Three-Dimensional Alexandrov spaces with positive or nonnegative Ricci curvature

TL;DR

This paper classifies three-dimensional Alexandrov spaces with positive or nonnegative Ricci curvature bounds, showing they are topologically either spherical space forms, suspensions of real projective planes, or other specific types.

Contribution

It provides a classification of closed three-dimensional Alexandrov spaces with Ricci curvature bounds in the d^* sense, identifying their topological types.

Findings

Spaces with d^*(2,3) are homeomorphic to spherical space forms or suspensions of b^2.

Spaces with d^*(0,3) are classified explicitly.

The work extends Ricci curvature classification to Alexandrov spaces in three dimensions.

Abstract

We study closed three-dimensional Alexandrov spaces with a lower Ricci curvature bound in the $\mathsf{CD}^*(K,N)$ sense, focusing our attention on those with positive or nonnegative Ricci curvature. First, we show that a closed three-dimensional $\mathsf{CD}^*(2,3)$-Alexandrov space must be homeomorphic to a spherical space form or to the suspension of $\mathbb{R}P^2$. We then classify closed three-dimensional $\mathsf{CD}^*(0,3)$-Alexandrov spaces.