We provide a general theory for parallel transport on non-collapsed 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 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.

The aim of this note is to explain in which sense an axiomatic Sobolev space over a general metric measure space (à la Gol’dshtein–Troyanov) induces – under suitable locality assumptions – a first-order differential structure.

UR - http://www.sciencedirect.com/science/article/pii/S0723086918300975 ER - TY - RPRT T1 - Quasi-continuous vector fields on RCD spaces Y1 - 2019 A1 - Clément Debin A1 - Nicola Gigli A1 - Enrico Pasqualetto ER - TY - JOUR T1 - The Serre–Swan theorem for normed modules JF - Rendiconti del Circolo Matematico di Palermo Series 2 Y1 - 2019 A1 - Danka Lučić A1 - Enrico Pasqualetto VL - 68 UR - https://doi.org/10.1007/s12215-018-0366-6 ER - TY - RPRT T1 - Differential of metric valued Sobolev maps Y1 - 2018 A1 - Nicola Gigli A1 - Enrico Pasqualetto A1 - Elefterios Soultanis ER - TY - RPRT T1 - On the notion of parallel transport on RCD spaces Y1 - 2018 A1 - Nicola Gigli A1 - Enrico Pasqualetto ER - TY - RPRT T1 - Behaviour of the reference measure on RCD spaces under charts Y1 - 2016 A1 - Nicola Gigli A1 - Enrico Pasqualetto ER - TY - RPRT T1 - Equivalence of two different notions of tangent bundle on rectifiable metric measure spaces Y1 - 2016 A1 - Nicola Gigli A1 - Enrico Pasqualetto ER -