Tangent lines and Lipschitz differentiability spaces

TL;DR

This paper investigates the existence and structure of tangent lines in tangent spaces of metric spaces, establishing their presence in Lipschitz differentiability spaces and their relation to local dimension and curvature conditions.

Contribution

It demonstrates that tangent spaces of Lipschitz differentiability spaces contain at least as many tangent lines as the local dimension, and under curvature conditions, these lines span an n-dimensional subset.

Findings

Tangent lines exist in tangent spaces at points of metric differentiability.

Number of tangent lines corresponds to the local measurable chart dimension.

Additional curvature assumptions lead to tangent lines spanning an n-dimensional part.

Abstract

We study the existence of tangent lines, i.e. subsets of the tangent space isometric to the real line, in tangent spaces of metric spaces. We first revisit the almost everywhere metric differentiability of Lipschitz continuous curves. We then show that any blow-up done at a point of metric differentiability and of density one for the domain of the curve gives a tangent line. Metric differentiability enjoys a Borel measurability property and this will permit us to use it in the framework of Lipschitz differentiability spaces. We show that any tangent space of a Lipschitz differentiability space contains at least $n$ distinct tangent lines, obtained as the blow-up of $n$ Lipschitz curves, where $n$ is the dimension of the local measurable chart. Under additional assumptions on the space, such as curvature lower bounds, these $n$ distinct tangent lines span an $n$-dimensional part of the tangent space.