Tangent Lines and Lipschitz Differentiability Spaces

Fabio Cavalletti, and Tapio Rajala. "Tangent Lines and Lipschitz Differentiability Spaces." Analysis and Geometry in Metric Spaces 4.1 (2016): 85-103, electronic only. <http://eudml.org/doc/277094>.

@article{FabioCavalletti2016,
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.},
author = {Fabio Cavalletti, Tapio Rajala},
journal = {Analysis and Geometry in Metric Spaces},
keywords = {metric geometry; Lipschitz differentiability spaces; tangent of metric spaces; Ricci curvature},
language = {eng},
number = {1},
pages = {85-103, electronic only},
title = {Tangent Lines and Lipschitz Differentiability Spaces},
url = {http://eudml.org/doc/277094},
volume = {4},
year = {2016},
}

TY - JOUR
AU - Fabio Cavalletti
AU - Tapio Rajala
TI - Tangent Lines and Lipschitz Differentiability Spaces
JO - Analysis and Geometry in Metric Spaces
PY - 2016
VL - 4
IS - 1
SP - 85
EP - 103, electronic only
AB - 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.
LA - eng
KW - metric geometry; Lipschitz differentiability spaces; tangent of metric spaces; Ricci curvature
UR - http://eudml.org/doc/277094
ER -