Three-manifolds with many flat planes

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 3 -manifold. We prove a rank rigidity theorem for complete 3 3 -manifolds, showing that having higher rank is equivalent to having reducible universal covering. We also study 3 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-analyticity assumptions.