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

## TL;DR

This paper investigates the interior angle sums of geodesic triangles in the homogeneous Thurston geometries $S^2 imes R$ and $H^2 imes R$, showing they can be greater or less than $\

## Contribution

It provides a detailed analysis of geodesic triangle angle sums in $S^2 imes R$ and $H^2 imes R$ geometries, expanding understanding of their geometric properties.

## Key findings

- In $S^2 imes R$, angle sums can be greater or equal to $\
- In $H^2 imes R$, angle sums can be less or equal to $\
- Utilizes the projective model of these geometries for analysis.

## 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}.