On the notion of parallel transport on $\sf RCD$ spaces

TL;DR

This paper introduces a new concept of parallel transport in RCD spaces, establishing its existence and uniqueness under certain conditions, advancing the understanding of geometric analysis in these metric measure spaces.

Contribution

It defines a general framework for parallel transport on RCD spaces and proves its existence and uniqueness without additional conditions.

Findings

Established a general notion of parallel transport on RCD spaces.

Proved existence of parallel transport under suitable assumptions.

Proved uniqueness of the parallel transport in this setting.

Abstract

We propose a general notion of parallel transport on $\sf RCD$ spaces, prove an unconditioned uniqueness result and existence under suitable assumptions on the space.