Fundamental groups of manifolds with positive sectional curvature and torus symmetry

TL;DR

This paper reviews and extends results on the fundamental groups of positively curved manifolds with torus symmetry, providing new classifications and insights into their structure.

Contribution

It offers a comprehensive overview and new findings on the fundamental groups of manifolds with positive sectional curvature under torus symmetry.

Findings

New classifications of fundamental groups under torus symmetry

Extension of existing results to broader categories

Enhanced understanding of symmetry's role in curvature constraints

Abstract

In 1965, S.-S. Chern posed a question concerning the extent to which fundamental groups of manifolds admitting positive sectional curvature look like spherical space form groups. The original question was answered in the negative by Shankar in 1998, but there are a number of positive results in the presence of symmetry. These classifications fall into categories according to the strength of their conclusions. We give an overview of these results in the case of torus symmetry and prove new results in each of these categories.