Dilatation structures with the Radon-Nikodym property

TL;DR

This paper introduces a generalized framework for dilatation structures with the Radon-Nikodym property, linking differentiability, length of curves, and sub-Riemannian geometry in metric spaces.

Contribution

It defines the Radon-Nikodym property for dilatation structures and proves its transfer between structures, extending classical geometric results to a broader setting.

Findings

Length of curves expressed as integral of tangent norms

Radon-Nikodym property transfers between dilatation structures

Generalization of differentiability in metric spaces

Abstract

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. To any pair of dilatation structures there is an associated notion of differentiability which generalizes the Pansu differentiability. This allows the introduction of the Radon-Nikodym property for dilatation structures, which is the straightforward generalization of the Radon-Nikodym property for Banach spaces. After an introducting section about length metric spaces and metric derivatives, is proved that for a dilatation structure with the Radon-Nikodym property the length of absolutely continuous curves expresses as an integral of the norms of the tangents to the curve, as in Riemannian geometry. Further it is shown that Radon-Nikodym property transfers from any "upper" dilatation structure looking down to a "lower" dilatation structure, theorem \ref{ttransfer}. 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.