Flat manifolds and reducibility

TL;DR

This paper explores the algebraic and geometric properties of holonomy-invariant sublattices and foliations in compact flat manifolds, extending classical results on reducibility and providing detailed descriptions of leaf geometries and examples.

Contribution

It analyzes the structure of holonomy-invariant sublattices and the geometry of associated foliations, including the existence of complements and descriptions of leaf geometries.

Findings

Holonomy-invariant sublattices form a closed class under spans and intersections.

Such sublattices admit (usually nonorthogonal) complements.

Descriptions of the intrinsic geometry of generic leaves are provided.

Abstract

Hiss and Szczepa\'nski proved in 1991 that the holonomy group of any compact flat Riemannian manifold, of dimension at least two, acts reducibly on the rational span of the Euclidean lattice associated with the manifold via the first Bieberbach theorem. Geometrically, their result states that such a manifold must admit a nonzero proper parallel distribution with compact leaves. We study algebraic and geometric properties of the sublattice-spanned holonomy-invariant subspaces that exist due to the above theorem, and of the resulting compact-leaf foliations of compact flat manifolds. The class consisting of the former subspaces, in addition to being closed under spans and intersections, also turns out to admit (usually nonorthogonal) complements. As for the latter foliations, we provide descriptions, first -- and foremost -- of the intrinsic geometry of their generic leaves in terms of that of the original flat manifold and, secondly -- as an essentially obvious afterthought -- of the leaf-space orbifold. The general conclusions are then illustrated by examples in the form of generalized Klein bottles.