00401nas a2200109 4500008004100000245005500041210005200096100002200148700002400170700001800194856007900212 2021 eng d00aOn master test plans for the space of BV functions0 amaster test plans for the space of BV functions1 aNobili, Francesco1 aPasqualetto, Enrico1 aSchultz, Timo uhttps://www.math.sissa.it/publication/master-test-plans-space-bv-functions06995nas a2200121 4500008004100000245006500041210005800106520655200164100002106716700001806737700002406755856009406779 2021 eng d00aParallel transport on non-collapsed $\mathsfRCD(K,N)$ spaces0 aParallel transport on noncollapsed mathsfRCDKN spaces3 a

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.

1 aCaputo, Emanuele1 aGigli, Nicola1 aPasqualetto, Enrico uhttps://www.math.sissa.it/publication/parallel-transport-non-collapsed-mathsfrcdkn-spaces00784nas a2200157 4500008004100000022001400041245006600055210006600121520024700187653002800434653002100462653003000483100001800513700002400531856007100555 2019 eng d a0723-086900aDifferential structure associated to axiomatic Sobolev spaces0 aDifferential structure associated to axiomatic Sobolev spaces3 a

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.

10aAxiomatic Sobolev space10aCotangent module10aLocality of differentials1 aGigli, Nicola1 aPasqualetto, Enrico uhttp://www.sciencedirect.com/science/article/pii/S072308691830097500394nas a2200109 4500008004100000245004900041210004800090100002000138700001800158700002400176856008400200 2019 eng d00aQuasi-continuous vector fields on RCD spaces0 aQuasicontinuous vector fields on RCD spaces1 aDebin, Clément1 aGigli, Nicola1 aPasqualetto, Enrico uhttps://www.math.sissa.it/publication/quasi-continuous-vector-fields-rcd-spaces00411nas a2200145 4500008004100000022001400041245004800055210004400103260000800147300001400155490000700169100001900176700002400195856004600219 2019 eng d a1973-440900aThe Serre–Swan theorem for normed modules0 aSerre–Swan theorem for normed modules cAug a385–4040 v681 aLučić, Danka1 aPasqualetto, Enrico uhttps://doi.org/10.1007/s12215-018-0366-600395nas a2200109 4500008004100000245004700041210004700088100001800135700002400153700002600177856008200203 2018 eng d00aDifferential of metric valued Sobolev maps0 aDifferential of metric valued Sobolev maps1 aGigli, Nicola1 aPasqualetto, Enrico1 aSoultanis, Elefterios uhttps://www.math.sissa.it/publication/differential-metric-valued-sobolev-maps00361nas a2200097 4500008004100000245005400041210004700095100001800142700002400160856007900184 2018 eng d00aOn the notion of parallel transport on RCD spaces0 anotion of parallel transport on RCD spaces1 aGigli, Nicola1 aPasqualetto, Enrico uhttps://www.math.sissa.it/publication/notion-parallel-transport-rcd-spaces00407nas a2200097 4500008004100000245006600041210006600107100001800173700002400191856009400215 2016 eng d00aBehaviour of the reference measure on RCD spaces under charts0 aBehaviour of the reference measure on RCD spaces under charts1 aGigli, Nicola1 aPasqualetto, Enrico uhttps://www.math.sissa.it/publication/behaviour-reference-measure-rcd-spaces-under-charts00471nas a2200097 4500008004100000245009600041210006900137100001800206700002400224856012500248 2016 eng d00aEquivalence of two different notions of tangent bundle on rectifiable metric measure spaces0 aEquivalence of two different notions of tangent bundle on rectif1 aGigli, Nicola1 aPasqualetto, Enrico uhttps://www.math.sissa.it/publication/equivalence-two-different-notions-tangent-bundle-rectifiable-metric-measure-spaces