2021/08/17 by Emanuele Caputo, Nicola Gigli, Caputo, Emanuele +3
Mathematics · #46G12 #49J52 #53C23 #Differential Geometry (math.DG) #FOS: Mathematics #Functional Analysis (math.FA) #Geometric Analysis and Curvature Flows #Geometry and complex manifolds #Markov Chains and Monte Carlo Methods #Metric Geometry (math.MG)
paper · pdf · doi:10.48550/arxiv.2108.07531
openalex publication_date 2021/08/17 · openalex created_date 2025/10/10 · openalex updated_date 2026/07/28
We provide a general theory for parallel transport on non-collapsed \sf RCD spaces obtaining both existence and uniqueness results. Our theory covers the case of geodesics and, more generally, of curves obtained via the flow of sufficiently regular time dependent vector fields: the price that we pay for this generality is that we cannot study parallel transport along a single such curve, but only along almost all of these (in a sense related to the notions of Sobolev vector calculus and Regular Lagrangian Flow in the nonsmooth setting). The class of \sf ncRCD spaces contains finite dimensional Alexandrov spaces with curvature bounded from below, thus our construction provides a way of speaking about parallel transport in this latter setting alternative to the one proposed by Petrunin (1998). The precise relation between the two approaches is yet to be understood.