Two orientations

All trivializations of an Euclidean line bundle π: ? → B over a connected base B split in two classes which can be naturally named orientations of π. In the case of an orienting sheaf of a manifold or a vector bundle, they admit a natural interpretation as orientations of these objects. This approach establishes an extension of standard classical constructions to all manifolds and vector bundles independently of orientability restrictions in the usual sense.