The topological holonomy group and the complexity of horizontality

TL;DR

This paper investigates the complexity of horizontality in twistor spaces associated with vector bundles over the 2-torus, classifies topological holonomy groups in SO(3), and explores their density in SO(4).

Contribution

It provides a classification of topological holonomy groups in SO(3) for certain vector bundles and reveals the existence of dense holonomy groups in SO(4).

Findings

Existence of many holonomy groups generated by two finite order elements.

Presence of noncommutative pairs with infinite order elements.

Holonomy groups dense in SO(4).

Abstract

Based on [1], we study the complexity of horizontality in each twistor space $\hat{E}_{\varepsilon}$ 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 $SO(4)$.