Title: Three-manifolds with constant vector curvature

Authors: Benjamin Schmidt and Jon Wolfson

Issue: Volume 63 (2014), Issue 6, 1757-1783


A connected Riemannian manifold $M$ has {\it constant vector curvature $\epsilon$}, denoted by $\operatorname{cvc}(\epsilon)$, if every tangent vectorlinebreak $v \in TM$ lies in a 2-plane with sectional curvature $\epsilon$.  When the sectional curvatures satisfy an additional bound $\sec \leq \epsilon$ or $\sec \geq \epsilon$, we say that $\epsilon$ is an \textit{extremal} curvature.     

In this paper, we study three-manifolds with constant vector curvature.  Our main results show that finite volume $\operatorname{cvc}(\epsilon)$ three-manifolds with extremal curvature $\epsilon$ are locally homogenous when $\epsilon=-1$, and admit a local product decomposition when $\epsilon=0$.  As an application, we deduce a hyperbolic rank-rigidity theorem.