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arXiv · 2108.07531

Parallel transport on non-collapsed $\mathsf{RCD}(K,N)$ spaces

Emanuele Caputo·Nicola Gigli·Enrico Pasqualetto
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Abstract

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.

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Emanuele Caputo, Nicola Gigli, Enrico Pasqualetto. 2021-08-17. Parallel transport on non-collapsed $\mathsf{RCD}(K,N)$ spaces. https://arxiv.org/abs/2108.07531

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