Dirac flow on the 3-sphere

We illustrate some well-known facts about the evolution of the 3-sphere (S3, g) generated by the Ricci flow. We define the Dirac flow and study the properties of the metric \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar g = dt^2 + g(t)$$\end{document}, where g(t) is a solution of the Dirac flow. In the case of a metric g conformally equivalent to the round metric on S3 the metric \documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$$\bar g$$\end{document} is of constant curvature. We study the properties of solutions in the case when g depends on two functional parameters. The flow on differential 1-forms whose solution generates the Eguchi–Hanson metric was written down. In particular cases we study the singularities developed by these flows.