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

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.