Complete curvature homogeneous metrics on ${\rm SL}_2(\mathbb R)$

Benjamin Schmidt and Jon Wolfson

Vol. 273 (2015), No. 2, 499–509

DOI: 10.2140/pjm.2015.273.499

Abstract

A construction is described that associates to each positive smooth function $F:S^1\to ℝ$ a smooth Riemannian metric $g_F$ on ${SL}_2\left(ℝ\right)\cong {ℝ}^2\times S^1$ that is complete and curvature homogeneous. The construction respects moduli: positive smooth functions $F$ and $G$ lie in the same $Diff\left(S^1\right)$ orbit if and only if the associated metrics $g_F$ and $g_G$ lie in the same $Diff\left({SL}_2\left(ℝ\right)\right)$ orbit.

The constructed metrics all have curvature tensor modeled on the same algebraic curvature tensor. Moreover, the following are shown to be equivalent: $F$ is constant, $g_F$ is left-invariant, and $\left({SL}_2\left(ℝ\right),g_F\right)$ Riemannian covers a finite volume manifold. Applications of the construction are discussed.

Keywords

curvature homogeneous space, homogeneous space, constant vector curvature

Mathematical Subject Classification 2010

Primary: 53C21

Secondary: 22F30

Milestones

Received: 31 March 2014

Accepted: 12 May 2014

Published: 23 December 2014