COMPLETE CURVATURE HOMOGENEOUS METRICS ON SL2(R)

A construction is described that associates to each positive smooth function F : S-1 -> R a smooth Riemannian metric g(F) on SL2(R) congruent to R-2 x S-1 that is complete and curvature homogeneous. The construction respects moduli: positive smooth functions F and G lie in the same Diff (S-1) orbit if and only if the associated metrics g(F) and g(G) lie in the same Diff(SL2(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 (SL2(R), g(F)) Riemannian covers a finite volume manifold. Applications of the construction are discussed.