Pythagorean Theorem & curvature with lower or upper bound

TL;DR

This paper extends the Pythagorean Theorem to compare curvature bounds in Alexandrov spaces and Riemannian manifolds, providing a new tool for analyzing geometric curvature constraints.

Contribution

It introduces a comparison version of the Pythagorean Theorem to determine curvature bounds in Alexandrov spaces, including Riemannian manifolds.

Findings

Provides a new comparison theorem for curvature bounds

Applies to Alexandrov spaces and Riemannian manifolds

Enhances understanding of geometric curvature constraints

Abstract

In this paper, we give a comparison version of Pythagorean Theorem to judge the lower or upper bound of the curvature of Alexandrov spaces (including Riemannian manifolds).