An introduction to Reshetnyak's theory of subharmonic distances

TL;DR

This paper introduces Reshetnyak's theory of subharmonic distances, providing foundational concepts and main results, emphasizing their distinction from bounded curvature distances, and serving as a comprehensive overview for readers.

Contribution

It offers a clear, accessible overview of Reshetnyak's subharmonic distances theory, highlighting its autonomous nature and key results, complementing existing translations.

Findings

Subharmonic distances form a distinct autonomous theory.

The theory clarifies differences from bounded curvature distances.

Main results are summarized for foundational understanding.

Abstract

The aim of the present text is to provide some basics around Reshetnyak's theory of subharmonic distances, together with an overview of the main results. This text is intended to be a complement to the English translation of \cite{R1954}, \cite{Res1959}, \cite{R60I}, \cite{R60II}, \cite{R61}, \cite{R61b}, \cite{R62}, \cite{R63}, \cite{R63III} and \cite{huber}. They all will be published in a same volume \cite{livre}. While subharmonic distances are often confused with distances of bounded curvature, we present them as a complete autonomous theory. In turn, there is no specific prerequisite for the present text.