6533b872fe1ef96bd12d30e6
RESEARCH PRODUCT
Tangent lines and Lipschitz differentiability spaces
Tapio RajalaFabio Cavallettisubject
Pure mathematicsLipschitz differentiability spaces; metric geometry; Ricci curvature; tangent of metric spaces01 natural sciencesMathematics - Metric GeometrySettore MAT/05 - Analisi MatematicaTangent lines to circles0103 physical sciencesTangent spaceClassical Analysis and ODEs (math.CA)FOS: Mathematicsmetric geometryDifferentiable function0101 mathematicsReal lineMathematicstangent of metric spacesQA299.6-433Applied Mathematics010102 general mathematicsTangentLipschitz differentiability spacesMetric Geometry (math.MG)Lipschitz continuityFunctional Analysis (math.FA)Mathematics - Functional AnalysisMetric spaceRicci curvatureMathematics - Classical Analysis and ODEsMetric (mathematics)010307 mathematical physicsGeometry and TopologyMathematics::Differential GeometryAnalysisdescription
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.
| year | journal | country | edition | language |
|---|---|---|---|---|
| 2015-03-03 |