MARIUS BULIGA

 Back to my homepage Spaces with dilations / Sub-Riemannian geometry / Non-euclidean analysis
 PAPERS Sub-riemannian geometry from intrinsic viewpoint. These are the notes prepared for the course "Metric spaces with dilations and sub-riemannian geometry from intrinsic point of view", CIMPA research school on sub-riemannian geometry (2012). Gromov proposed to extract the (differential) geometric content of a sub-riemannian space exclusively from its Carnot-Carath\'eodory distance. One of the most striking features of a regular sub-riemannian 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 one-parameter group of automorphisms. Siebert result has not a metric character. In these notes I show that sub-riemannian 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. Lambda-Scale, a lambda calculus for spaces with dilations (2012). Lambda-Scale 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 sub-riemannian spaces as length dilation structures constructed via coherent projections Commun. Math. Anal. 11 (2011), No. 2, pp. 70-111, (pdf) We introduce length dilation structures on metric spaces, tempered dilation structures and coherent projections and explore the relations between these objects and the Radon-Nikodym property and Gamma-convergence of length functionals. Then we show that the main properties of sub-riemannian 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), 91-136. 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 sub-riemannian symmetric spaces. in: Geometry. Exploratory Workshop on Differential Geometry and its Applications, eds. D. Andrica, S. Moroianu, Cluj-Napoca 2011, 21-35 (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 sub-riemannian symmetric spaces. Keywords: braided sets, quandles; emergent algebras; dilatation structures (spaces with dilations); sub-riemannian 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 Kirchheim-Magnani counterexample to metric differentiability (2007) In this short note we give an interpretation of the Kirchheim-Magnani counterexample to metric differentiability in terms of dilatation structures. Self-similar dilatation structures and automata (2007), Proceedings of the 6-th Congress of Romanian Mathematicians, Bucharest, 2007, vol. 1, 557-564 (2008) We show that on the boundary of the dyadic tree, any self-similar dilatation structure induces a web of interacting automata. This is a short version, for publication, of the paper arXiv:math/0612509v2. Dilatation structures in sub-riemannian geometry (2007) in: Contemporary Geometry and Topology and Related Topics, Cluj-Napoca, Cluj University Press (2008), 89-105 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 sub-riemannian manifolds admit dilatation structures. From the existence of normal frames proved by Bellaiche we deduce the rest of the properties of regular sub-riemannian 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, 349-352 (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 sub-riemannian geometry and geometric group theory. Easy to read. Dilatation structures with the Radon-Nikodym 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 Radon-Nikodym property for dilatation structures is the straightforward generalization of the Radon-Nikodym property for Banach spaces. It is shown that Radon-Nikodym 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 sub-Riemannian manifolds are derivable almost everywhere, as proved by Margulis-Mostow, 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 self-similarity 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 self-similar 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 self-similar 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, self-similarity 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 self-similarity. By way of examples this is explained here with the help of the middle-thirds Cantor set. Linear and self-similar dilatation structures are introduced and studied on ultrametric spaces, especially on the boundary of the dyadic tree (same as the middle-thirds 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 self-similar dilatation structure on the Cantor set, induced by the iterated functions system which defines the Cantor set. Applications to self-similar groups are reserved for a further paper. Dilatation structures I. Fundamentals. J. Gen. Lie Theory Appl. Vol 1 (2007), No 2, 65-95. 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. Sub-Riemannian 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 sub-Riemannian geometry and its implications in Lie groups theory: "Sub-Riemannian geometry and Lie groups. Part I", and "Tangent bundles to sub-Riemannian 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 sub-Riemannian 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 self-adjoint representations of algebras naturally associated with the structure of the tangent space. Curvature of sub-Riemannian 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 sub-Riemannian 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 sub-Riemannian 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 "Sub-Riemannian geometry and Lie groups. Part I". It contains various constructions of tangent bundles and non-Euclidean analysis for Lie groups endowed with left invariant distributions, seen as sub-Riemannian manifolds. Sub-Riemannian geometry and Lie groups. Part I, (2002) These pages covers my expository talks during the seminar "Sub-Riemannian 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 bi-Lipschitz homeomorphisms on the Heisenberg group(2002) The purpose of this note is to make some connection between the sub-Riemannian geometry on Carnot-Caratheodory 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 Carnot-Caratheodory group. Symplectic, Hofer and sub-Riemannian geometry(2002) Elementary sub-Riemannian 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 bi-Lipschitz 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 non-degeneracy of the Hofer distance is a direct consequence of this fact.