Generating and Adding Flows on Locally Complete Metric Spaces

TL;DR

This paper extends the theory of arc fields on locally complete metric spaces by proving existence and uniqueness of solution curves for time-dependent cases and their sums, generalizing previous results on vector fields.

Contribution

It introduces the concept of time-dependent arc fields and establishes their solution curve properties, expanding the mathematical framework for metric space analysis.

Findings

Existence and uniqueness of solution curves for time-dependent arc fields.

Definition and analysis of the sum of two time-dependent arc fields.

Extension of Cauchy-Lipschitz type results to metric spaces.

Abstract

As a generalization of a vector field on a manifold, the notion of an arc field on a locally complete metric space was introduced in \cite{BC}. In that paper, the authors proved an analogue of the Cauchy-Lipschitz Theorem i.e they showed the existence and uniqueness of solution curves for a time independent arc field. In this paper, we extend the result to the time dependent case, namely we show the existence and uniqueness of solution curves for a time dependent arc field. We also introduce the notion of the sum of two time dependent arc fields and show existence and uniqueness of solution curves for this sum.