Closed ray nil-affine manifolds and parabolic geometries

Raphaël V Alexandre (1, 2)

1 OURAGAN - OUtils de Résolution Algébriques pour la Géométrie et ses ApplicatioNs
2 IMJ-PRG (UMR_7586) - Institut de Mathématiques de Jussieu - Paris Rive Gauche

Ray nil-affine geometries are defined on nilpotent spaces. They occur in every parabolic geometry and in those cases, the nilpotent space is an open dense subset of the corresponding flag manifold. We are interested in closed manifolds having a ray nil-affine structure. We show that under a rank one condition on the isotropy, closed manifolds are either complete or their developing map is a cover onto the complement of a nil-affine subspace. We prove that if additionally there is a parallel volume or if the automorphism group acts non properly then closed manifolds are always complete. This paper is a sequel to a previous work on ray manifolds in affine geometry.

Domaines: Géométrie différentielle [math.DG]; Topologie géométrique [math.GT]

Raphaël V Alexandre. Closed ray nil-affine manifolds and parabolic geometries. 2021. ⟨hal-03433549⟩