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

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

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

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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.