Communications in Mathematical Analysis

A Characterization of Sub-Riemannian Spaces as Length Dilation Structures Constructed Via Coherent Projections

Marius Buliga

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Abstract

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. Thus we get an intrinsic, synthetic, axiomatic description of sub-riemannian geometry, which transforms the classical construction of a Carnot-Carathéodory distance on a regular sub-riemannian manifold into a model for this abstract sub-riemannian geometry.

Article information

Source
Commun. Math. Anal., Volume 11, Number 2 (2011), 70-111.

Dates
First available in Project Euclid: 25 February 2011

Permanent link to this document
https://projecteuclid.org/euclid.cma/1298669956

Mathematical Reviews number (MathSciNet)
MR2780883

Zentralblatt MATH identifier
1214.51004

Subjects
Primary: 51K10: Synthetic differential geometry 53C17: Sub-Riemannian geometry 53C23: Global geometric and topological methods (à la Gromov); differential geometric analysis on metric spaces

Keywords
Spaces with dilations self-similar spaces metric spaces with Radon-Nikodym property sub-riemannian geometry Gamma-convergence

Citation

Buliga, Marius. A Characterization of Sub-Riemannian Spaces as Length Dilation Structures Constructed Via Coherent Projections. Commun. Math. Anal. 11 (2011), no. 2, 70--111. https://projecteuclid.org/euclid.cma/1298669956


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