a uniform
idempotent right quasigroup (irq). Then a emergent algebra is a
algebra $\mathcal{A}$ over the uniform irq $X$ such that all
operations and algebraic relations from $\mathcal{A}$ can be
constructed or deduced from combinations of operations in the
uniform irq, possibly by taking limits which are uniform with
respect to a set of parameters. In this approach, the usual
compatibility condition between algebraic information
$\mathcal{A}$ and differential information $\mathcal{D}$,
expressed as the differentiability of operations from
$\mathcal{A}$ with respect to $\mathcal{D}$, is replaced by the
"emergence" of algebraic operations and relations from the minimal
structure of a uniform irq. Two applications are considered: we
prove a bijection between contractible groups and distributive
uniform irqs and that symmetric spaces in the sense of Loos may be
seen as uniform quasigroups with a distributivity property.
 On the KirchheimMagnani counterexample to metric differentiability (2007)
 Selfsimilar dilatation
structures and automata (2007), Proceedings of the 6th Congress of Romanian Mathematicians,
Bucharest, 2007, vol. 1, 557564 (2008)
 We show that on the boundary of the dyadic tree, any selfsimilar dilatation
structure induces a web of interacting automata. This is a short
version, for publication, of the paper
arXiv:math/0612509v2.
 Dilatation structures in subriemannian geometry (2007)
in: Contemporary Geometry and Topology and Related Topics,
ClujNapoca, Cluj University Press (2008), 89105
 Based on the notion of dilatation structure
arXiv:math/0608536,
we give an intrinsic treatment to subriemannian geometry, started
in the paper arXiv:0706.3644.
Here we prove that regular subriemannian manifolds admit
dilatation structures. From the existence of normal frames proved by
Bellaiche we deduce the rest of the properties of regular subriemannian
manifolds by using the formalism of dilatation structures
 Spatiile neolonome ale lui Vranceanu din punctul de vedere al geometriei distantei
(2007), Gazeta Matematica A, No. 4, 349352 (2008)
 In romanian, english title: Vranceanu' nonholonomic spaces
from the viewpoint of distance geometry. Mostly a review paper
concerning the appearance of nonholonomic spaces, discovered by
Vranceanu in 1926, in different places, such as subriemannian
geometry and geometric group theory. Easy to read.
 Dilatation structures with the RadonNikodym property (2007)
 In this paper I explain what is a pair of dilatation structures, one looking down to another.
Such a pair of dilatation structures leads to the intrinsic definition of a distribution as a field of
topological filters.
The RadonNikodym property for dilatation structures is the straightforward generalization of the RadonNikodym
property for Banach spaces.
It is shown that RadonNikodym property transfers from any "upper" dilatation structure looking down to a "lower"
dilatation structure. Im my opinion this result explains intrinsically the fact that absolutely continuous curves
in regular subRiemannian manifolds are derivable almost everywhere, as proved by MargulisMostow, Pansu
(for Carnot groups) or Vodopyanov.
 Linear dilatation structures and inverse
semigroups (2007)
 Here we prove that for dilatation structures linearity (see
arXiv:0705.1440v1) is equivalent to a statement about the inverse semigroup generated by the family of dilatations of the space.
The result is new for Carnot groups and the proof seems to be new even for vector spaces.
 Contractible groups and linear
dilatation structures (2007)
 A dilatation structure on a metric space,
math.MG/0608536, is a notion in
between a group and a differential structure, accounting for the approximate
selfsimilarity of the metric space.
The basic objects of a dilatation structure are dilatations
(or contractions). The axioms of a dilatation structure set the
rules of interaction between different dilatations.
 Linearity is also a property which can be explained with the
help of a dilatation structure. In this paper we show that we can
speak about two kinds of linearity: the linearity of a function
between two dilatation structures and the linearity of the
dilatation structure itself.
 Our main result here is a characterization of contractible
groups in terms of dilatation structures. To a normed conical group
(normed contractible group) we can naturally associate a linear
dilatation structure. Conversely, any linear and strong dilatation
structure comes from the dilatation structure of a normed
contractible
group.
 Linearization
of selfsimilar groups by dilatation structures
(2007), in preparation, available from the site of the conference
Geometric linearization of
graphs and groups, Centre
Interfacultaire Bernoulli, EPFL, Lausanne, Switzerland.
 In this article we study dilatation structures on the boundary
of the dyadic tree. We answer then to the question: given a
selfsimilar group of isometries of the dyadic tree, is there any
dilatation structure which makes the group linear? Here linearity
is meant in the generalized sense of dilatation structures: a
transformation is linear if it commutes (in the right way) with
dilatations.
 Dilatation
structures II. Linearity, selfsimilarity and the Cantor set
(2006)
 In this paper we continue the study of dilatation
structures, introduced in
math.MG/0608536. A dilatation structure on a metric space is a kind
of enhanced selfsimilarity. By way of examples this is explained here
with the help of the middlethirds Cantor set.
Linear and selfsimilar dilatation structures are introduced and
studied on ultrametric spaces, especially on the boundary of the dyadic
tree (same as the middlethirds Cantor set).
Some other examples of dilatation structures, which share some common
features, are given. As an application we prove that there is more than
one
linear and
selfsimilar dilatation structure on the Cantor set, induced by the
iterated functions system which defines the Cantor set.
Applications to selfsimilar groups are reserved for a further paper.
 Dilatation
structures I. Fundamentals. J. Gen. Lie Theory Appl. Vol 1 (2007), No 2,
6595.
Here is a link to the journal
site.
 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 series of papers which show that such a structure allows to do
analysis, in the sense of differential calculus, on a metric space. We
also describe a formal, universal calculus with binary decorated planar
trees, which underlies any dilatation structure.
 SubRiemannian
geometry and Lie groups. Part II. Curvature of metric
spaces, coadjoint orbits and associated representations (2004)
(current version pdf)
 This
paper is the third in a series dedicated to the fundamentals of
subRiemannian geometry and its implications in Lie groups theory:
"SubRiemannian geometry and Lie groups. Part I", and "Tangent bundles
to subRiemannian groups".
 The dilatation
structure is the basic object in the study of differential properties
of metric spaces of a certain type. (For example any Riemannian or
subRiemannian manifold, with or without conical singularities, has a
dilatation structure. But there are metric spaces which are not
admitting metric tangent spaces, but they admit dilatation structures).
This structure tells us what is the good notion of analysis on that
space. There is an infinite class of different such analysis and the
classical one, which we call Euclidean, is only one of them.
 We establish here a bridge between:
 curvature notion in spaces which admit metric
tangent spaces at any point, and
 selfadjoint representations of algebras naturally
associated with the structure of the tangent space.
 Curvature
of subRiemannian spaces (2003) (current version ps or pdf)
 To any metric space there is an associated metric
profile. The rectifiability of the metric profile gives a good notion
of curvature of a subRiemannian space. We shall say that a curvature
class is the rectifiability class of the metric profile. We classify
then the curvatures by looking to homogeneous metric spaces. The
classification problem is solved for contact 3 manifolds, where we
rediscover a 3 dimensional family of homogeneous contact manifolds,
with a distinguished 2 dimensional family of contact manifolds which
don't have a natural group structure.
 Tangent
bundles to subRiemannian groups (2003) (current version ps or pdf )
 Parts of the paper have been
presented in two lectures during the Séminaire Borel 2003
"Tangent spaces to metric spaces",
IIIème Cycle Romand de Mathématiques, Bern, Switzerland.
It is a continuation of "SubRiemannian geometry and Lie groups. Part I". It contains various
constructions of tangent bundles and nonEuclidean analysis for Lie
groups endowed with left invariant distributions, seen as
subRiemannian manifolds.
 SubRiemannian
geometry and Lie groups. Part I, (2002)
 These pages covers my expository talks during the
seminar "SubRiemannian geometry and Lie groups" organised by the
author and Tudor Ratiu at the Mathematics Department, EPFL, 2001.
However, this is the first part of three, dedicated to this subject. It
covers, with mild modifications, an elementary introduction to the
field.
 Volume
preserving biLipschitz homeomorphisms on the Heisenberg
group(2002)
 The purpose of this note is to make some connection
between the subRiemannian geometry on CarnotCaratheodory groups and
symplectic geometry. We shall concentrate here on the Heisenberg group,
although it is transparent that almost everything can be done on a
general CarnotCaratheodory group.
 Symplectic,
Hofer and subRiemannian geometry(2002)
 Elementary subRiemannian geometry on the Heisenberg
group H(n) provides a compact picture of symplectic geometry. Any
Hamiltonian diffeomorphism of the 2n dimensional Euclidean space lifts
to a volume preserving biLipschitz homeomorphisms of H(n), with the
use of its generating function. Any curve of a flow of such
homeomorphisms deviates from horizontality by the Hamiltonian of the
flow. From the metric point of view this means that any such curve has
Hausdorff dimension 2 and the (area) density equals the Hamiltonian.
The nondegeneracy of the Hofer distance is a direct consequence of
this fact.
