only if a real line bundle is orientable, both conditions are then equivalent to a trivial determinant line bundle.
* The complex determinant line bundle preserves the first Chern class, which for complex line bundles over topological spaces with the homotopy type of a CW complex is a group isomorphism.
* The pullback bundle commutes with the determinant line bundle. For a continuous map between paracompact spaces and as well as a vector bundle , one has:
*:

: Proof: Assume is