The deformation spaces of geodesic triangulations of flat tori

We prove that the deformation space of geodesic triangulations of a flat torus is homotopically equivalent to a torus. This solves an open problem proposed by Connelly et al. in 1983 in the case of flat tori. A key tool of the proof is a generalization of Tutte's embedding theorem for flat tori. While this paper was under preparation, Erickson and Lin proved a similar result, which works for all convex drawings.