2019/06/30 by Matías del Hoyo, Matias del Hoyo, Mateus de Melo
Mathematics · #Geodesic #Geometric Analysis and Curvature Flows #Geometric and Algebraic Topology #Homotopy and Cohomology in Algebraic Topology #Mathematical analysis #Mathematics #Orbit (dynamics) #Pure mathematics #math.DG
paper · pdf · doi:10.1007/s00031-020-09596-y
published as Transformation Groups (2020) · 23 pages
arxiv created 2019/10/23 · openalex publication_date 2020/07/10 · arxiv updated 2020/08/04 · openalex created_date 2025/10/10 · openalex updated_date 2026/08/05
Metrics on Lie groupoids and differentiable stacks have been introduced recently, extending the Riemannian geometry of manifolds and orbifolds to more general singular spaces. Here we continue that theory, studying stacky curves on Riemannian stacks, measuring their length using stacky metrics, and introducing stacky geodesics. Our main results show that the length of stacky curves measure distances on the orbit space, characterize stacky geodesics as locally minimizing curves, and establish a stacky version of Hopf-Rinow Theorem. We include a concise overview that bypasses nonessential technicalities, and we lay stress on the examples of orbit spaces of isometric actions and leaf spaces of Riemannian foliations.