Three-manifolds with constant vector curvature
Benjamin SchmidtJon Wolfson
53C2453C3053C21Riemannian manifoldcurvaturehomogeneous space
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.
Indiana University Mathematics Journal
2014
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10.1512/iumj.2014.63.5436
10.1512/iumj.2014.63.5436
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Indiana Univ. Math. J. 63 (2014) 1757 - 1783
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