Abelian Livshits theorems and geometric applications

We introduce a notion of abelian cohomology in the context of smooth flows. This is an equivalence relation which is weaker than the standard cohomology equivalence relation for flows. We develop Livshits theory for abelian cohomology over transitive Anosov flows. In particular, we prove an abelian Livshits theorem for homologically full Anosov flows. Then we apply this theorem to strengthen marked length spectrum rigidity for negatively curved surfaces. We also present an application to rigidity of contact Anosov flows. Some new results on homologically full Anosov flows are also given.