  In this article, we give numerical restrictions on the Chern classes of
Ulrich bundles on higher-dimensional manifolds, which are inspired by the
results of Casnati in the case of surfaces. As a by-product, we prove that the
only projective manifolds whose tangent bundle is Ulrich are the twisted cubic
and the Veronese surface. Moreover, we prove that the cotangent bundle is never
Ulrich.
