Ulrich bundles with $c_2(\mathcal{E}{)}^2=0$ and connectedness of Ulrich subvarieties

We give an almost complete classification of Ulrich bundles $\mathcal{E}$ with $c_2(\mathcal{E}{)}^2=0$ on a variety $X$ of dimension $n\ge 4$. Moreover, we show that there are strong constraints on the geometry of $X$ and we study disconnected Ulrich subvarieties.