Nonorientable manifolds, complex and symplectic structures, and characteristic classes

A complex manifold or a symplectic manifold is automatically oriented. We investigate these structures in the context of non-orientable manifolds. Any smooth connected non-orientable manifold is equipped with a real line bundle of order two. Various structures which are defined only on oriented manifolds extend to non-orientable manifolds once they are twisted by this line bundle of order two. Our aim is to develop this theme.