Positively curved Riemannian orbifolds and Alexandrov spaces with circle symmetry in dimension 4

TL;DR

This paper classifies 4-dimensional positively curved Alexandrov spaces and Riemannian orbifolds with circle symmetry, providing a comprehensive understanding of their geometric structure under specific conditions.

Contribution

It offers a classification of positively curved 4-dimensional Alexandrov spaces and orbifolds with circle actions, under a conjectured geometric condition.

Findings

Classification of 4D positively curved Alexandrov spaces with circle symmetry

Classification of 4D positively curved Riemannian orbifolds with circle symmetry

Results depend on a conjectured condition on fixed point infinitesimal geometry

Abstract

We classify positively curved Alexandrov spaces of dimension 4 with an isometric circle action up to equivariant homeomorphism, subject to a certain additional condition on the infinitesimal geometry near fixed points which we conjecture is always satisfied. As a corollary, we also classify positively curved Riemannian orbifolds of dimension 4 with an isometric circle action.