The topological holonomy group and the complexity of horizontality

Abstract

Based on [1], we study the complexity of horizontality in each twistor space ${\hat{E}}_{\epsilon }$ associated with an oriented vector bundle $E$ of rank $4$ with a positive-definite metric over the $2$-torus $T^2$, and obtain classification of the topological holonomy groups in $SO(3)$. We observe that there exist many topological holonomy groups in $SO(3)$ generated by two finite order elements and equipped with noncommutative pairs which consist of infinite order elements. We find topological holonomy groups which are dense in $SO(4)$.

arXiv admin note: text overlap with arXiv:2402.02748 \\ The authors' comments: The two manuscripts arXiv:2402.02748 and arXiv:2507.20593 are incorporated into this manuscript, and accordingly, Anri Yonezaki becomes one of the authors