The Deformation Space of Geodesic Triangulations and Generalized Tutte's Embedding Theorem

TL;DR

This paper proves the contractibility of the deformation space of geodesic triangulations on negatively curved closed surfaces, extending Tutte's embedding theorem to hyperbolic surfaces and solving a decades-old open problem.

Contribution

It generalizes Tutte's embedding theorem to hyperbolic surfaces and establishes the contractibility of their geodesic triangulation deformation space.

Findings

Proved the contractibility of the deformation space for geodesic triangulations on hyperbolic surfaces.

Generalized Tutte's embedding theorem for closed surfaces of negative curvature.

Solved an open problem posed in 1983 by Connelly et al.

Abstract

We proved the contractibility of the deformation space of the geodesic triangulations on a closed surface of negative curvature. This solves an open problem proposed by Connelly et al. in 1983, in the case of hyperbolic surfaces. The main part of the proof is a generalization of Tutte's embedding theorem for closed surfaces of negative curvature.