Parallel transport on non-collapsed 𝖱𝖢𝖣(𝐾, 𝑁) spaces

Abstract We provide a general theory for parallel transport on non-collapsed <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="sans-serif">RCD</m:mi> </m:math> \mathsf{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 non-smooth setting). The class of <m:math xmlns:m="http://www.w3.org/1998/Math/MathML"> <m:mi mathvariant="sans-serif">ncRCD</m:mi> </m:math> \mathsf{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 in [Parallel transportation for Alexandrov space with curvature bounded below, Geom. Funct. Anal. 8 (1998), 1, 123–148]. The precise relation between the two approaches is yet to be understood.