  The non-commutative Bianchi I curved space-time vierbeins and spin
connections are derived. Moreover, the corresponding non-commutative Dirac
equation as well as its solutions are presented. As an application within the
quantum field theory approach using Bogoliubov transformations, the von Neumann
fermion-antifermion pair creation quantum entanglement entropy is studied. It
is shown that its behaviour is strongly dependent on the value of the
non-commutativity $\theta$ parameter, $k_\perp$-modes frequencies and the
structure of the curved space-time. Various discussions of the obtained
features are presented.
