The sum of the interior angles in geodesic and translation triangles of \(\widetilde{\mathrm{SL}_2(\mathbb{R})}\) geometry. (English) Zbl 1499.52028
Summary: We study the interior angle sums of translation and geodesic triangles in the universal cover of real \(2 \times 2\) matrices with unit determinant, as a Thurston geometry denoted by \(P\) of \(\widetilde{\mathrm{SL}_2(\mathbb{R})}\) geometry. We prove that the angle \(\sum^3_{i=1}(\alpha_i) \geq \pi\) for translation triangles and for geodesic triangles the angle sum can be larger, equal or less than \(\pi \).
MSC:
| 52C17 | Packing and covering in \(n\) dimensions (aspects of discrete geometry) |
| 52A55 | Spherical and hyperbolic convexity |
| 51M20 | Polyhedra and polytopes; regular figures, division of spaces |
Keywords:
Thurston geometries; \(\widetilde{\mathrm{SL}_2(\mathbb{R})}\) geometry; spherical geometry; translation triangles; geodesic trianglesReferences:
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