Three-manifolds with constant vector curvature

Details Three-manifolds with constant vector curvature Three-manifolds with constant vector curvature

2011-10-20

arxiv.org/abs/1110.4619v1

Mathematics

Differential Geometry

Scientific paper

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. We give descriptions of the moduli spaces of cvc(\epsilon) metrics for each case, \epsilon = -1, 0, or 1, under global conditions on M. For example, in the case \epsilon = -1 is extremal, we show, under the assumption that M has finite volume, that M is isometric to a locally homogeneous manifold. In the case that M is compact, \epsilon = 1 is extremal and there are no points in M with all sectional curvatures identically one, we describe the moduli space of cvc(1) metrics in terms of locally homogeneous manifolds and the solutions of an elliptic partial differential equation.

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