Projective nilmanifolds (Q1343327)

A nilmanifold is a compact quotient of a nilpotent Lie group by a discrete subgroup. An affine (resp. projective) structure on a \(C^\infty\)-manifold is an atlas whose charts take values in \(\mathbb{R}^n\) (resp. \(\mathbb{S}^n\)) and whose coordinate changes are affine (resp. projective) transformations. The author studies properties of affine and projective structures on nilmanifolds. The main results are: 1) On ``filiform'' nilmanifolds, any affine or projective structure is left invariant. 2) There exist nilmanifolds without projective structure.