Papers by Athanase Papadopoulos
Geometriae Dedicata, Jan 31, 2012
A bstract. In the Teichm uller space ofa hyperbolic surface of nite type,w e construct geodesic l... more A bstract. In the Teichm uller space ofa hyperbolic surface of nite type,w e construct geodesic lines for T hurston's asym m etric m etric having the property that w hen they are traversed in the reverse direction, they are also geodesic lines(up to reparam etrization). T he linesw e construct are specialstretch lines in the sense ofT hurston. T hey are directed by com plete geodesic lam inations that are not chain-recurrent, and they have a nice description in term s of Fenchel-N ielsen coordinates. A t the basis ofthe construction are certain m aps w ith controlled Lipschitz constants betw een right-angled hyperbolic hexagons having three non-consecutive edges ofthe sam e size. U sing these m aps,w e obtain Lipschitz-m inim izing m aps betw een hyperbolic particular pairs of pants and, m ore generally, betw een som e hyperbolic sufaces of nite type w ith arbitrary genus and arbitrary num ber ofboundary com ponents. T he Lipschitzm inim izing m aps that w e contruct are distinct from T hurston's stretch m aps. A M S M athem atics Subject C lassi cation: 32G 15 ;30F30 ;30F60.
HAL (Le Centre pour la Communication Scientifique Directe), 2022
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
On hyperbolic geometry and the history of its reception
HAL (Le Centre pour la Communication Scientifique Directe), 2010
arXiv (Cornell University), Mar 8, 2023
We first review the notion of timelike metric spaces. This is a metric theory developed by Herber... more We first review the notion of timelike metric spaces. This is a metric theory developed by Herbert Busemann, as a geometrical setting for the theory of general relativity. We review in particular the notions of timelike Hilbert and timelike spherical Hilbert geometry. We prove the following result on the timelike spherical Hilbert geometry of simplices: Let ∆ 2 be a simplex on the 2-sphere and∆ 2 the antipodal simplex. We show that the timelike spherical Hilbert geometry associated with the pair ∆ 2 ,∆ 2 is isometric to a union of six copies of vector spaces equipped with a timelike norm, isometrically and transitively acted upon by the group R 2 >0 × Z 3 × Z 2. This is a timelike spherical analogue of a well-known result (due to Busemann) stating that the Hilbert metric of a Euclidean simplex is isometric to a metric induced by a normed vector space. At the same time, this gives a new example of timelike space.
EMS Press eBooks, Apr 30, 2015
We develop a transitional geometry, that is, a family of geometries of constant curvatures which ... more We develop a transitional geometry, that is, a family of geometries of constant curvatures which makes a continuous connection between the hyperbolic, Euclidean and spherical geometries. In this transitional setting, several geometric entities like points, lines, distances, triangles, angles, area, curvature, etc. as well as trigonometric formulae and other properties transit in a continuous manner from one geometry to another.
HAL (Le Centre pour la Communication Scientifique Directe), 2015
For each right-angled hexagon in the hyperbolic plane, we construct a one-parameter family of rig... more For each right-angled hexagon in the hyperbolic plane, we construct a one-parameter family of right-angled hexagons with a Lipschitz map between any two elements in this family, realizing the smallest Lipschitz constant in the homotopy class of this map relative to the boundary. As a consequence of this construction, we exhibit new geodesics for the arc metric on the Teichmüller space of an arbitrary surface of negative Euler characteristic with nonempty boundary. We also obtain new geodesics for Thurston's metric on Teichmüller spaces of hyperbolic surfaces without boundary. Our results generalize results obtained in the two papers [8] and [9].
HAL (Le Centre pour la Communication Scientifique Directe), 2014
We provide hyperbolic analogues of some classical theorems in spherical geometry due to Menelaus,... more We provide hyperbolic analogues of some classical theorems in spherical geometry due to Menelaus, Euler, Lexell, Ceva and Lambert. Some of the spherical results are also made more precise. Our goal is to go through the works of some of the eminent mathematicians from the past and to include them in a modern perspective. Putting together results in the three constantcurvature geometries and highlighting the analogies between them is mathematically as well as aesthetically very appealing.
Springer eBooks, 2022
The use of general descriptive names, registered names, trademarks, service marks, etc. in this p... more The use of general descriptive names, registered names, trademarks, service marks, etc. in this publication does not imply, even in the absence of a specific statement, that such names are exempt from the relevant protective laws and regulations and therefore free for general use.
Euler and Maupertuis on the Figure of the Earth
Springer eBooks, 2022
Réseaux ferroviaires et courbes simples sur une surface (Train tracks and simple curves on surfaces)
HAL (Le Centre pour la Communication Scientifique Directe), 1983
HAL (Le Centre pour la Communication Scientifique Directe), Aug 17, 2022
Toute ma métaphysique sous-jacente, c'est d'essayer de transformer le conceptuel en géométrique, ... more Toute ma métaphysique sous-jacente, c'est d'essayer de transformer le conceptuel en géométrique, le logique en dynamique.
arXiv (Cornell University), Oct 17, 2011
We study the action of the elements of the mapping class group of a surface of finite type on the... more We study the action of the elements of the mapping class group of a surface of finite type on the Teichmüller space of that surface equipped with Thurston's asymmetric metric. We classify such actions as elliptic, parabolic, hyperbolic and pseudo-hyperbolic, depending on whether the translation distance of such an element is zero or positive and whether the value of this translation distance is attained or not, and we relate these four types to Thurston's classification of mapping classes. The study is parallel to the one made by Bers in the setting of Teichmüller space equipped with Teichmüller's metric, and to the one made by Daskalopoulos and Wentworth in the setting of Teichmüller space equipped with the Weil-Petersson metric.
HAL (Le Centre pour la Communication Scientifique Directe), Nov 18, 2021
Christiaan Huygens : Écrits sur la musique et le son
HAL (Le Centre pour la Communication Scientifique Directe), 2021
International audienceThe book contains all the texts written by Christiaan Huygens on music theo... more International audienceThe book contains all the texts written by Christiaan Huygens on music theory and the theory of sound, with translations from the Latin into French. These texts are accompanied by mathematical, physical, musical, biographical and historical notes, and several essays explaining the place of Huygens' works in the sciences and arts literature since the seventeenth century till our days.Le présent volume réunit tous les textes de Christiaan Huygens sur la musique et sur la théorie du son, avec une traduction en français de tous ceux qui sont originellement écrits en latin. Ces textes sont accompagnés de commentaires mathématiques, musicaux et historiques, d’une biographie de Christiaan Huygens, d’extraits de sa correspondance sur la musique, ainsi que de plusieurs essais expliquant la place de son œuvre dans la littérature musicale et dans la musique, depuis le dix-septième siècle jusqu’à nos jours.Cet ouvrage sera spécialement utile aux étudiants et chercheurs en musique et en acoustique, mais il s’adresse aussi à un public plus large, en particulier à toutes les personnes intéressées par l’alliage entre la science et l’art, par l’histoire des sciences et leur transmission, notamment par la vie culturelle dans l’espace européen du dix-septième siècle
Leohnard Euler, Ecrits sur la musique, Vol. 2
Le second volume des œuvres d'Euler sur la musique est divise en cinq parties. La premiere pa... more Le second volume des œuvres d'Euler sur la musique est divise en cinq parties. La premiere partie contient une biographie d’Euler ainsi qu’une introduction a la place de la musique dans l’œuvre generale de cet auteur. La seconde partie contient les trois memoires d'Euler sur la musique : Conjecture sur la raison de quelques dissonances generalement recues dans la musique (presente a l’Academie des sciences de Berlin en 1760), Du veritable caractere de la musique moderne (presente a la meme Academie en 1764) et le memoire De harmoniae veris principiis per speculum musicum repraesentatis (presente en 1773 a l’Academie des sciences de Saint-Petersbourg) traduit du latin. Ces trois memoires sont accompagnes d’introductions et de notes historiques et techniques. La troisieme partie presente des extraits des Lettres a une princesse d’Allemagne qui concernent la musique et le son, ainsi que des notes et commentaires sur ces extraits. La quatrieme partie est centree sur la correspon...
The boundary of a torsion-free hyperbolic group as a semi-Markovian space
On the legacy of Ibn Al-Haytham: an exposition based on the work of Roshdi Rashed
We report of the work of Ibn al-Haytham, an Arabic scholar who was settled in Cairo in the eleven... more We report of the work of Ibn al-Haytham, an Arabic scholar who was settled in Cairo in the eleventh century and who worked in several fields, including mathematics, physics and philosophy. We review some of of his work on optics, astronomy, number theory and especially spherical geometry. Our report is mostly based on the critical editions published by Roshdi Rashed, a specialist of Ibn al-Haytham and the world expert on Arabic and Greek mathematics and their interaction. We also provide the reader with a report on the life of Ibn al-Haytham, his influence, and the general background in which he flourished.
We comment on some combinatorial aspects of Nasir al-Din al-Tusi's treatise on the quadrilate... more We comment on some combinatorial aspects of Nasir al-Din al-Tusi's treatise on the quadrilateral, a 13th century work on spherical trigonometry. The final version of this paper will appear in Ganita Bharati (the Bulletin of the Indian Society for History of Mathematics).
A Commentary on Teichmüller’s paper Vollständige Lösung einer Extremalaufgabe der quasikonformen Abbildung
Handbook of Teichmüller Theory, Volume VI
We comment on Teichmuller 's paper "Vollstandige Losung einer Extremalaufgabe der quasik... more We comment on Teichmuller 's paper "Vollstandige Losung einer Extremalaufgabe der quasikonformen Abbildung" (Complete solution of an ex-tremal problem of the quasiconformal mapping),, published in 1941. In this paper, Teichmuller gives a proof of the existence of extremal quasiconformal mappings in the case of the pentagon (disc with five distinguished points on the boundary). The final version of this paper will appear as a chapter in Volume VI of the Handbook of Teichmuller theory.
Arc length as a global conformal parameter for analytic curves
Journal of Mathematical Analysis and Applications, 2017
Abstract We show that arc length is a global conformal parameter for analytic curves and that thi... more Abstract We show that arc length is a global conformal parameter for analytic curves and that this parameter can be used to decide whether the domain of definition of an analytic curve can be extended or not. The maximal extension with respect to the arc length parameter is the largest possible extension (over all parametrizations of the curve). Our proof is elementary, simple and short. Several examples are given in the plane, and the results remain true for curves in an arbitrary Euclidean space R k .
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Papers by Athanase Papadopoulos