Three-manifolds with constant vector curvature

Abstract

A connected Riemannian manifold M has constant vector curvature ε, denoted by cvc(ε), if every tangent vector v ∈ T M lies in a 2-plane with sectional curvature ε.When the sectional curvatures satisfy an additional bound sec ≤ ε or sec ≥ ε, we say that ε is an extremal curvature.In this paper we study three-manifolds with constant vector curvature.Our main results show that finite volume cvc(ε) three-manifolds with extremal curvature ε are locally homogenous when ε = -1 and admit a local product decomposition when ε = 0.As an application, we deduce a hyperbolic rankrigidity theorem.