Search results for "Hyperbolic geometry"

La théorie des lignes parallèles de Johann Heinrich Lambert

2014

International audience; The memoir "Theory of parallel lines" (1766) by Johannes Heinrich Lambert is one of the founding texts of hyperbolic geometry, even though his author conceived it as an attempt to show that this geometry does not exist. In fact, Lambert developed that theory with the hope of finding a contradiction. In doing so, he obtained several fundamental results of hyperbolic geometry. This was sixty years before the first writings of Lobachevsky and Bolyai appeared in print. This book contains the first complete translation of Lambert's memoir as well as mathematical and historical commentaries.

A classification of $\protect \mathbb{R}$-Fuchsian subgroups of Picard modular groups

2018

A Survey of Some Arithmetic Applications of Ergodic Theory in Negative Curvature

2017

This paper is a survey of some arithmetic applications of techniques in the geometry and ergodic theory of negatively curved Riemannian manifolds, focusing on the joint works of the authors. We describe Diophantine approximation results of real numbers by quadratic irrational ones, and we discuss various results on the equidistribution in \(\mathbb{R}\), \(\mathbb{C}\) and in the Heisenberg groups of arithmetically defined points. We explain how these results are consequences of equidistribution and counting properties of common perpendiculars between locally convex subsets in negatively curved orbifolds, proven using dynamical and ergodic properties of their geodesic flows. This exposition…