Corpus ID: 237142320

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

@article{Caputo2021ParallelTO,
  title={Parallel transport on non-collapsed \$\mathsf\{RCD\}(K,N)\$ spaces},
  author={Emanuele Caputo and Nicola Gigli and Enrico Pasqualetto},
  journal={arXiv: Differential Geometry},
  year={2021}
}
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… Expand

References

SHOWING 1-10 OF 39 REFERENCES
Well posedness of Lagrangian flows and continuity equations in metric measure spaces
We establish, in a rather general setting, an analogue of DiPerna-Lions theory on well-posedness of flows of ODE's associated to Sobolev vector fields. Key results are a well-posedness result for theExpand
Parallel Transportation for Alexandrov Space with Curvature Bounded Below
Abstract. In this paper we construct a "synthetic" parallel transportation along a geodesic in Alexandrov space with curvature bounded below, and prove an analog of the second variation formula forExpand
Gradient flows and Evolution Variational Inequalities in metric spaces. I: Structural properties
This is the first of a series of papers devoted to a thorough analysis of the class of gradient flows in a metric space $(X,\mathsf{d})$ that can be characterized by Evolution VariationalExpand
Korevaar–Schoen’s energy on strongly rectifiable spaces
We extend Korevaar–Schoen’s theory of metric valued Sobolev maps to cover the case of the source space being an $$\mathsf{RCD}$$ RCD space. In this situation it appears that no version of theExpand
Sobolev spaces in metric measure spaces: reflexivity and lower semicontinuity of slope
In this paper we make a survey of some recent developments of the theory of Sobolev spaces $W^{1,q}(X,\sfd,\mm)$, $1<q<\infty$, in metric measure spaces $(X,\sfd,\mm)$. In the final part of the paperExpand
Calculus and heat flow in metric measure spaces and applications to spaces with Ricci bounds from below
This paper is devoted to a deeper understanding of the heat flow and to the refinement of calculus tools on metric measure spaces $(X,\mathsf {d},\mathfrak {m})$. Our main results are: A generalExpand
Ricci curvature for metric-measure spaces via optimal transport
We dene a notion of a measured length space X having nonnegative N-Ricci curvature, for N 2 [1;1), or having1-Ricci curvature bounded below byK, forK2 R. The denitions are in terms of theExpand
On the differential structure of metric measure spaces and applications
The main goals of this paper are: i) To develop an abstract differential calculus on metric measure spaces by investigating the duality relations between differentials and gradients of SobolevExpand
Constancy of the Dimension for RCD( K , N ) Spaces via Regularity of Lagrangian Flows
We prove a regularity result for Lagrangian flows of Sobolev vector fields over RCD(K,N) metric measure spaces, regularity is understood with respect to a newly defined quasi-metric built from theExpand
Metric measure spaces with Riemannian Ricci curvature bounded from below
In this paper we introduce a synthetic notion of Riemannian Ricci bounds from below for metric measure spaces (X,d,m) which is stable under measured Gromov-Hausdorff convergence and rules out FinslerExpand
...
1
2
3
4
...