Calculus of Variations and Geometric Measure Theory

B. Han - A. Mondino

Angles between curves in metric measure spaces

created by mondino on 18 Jan 2017
modified on 11 Sep 2017


Published Paper

Inserted: 18 jan 2017
Last Updated: 11 sep 2017

Journal: Analysis and Geometry in Metric Spaces
Volume: 5
Number: 1
Pages: 47--68
Year: 2017


The goal of the paper is to study the angle between two curves in the framework of metric (and metric measure) spaces. More precisely, we give a new notion of angle between two curves in a metric space. Such a notion has a natural interplay with optimal transportation and is particularly well suited for metric measure spaces satisfying the curvature-dimension condition. Indeed one of the main results is the validity of the cosine formula on $RCD^{*}(K,N)$ metric measure spaces. As a consequence, the new introduced notions are compatible with the corresponding classical ones for Riemannian manifolds, Ricci limit spaces and Alexandrov spaces.