Three-manifolds with constant vector curvature

TL;DR

This paper investigates three-dimensional Riemannian manifolds with constant vector curvature, revealing they are locally homogeneous or decompose locally into products depending on the curvature value, with implications for hyperbolic rank rigidity.

Contribution

It characterizes finite volume cvc( extepsilon) three-manifolds with extremal curvature, showing they are locally homogeneous or decompose locally, and applies this to hyperbolic rank rigidity.

Findings

Finite volume cvc( extepsilon) three-manifolds with extremal curvature are locally homogeneous for extepsilon=-1.

Such manifolds admit a local product structure when extepsilon=0.

Derived a hyperbolic rank-rigidity theorem from these results.

Abstract

A connected Riemannian manifold M has constant vector curvature \epsilon, denoted by cvc(\epsilon), if every tangent vector v in TM lies in a 2-plane with sectional curvature \epsilon. By scaling the metric on M, we can always assume that \epsilon = -1, 0, or 1. When the sectional curvatures satisfy the additional bound that each sectional curvature is less than or equal to \epsilon, or that each sectional curvature is greater than or equal to \epsilon, we say that, \epsilon, is an extremal curvature. In this paper we study three-manifolds with constant vector curvature. Our main results show that finite volume 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.