On the geometry and topology of positively curved Eschenburg orbifolds

The present article explores the relationship between positive sectional curvature and the geometric and topological properties of Eschenburg 6-orbifolds. First, we prove that positive sectional curvature imposes restrictions on their singular sets, thereby confirming a conjecture posed by Florit and Ziller. Then we compute the orbifold cohomology rings of those with a specific singular locus. This reveals a distinctive behaviour in the cohomology groups of positively curved Eschenburg orbifolds compared to their non-negatively curved counterparts. Furthermore, we compute the orbifold cohomology rings of all Eschenburg orbifolds.