On varieties with Ulrich twisted tangent bundles

We study varieties $X\subseteq {\mathbb{P}}^N$ of dimension $n$ such that $T_X(k)$ is an Ulrich vector bundle for some $k\in \mathbb{Z}$. First we give a sharp bound for $k$ in the case of curves. Then we show that $k\le n+1$ if $2\le n\le 12$. We classify the pairs $(X,{\mathcal{O}}_X(1))$ for $k=1$ and we show that, for $n\ge 4$, the case $k=2$ does not occur.