Interior angle sums of geodesic triangles in $S^2 \times R$ and $H^2 \times R$ geometries

Abstract

In the present paper we study $S^2 \times R$ and $H^2 \times R$ geometries, which are homogeneous Thurston 3-geometries. We analyse the interior angle sums of geodesic triangles in both geometries and prove, that in $S^2 \times R$ space it can be larger or equal than $\pi$ and in $H^2 \times R$ space the angle sums can be less or equal than $\pi$. In our work we will use the projective model of $S^2 \times R$ and $H^2 \times R$ geometries described by E. Moln\'ar in \cite{M97}.