The radiance obstruction and parallel forms on affine manifolds

Source abstract

A manifold M M is affine if it is endowed with a distinguished atlas whose coordinate changes are locally affine. When they are locally linear M M is called radiant. The obstruction to radiance is a one-dimensional class c M {c_M} with coefficients in the flat tangent bundle of M M . Exterior powers of c M {c_M} give information on the existence of parallel forms on M M , especially parallel volume forms. As applications, various kinds of restrictions are found on the holonomy and topology of compact affine manifolds.