Projective bundles over small covers and topological triviality problem

This paper investigates the projectivization of real vector bundles over small covers. We first give a necessary and sufficient condition for such a projectivization to be a small cover. Then associated with moment-angle manifolds, we further study the structure of such a projectivization as a small cover. As an application, we characterize the real projective bundles over 2-dimensional small covers by interpreting the fibre sum operation to some combinatorial operation. Finally, we study when the projectivization of Whitney sum of the tautological line bundle and the tangent bundle over real projective space is diffeomorphic to the product of two real projective spaces.