Nonorientable manifolds, complex and symplectic structures, and characteristic classes (Q2360540)

The authors define an almost complex structure on a non-orientable manifold \(M\) and a corresponding Nijenhuis tensor. They give an integrability condition and extend the notions of complex and holomorphic bundle to this case. They also consider a symplectic structure on \(M\). They use the orientation bundle \(L\), a real line bundle of order two, to twist differential forms and study \(L\)-symplectic forms and the Poisson bracket, \(L\)-contact forms, \(L\)-complex structures and \(L\)-holomorphic bundles, thus extending results from the classical case to the non-orientable case.