Tangent Lines and Lipschitz Differentiability Spaces
Cavalletti, F., & Rajala, T. (2016). Tangent Lines and Lipschitz Differentiability Spaces. Analysis and Geometry in Metric Spaces, 4 (1). doi:10.1515/agms-2016-0004
Published inAnalysis and Geometry in Metric Spaces
© the Authors, 2013. This is an open access article distributed under the terms of the Creative Commons Attribution-Non-Commercial-NoDerivs license.
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 rst revisit the almost everywhere metric di erentiability of Lipschitz continuous curves. We then show that any blow-up done at a point of metric di erentiability and of density one for the domain of the curve gives a tangent line. Metric di erentiability enjoys a Borel measurability property and this will permit us to use it in the framework of Lipschitz di erentiability spaces. We show that any tangent space of a Lipschitz di erentiability 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.