On Ulrich bundles on projective bundles

In this article, the existence of Ulrich bundles on projective bundles P(E)→X\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {P}}}({{\mathcal {E}}}) \rightarrow X$$\end{document} is discussed. In the case, that the base variety X is a curve or surface, a close relationship between Ulrich bundles on X and those on P(E)\documentclass[12pt]{minimal} \usepackage{amsmath} \usepackage{wasysym} \usepackage{amsfonts} \usepackage{amssymb} \usepackage{amsbsy} \usepackage{mathrsfs} \usepackage{upgreek} \setlength{\oddsidemargin}{-69pt} \begin{document}$${{\mathbb {P}}}({{\mathcal {E}}})$$\end{document} is established for specific polarisations. This yields the existence of Ulrich bundles on a wide range of projective bundles over curves and some surfaces.