The space of Schwarz-Klein triangles

TL;DR

This paper characterizes the space of spherical triangles, showing it as a smooth 3-manifold with a specific topological structure, and describes how angles and sides relate analytically within this space.

Contribution

It provides a detailed geometric and topological description of the space of spherical triangles, including its manifold structure and analytic embedding.

Findings

The space of spherical triangles is a smooth connected orientable 3-manifold.

It is homotopy equivalent to the 1-skeleton of a cubic partition of the first octant.

Angles and sides are real analytic functions on this manifold, embedding it into a6^6.

Abstract

We describe the space of spherical triangles (in the sense of Schwarz and Klein). It is a smooth connected orientable 3-manifold, homotopy equivalent to the 1-skeleton of the cubic partition of the closed first octant in $\mathbf{R}^3$. The angles and sides are real analytic functions on this manifold which embed it to $\mathbf{R}^6$.