Horizontality with infinite complexity in the twistor spaces on tori

Abstract

We study the complexity of horizontality in the twistor space $\hat{E}$ associated with an oriented vector bundle $E$ of rank $4$ with a positive-definite metric over a torus. If the horizontality has finite complexity of degree $d>2$ for an element of a fiber of $\hat{E}$, then the complexity is expressed in terms of a finite subgroup of $SO(3)$ ([3]). In the present paper, we observe that if the horizontality has infinite complexity derived from one of the cases studied in [3], then the complexity is expressed by a dense subset of $S^2$.