Appears in collection : Geometry in non-positive curvature and Kähler groups

This talk is based on joint works with Anna Wienhard and François Labourie. In a number of situations, an action on a flag manifold is well controlled via a structure mimicking the orientability of the circle. After reviewing the classical examples of such structures, we will propose an axiomatization of them leading to a complete classification. We will then describe some properties of corresponding positive actions or representations.