CURVATURES ON THE ABBENA-THURSTON MANIFOLD

Abstract

Let H be the 3-dimensional Heisenberg group, (<TEX>$G=H{\times}S^1$</TEX>, g) a product Riemannian manifold of Riemannian manifolds H and S with arbitrarily given left invariant Riemannian metrics respectively, and <TEX>${\Gamma}$</TEX> the discrete subgroup of G with integer entries. Then, on the Riemannian manifold (<TEX>$M:=G/{\Gamma}$</TEX>, <TEX>${\Pi}^*g=\bar{g}$</TEX>), <TEX>${\Pi}:G{\rightarrow}G/{\Gamma}$</TEX>, we evaluate the scalar curvature and the Ricci curvature.