Linear foliations on affine manifolds

TL;DR

This paper investigates affine manifolds with linear foliations, establishing conditions under which such manifolds are homeomorphic to tori or fiber bundles, based on properties like completeness, holonomy, and the nature of the developing map.

Contribution

It characterizes the topology of affine manifolds with linear foliations, showing they are homeomorphic to tori or fiber bundles under specific geometric conditions.

Findings

Compact affine manifolds with codimension 1 linear foliations are homeomorphic to tori if leaves are simply connected.

3D affine manifolds with such foliations have finite covers that are bundles over the circle when the developing map is injective.

Injectivity and convexity of the developing map imply a bundle structure over the circle.

Abstract

In this paper, we study affine manifolds endowed with linear foliations. These are foliations defined by vector subspaces invariant by the linear holonomy. We show that an $n$-dimensional compact, complete, and oriented affine manifold endowed with a codimension $1$ linear foliation ${\cal F}$ is homeomophic to the $n$-dimensional torus if the leaves of ${\cal F}$ are simply connected. Let $(M,\nabla_M)$ be a $3$-dimensional compact affine manifold endowed with a codimension $1$ linear foliation. We prove that $(M,\nabla_M)$ has a finite cover which is homeomorphic to the total space of a bundle over the circle if its developing map is injective, and has a convex image.