Dilatation structures in sub-riemannian geometry
نویسنده
چکیده
Based on the notion of dilatation structure [2], we give an intrinsic treatment to sub-riemannian geometry, started in the paper [4]. Here we prove that regular sub-riemannian manifolds admit dilatation structures. From the existence of normal frames proved by Belläıche we deduce the rest of the properties of regular sub-riemannian manifolds by using the formalism of dilatation structures.
منابع مشابه
Dilatation structures with the Radon-Nikodym property
The notion of a dilatation structure stemmed out from my efforts to understand basic results in sub-Riemannian geometry, especially the last section of the paper by Belläıche [2] and the intrinsic point of view of Gromov [5]. In these papers, as in other articles devoted to sub-Riemannian geometry, fundamental results admiting an intrinsic formulation were proved using differential geometry too...
متن کاملOn the Kirchheim-Magnani counterexample to metric differentiability
In Kirchheim-Magnani [7] the authors construct a left invariant distance ρ on the Heisenberg group such that the identity map id is 1-Lipschitz but it is not metrically differentiable anywhere. In this short note we give an interpretation of the Kirchheim-Magnani counterexample to metric differentiability. In fact we show that they construct something which fails shortly from being a dilatation...
متن کاملSub-riemannian geometry from intrinsic viewpoint
Gromov proposed to extract the (differential) geometric content of a sub-riemannian space exclusively from its Carnot-Carathéodory 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 co...
متن کاملA 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 Gamma-convergence of length functionals. Then we show that the main properties of sub-riemannian spaces can be obtained from pairs of length dilatation structures, the first being a tempered one and...
متن کاملSymmetries of Flat Rank Two Distributions and Sub-riemannian Structures
Flat sub-Riemannian structures are local approximations — nilpotentizations — of sub-Riemannian structures at regular points. Lie algebras of symmetries of flat maximal growth distributions and sub-Riemannian structures of rank two are computed in dimensions 3, 4, and 5. 1. Sub-Riemannian structures A sub-Riemannian geometry is a triple (M,∆, 〈·, ·〉), where M is a smooth manifold, ∆ ⊂ TM is a s...
متن کامل