Examples of $CD(0,N)$ spaces with non-constant dimension

TL;DR

This paper constructs examples of $CD(0,N)$ spaces with varying local dimensions, showing that classical geometric and optimal transport properties may fail without additional assumptions, and explores the distinctions between different $CD(0,N)$ conditions.

Contribution

It provides explicit examples of $CD(0,N)$ spaces with non-constant dimension and analyzes the failure of expected geometric and transport properties.

Findings

Topological splitting can fail in $CD(0,N)$ spaces.

Non-branching conditions may not hold in $CD(0,N)$ spaces.

Existence of optimal transport maps is not guaranteed without non-branching.

Abstract

In this work, we generalize the results obtained in (J. Geom. Anal., 32(6):Paper No.173, 32, 2022), presenting some examples of $CD(0,N)$ spaces having different dimensions in different regions, deducing in particular that the topological splitting may fail in $CD(0,N)$ spaces. We also observe that any reasonable non-branching condition may fail in $CD(0,N)$ spaces and that the existence of an optimal transport map, between two absolutely continuous marginals, is not guaranteed by the $CD(0,N)$ condition, without requiring a non-branching assumption. Moreover, we show that the strict $CD(0,N)$ condition is strictly stronger than the classical $CD(0,N)$ one and it is not stable with respect to the measured Gromov-Hausdorff convergence.