Ph.D. Thesis

Inserted: 29 aug 2024
Last Updated: 29 aug 2024

Year: 2021
Links: PDF

Abstract:

The thesis mainly contains the construction of parallel transport on non-collapsed $\mathsf{RCD}(K,N)$ spaces, done in collaboration with N. Gigli and E. Pasqualetto. We obtain 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. In this generality, we don't study parallel transport along a single such curve, but along a generic collection of such integral curves. The notion of flow under consideration is the one of regular Lagrangian flow, after the axiomatization in the nonsmooth setting by Ambriosio and Trevisan. A preliminary introduction on calculus on metric measure spaces and the theory of flows of Sobolev vector fields, both in euclidean and nonsmooth setting, is also included.