Abstract
We study the topology and geometry of the space $$\varUpsilon $$ of shapes of abstract tetrahedra, i.e. flat metrics on the 2-sphere with four conical singularities, up to equivalence by orientation-preserving similarities. There is no labeling of the conical singularities; two of them with the same cone-angle can be interchanged by a similarity. We are dealing with the geometric structure introduced by Thurston through the area viewed as a quadratic form. $$\varUpsilon $$ is shown to be a quotient of a real analytic fibration over the 3-dimensional space of cone-angles, with each fiber being a thrice-punctured 2-sphere of constant curvature. In particular, $$\varUpsilon $$ is shown to be a contractible orbifold of dimension five. It is also observed that the metric given by the area form degenerates transversally to the fibers. A description of the real analytic structure of $$\varUpsilon $$ is given by hypergeometric functions.
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