Closed ray nil-affine manifolds and parabolic geometries

TL;DR

This paper investigates closed manifolds with ray nil-affine structures, establishing conditions under which they are complete or have developing maps covering complements of subspaces, with implications for parabolic geometries.

Contribution

It characterizes the completeness and developing map properties of closed ray nil-affine manifolds under specific geometric conditions.

Findings

Closed manifolds are either complete or their developing map covers the complement of a nil-affine subspace.

Presence of a parallel volume or non-proper automorphism group action implies completeness.

The results extend previous work on ray manifolds in affine geometry.

Abstract

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.