Three-manifolds with many flat planes

TL;DR

This paper investigates the geometric structure of 3-manifolds with many flat (zero curvature) planes, establishing rigidity results and exploring the implications of various conditions on their universal coverings.

Contribution

It proves a rank rigidity theorem linking higher rank to reducible universal coverings and examines manifolds where every tangent vector lies in a flat plane.

Findings

Higher rank implies reducible universal covering.

Examples of manifolds with all tangent vectors in flat planes.

Finite volume and real-analytic conditions influence manifold structure.

Abstract

We discuss the rigidity (or lack thereof) imposed by different notions of having an abundance of zero curvature planes on a complete Riemannian 3-manifold. We prove a rank rigidity theorem for complete 3-manifolds, showing that having higher rank is equivalent to having reducible universal covering. We also study 3-manifolds such that every tangent vector is contained in a flat plane, including examples with irreducible universal covering, and discuss the effect of finite volume and real-analiticity assumptions.