Submetries vs. submersions

TL;DR

This paper investigates submetries between Alexandrov spaces, highlighting how certain properties of Riemannian submersions do not hold in the absence of smoothness, thus exploring the geometric structure of these non-smooth spaces.

Contribution

The paper provides a detailed analysis of submetries in Alexandrov spaces and demonstrates the failure of typical Riemannian submersion features without smoothness.

Findings

Certain features of Riemannian submersions do not extend to Alexandrov spaces.

Examples illustrating the failure of smoothness-dependent properties.

Insights into the geometric structure of Alexandrov spaces.

Abstract

We study submetries between Alexandrov spaces and show how some of the usual features of Riemannian submersions fail due to the lack of smoothness.