# CURVATURES ON THE ABBENA-THURSTON MANIFOLD

• Han, Ju-Wan (Department of Applied Mathematics, Pukyong National University) ;
• Kim, Hyun Woong (Department of Applied Mathematics, Pukyong National University) ;
• Pyo, Yong-Soo (Department of Applied Mathematics, Pukyong National University)
• Received : 2016.01.03
• Accepted : 2016.02.03
• Published : 2016.06.25
• 83 16

#### Abstract

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

#### Keywords

Heisenberg group;Abbena-Thurston manifold;scalar curvature;homogeneous Riemannian manifold;Ricci curvature

#### Acknowledgement

Supported by : Pukyong National University

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