  We prove that the deformation space of geodesic triangulations of a flat
torus is homotopy 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. When this paper
is under preparation, Erickson and Lin proved a similar result, which works for
all convex drawings.
