Geometric triangulations and flips

We prove that for a given flat surface with conical singularities, any pair of geometric triangulations can be connected by a chain of flips.Comment: 5 page

Stiff connections in pseudo-Euclidean manifolds

For a smooth manifold endowed with a (similarity) pseudo-Euclidean structure, a stiff connection $\nabla$ is a symmetric affine connection such that geodesics of $\nabla$ are straight lines of the pseudo-Euclidean structure while the first-order infinitesimal holonomy at each point is an infinitesimal isometry. In this paper, we give a complete classification of stiff connections in a local chart, identify canonical models and start investigating the global geometry of (similarity) pseudo-Euclidean manifolds endowed with a stiff connection. In the conformal class of the pseudo-Euclidean metric g, a stiff connection $\nabla$ defines a pseudo-Riemannian metric h such that unparameterized geodesics of $\nabla$ coincide with unparameterized geodesics of g but have a constant speed with respect to the so-called isochrone metric h. In particular, we obtain a new natural connection on the open unit ball that provides a compromise between properties of Cayley-Klein and Poincar\'e hyperbolic models.Comment: 51 pages, 5 figures, 1 tabl

Similarity surfaces, connections, and the measurable Riemann mapping theorem

This article studies a particular process that approximates solutions of the Beltrami equation (straightening of ellipse fields, a.k.a. measurable Riemann mapping theorem) on $\mathbb{C}$. It passes through the introduction of a sequence of similarity surfaces constructed by gluing polygons, and we explain the relation between their conformal uniformization and the Schwarz-Christoffel formula. Numerical aspects, in particular the efficiency of the process, are not studied, but we draw interesting theoretical consequences. First, we give an independent proof of the analytic dependence, on the data (the Beltrami form), of the solution of the Beltrami equation (Ahlfors-Bers theorem). For this we prove, without using the Ahlfors-Bers theorem, the holomorphic dependence, with respect to the polygons, of the Christoffel symbol appearing in the Schwarz-Christoffel formula. Second, these Christoffel symbols define a sequence of parallel transports on the range, and in the case of a data that is $C^2$ with compact support, we prove that it converges to the parallel transport associated to a particular affine connection, which we identify.Comment: 95 pages, 17 figure