Topology and Dynamics of compact plane waves

TL;DR

This paper investigates the structure of compact locally homogeneous plane waves, showing they are mostly standard or semi-standard quotients, and explores conditions for standardness, with implications for the flow of isotropic vector fields.

Contribution

It characterizes compact quotients of homogeneous plane waves as standard or semi-standard and identifies conditions for standardness, advancing understanding of their geometric and dynamical properties.

Findings

Compact quotients are essentially standard or semi-standard.

Conditions are identified for a quotient to be standard.

The flow of the parallel isotropic vector field is equicontinuous.

Abstract

We study compact locally homogeneous plane waves. Such a manifold is a quotient of a homogeneous plane wave $X$ by a discrete subgroup of its isometry group. This quotient is called standard if the discrete subgroup is contained in a connected subgroup of the isometry group that acts properly cocompactly on $X$. We show that compact quotients of homogeneous plane waves are ``essentially" standard; more precisely, we show that they are standard or `semi-standard'. We find conditions which ensure that a quotient is not only semi-standard but even standard. As a consequence of these results, we obtain that the flow of the parallel isotropic vector field of a compact locally homogeneous plane wave is equicontinuous.