Complete curvature homogeneous metrics on \(\mathrm{SL}_2(\mathbb{R})\) (Q2346669)

The authors describe a construction that associates to each positive smooth function \(F: \mathbb{S}^1\to\mathbb{R}\) a smooth Riemannian metric \(g_F\) on \(\mathrm{SL}_2 (\mathbb{R})\cong \mathbb{R}^2\times\mathbb{S}^1\) that is complete and curvature homogeneous. The construction respects moduli: positive smooth functions \(F\) and \(G\) lie in the same \(\mathrm{Diff}(\mathbb{S}^1)\)-orbit if and only if the associated metrics \(g_F\) and \(g_G\) lie in the same \(\mathrm{Diff}(\mathrm{SL}_2(\mathbb{R}))\)-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 \((\mathrm{SL}_2 (\mathbb{R}), g_F)\) Riemannian covers a finite volume manifold. Applications of the construction are discussed.