New definitions of Alexandrov space and applications

TL;DR

This paper introduces weaker conditions for defining Alexandrov spaces using novel concepts like imaginary angles and support derivatives, leading to simplified proofs of key theorems in the field.

Contribution

It proposes new, less restrictive definitions of Alexandrov spaces and applies these to streamline proofs of fundamental theorems.

Findings

Weaker conditions suffice for defining Alexandrov spaces.

New proofs of the Doubling and Globalization Theorems.

Introduction of imaginary angles and support derivatives.

Abstract

In this paper we show that, in the definition of Alexandrov spaces with lower or upper curvature bound, the original conditions can be replaced with much weaker ones. For the purpose, we introduce `imaginary' comparison angles (and `imaginary' angles), and the right or left bounded second derivative in the support sense. As applications, we provide new proofs for the Doubling Theorem, and the Globalization Theorem for complete or geodesic Alexandrov spaces with lower curvature bound.