arXiv subjects

Ken'Ichi Ohshika

Publications and source records attributed to Ken'Ichi Ohshika.

16 recordsLinked to original sources

arXiv2026

On spaces of Euclidean triangles and triangulated Euclidean surfaces

In this paper, we introduce an asymmetric distance function on the space of marked Euclidean triangles of normalised area, and we prove several properties of this metric, which turns out to be (a restriction of) a non-symmetric version of the classical Thompson distance. We give a description of the geodesics of this metric, we show that it is Finsler, and we give a formula for its infinitesimal Finsler structure. We then introduce and study a Finsler metric of the space of singular Euclidean structures on a surface adapted to an underlying fixed triangulation, and we also study its geodesics and its Finsler infinitesimal structure. We then develop a theory of completeness and completion of asymmetric metrics which is adapted to our setting, and we use this theory in the study of the completeness of the metric we introduced on the space of triangles. In doing so, we establish a bridgebetween one aspect of Thurston's theory of metrics on spaces of surfaces and Thompson's metrics. The final version of this paper will appear in Monatshefte f{ü}r Mathematik

math.GT↗

arXiv2026

On the construction of geographical maps: Lagrange, Chebyshev, Darboux and Milnor

Lagrange, Chebyshev, and Darboux, in 1779, 1856, and 1911, respectively, wrote articles all bearing the same title, \emph{On the Construction of Geographical Maps}. In 1969, Milnor wrote a paper in which he refers to Chebyshev's paper, of which he provides a new formulation and proof. In the present article, we review the results of all these papers, explaining the main ideas they contain and pointing out connections between them. We give complete proofs of the statements by Darboux and Milnor, both of which aim to make explicit and provide a proof of Chebyshev's result, but whose contents are different. Although Chebyshev did not state explicitly what the word ``best'' means, his conclusion, like that of Darboux and of Milnor, is that a best geographical map is characterised by the fact that its conformal factor is constant on the boundary of the region represented. Our statement and proof of Milnor's theorem work in a more general setting than the one he gives. The final version of this paper will appear in the Handbook of Mathematics in the Arts and Sciences (second edition), ed. Bharath Sriraman, Springer, 2027.

math.DG↗

arXiv2026

Convex structures of the unit tangent spheres in Teichm{ü}ller space

We analyse the convex structure of the Finsler infinitesimal balls of the Thurston metric on Teichm{ü}ller space. We analyse the convex structure of the Finsler unit ball in the tangent space at each point of Teichm\''uller space of a closed surface of genus $\geq 2$ equipped with Thurston's metric. We obtain a characterisation of faces, exposed faces and extreme points of such a unit sphere. In particular, we prove that every face has a unique naturally associated chain-recurrent geodesic lamination such that this face consists of the unit tangent vectors which are linear combinations of stretch vectors along maximal chain-recurrent geodesic laminations containing the given one. We show that a face is exposed if and only if its associated chain-recurrent geodesic lamination is the support of a measured lamination. Furthermore, we show that a point on a tangent unit sphere is an extreme point if and only if it is a stretch vector along some maximal chain-recurrent geodesic lamination. The last result gives an affirmative answer to a conjecture whose answer was known positively in the case where the surface is either the once-punctured torus or the 4-punctured sphere. Our main results also provide an alternative approach to the topological part of the infinitesimal rigidity result concerning Thurston's metric and the equivariance property of stretch vectors.

math.GT↗

arXiv2026

Convergence-Symmetric Metric Spaces

We study some basic properties of spaces which satisfy the usual axioms of a metric space but without the symmetry axiom. We realised that such a study is needed in view of the relatively recent appearance of several papers on natural asymmetric metrics to which the available theories do not apply. One prominent example is Thurston's metric on the Teichm{ü}ller spaces of hyperbolic surfaces of finite type, introduced by Thurston in 1985, with several variants and generalisations. Another asymmetric metric is the earthquake metric, also introduced by Thurston and about which several basic questions remain open. Other asymmetric metrics we consider here include the Funk metric, the Apollonian metric and the left Hausdorff metric. We are particularly interested in questions of completeness and completion and in associated notions of boundary at infinity for these asymmetric metrics. In this paper, after discussing several examples, we prove results such as a Banach fixed point theorem, an Arzel{à}--Ascoli theorem and a Hopf--Rinow theorem adapted to this asymmetric setting. We introduce a property we call ``convergence-symmetry'' which turns out to be crucial in the study of metric completeness of some asymmetric metrics. This property is stronger than a property which was formulated by Herbert Busemann around 1970, and which we call the ``Busemann condition''. Every convergence-symmetric metric space has a unique minimal completion which is convergence-symmetric. This does not hold for spaces satisfying Busemann's condition. Several examples we consider satisfy Busemann's condition but are not convergence-symmetric. We have included throughout the paper a certain number of open questions.

math.MG↗

arXiv2026

The combinatorial structure of the unit tangent spheres and cotangent spheres of Teichm{ü}ller space with Thurston's Finsler metric

We prove several new results on the combinatorial structures of the unit spheres of the norms induced by Thurston's metric on the tangent and cotangent spaces of the Teichm{ü}ller space of a closed surface of negative Euler characteristic. These results include a formula for the dimension of every face of a unit sphere in the tangent space in terms of an invariant of the chain-recurrent lamination representing the face. We then prove that the combinatorial structure of such a unit sphere is independent of the underlying point in Teichm{ü}ller space. Provided the genus of the surface is $\ge$ 2, we show that there is a natural isomorphism between the extended mapping class group of the surface and the group of combinatorial automorphisms of such a unit sphere. In the case of genus 2, we obtain a natural epimorphism between the two groups whose kernel is the class of the hyperelliptic involution. Regarding the unit spheres of Thurston's metric in the cotangent spaces, we obtain a formula describing the codimensions of faces of such a sphere in terms of corresponding projective measured laminations. We then give a necessary and sufficient condition for a face to be exposed, and of a face to correspond to a projectively weighted multi-curve. Some of the results obtained answer open questions.

math.GT↗

arXiv2026

The horocyclic metric on Teichm{ü}ller spaces

In his paper Minimal stretch maps between hyperbolic surfaces, William Thurston defined a norm on the tangent space to Teichm{ü}ller space of a hyperbolic surface, which he called the earthquake norm. This norm is obtained by assigning a length to a tangent vector after such a vector is considered as an infinitesimal earthquake deformation of the surface. This induces a Finsler metric on the Teichm{ü}ller space, called the earthquake metric. This theory was recently investigated by Huang, Ohshika, Pan and Papadopoulos. In the present paper, we study this metric from the conformal viewpoint and we adapt Thurston's theory to the case of Riemann surfaces of arbitrary genus with marked points. A complex version of the Legendre transform defined for Finsler manifolds gives an analogue of the Wolpert duality for the Weil-Petersson symplectic form, which establishes a complete analogue of Thurston's theory of the earthquake norm in the conformal setting. This paper will appear in the Annales de l'Institut Fourier

math.CV↗

arXiv2026

On the Lambert conformal conical projection and the general map of the Russian Empire

The problem of drawing geographical maps is the one of mapping a subset of the sphere, representing a country or some other region on the surface of the Earth, into the Euclidean plane, minimising certain distortion properties that are specified in advance. It is known that from the purely mathematical point of view, this is an extremely difficult problem. One of Leonhard Euler's duties during his first stay at the Imperial Academy of Sciences of Saint Petersburg (1727-1741) was to help establishing maps of the Russian Empire. He worked on this project under the direction of the famous French geographer Joseph-Nicolas Delisle, who was the head of the astronomy and geography departments of the Academy. The general map of the Russian Empire, together with several maps of its particular regions were published under Euler's direction in the so-called Russian Atlas in 1745. In his later memoir ``De proiectione geographica De Lisliana in mappa generali imperii russici usitata'', written in 1777, Euler developed the mathematical theory of the method used by Delisle on a heuristic basis, which he himself used for drawing the general map of the Russian Empire. This method usually carries now the name Delisle--Euler map. In a previous paper, the first two authors of the present paper compared the Delisle--Euler map with several other maps of the conical type, with respect to various mathematical distorsion properties. They showed that this map is the best one from all the points of view considered, when it is applied to the drawing of the Russian Empire. In the present paper, we compare the Euler--Delisle map with a map which was not considered in the paper mentioned, namely, the so-called Lambert conformal conical projection, applied to the same region of the Earth. We show that the latter is better in several respects than all the other maps considered in the previous paper, including the Delisle--Euler map.

math.DG↗

arXiv2025

On families of Finsler metrics

In this paper, we answer some natural questions on symmetrisation and more general combinations of Finsler metrics, with a view towards applications to Funk and Hilbert geometries and to metrics on Teichm{ü}ller spaces. For a general non-symmetric Finsler metric on a smooth manifold, we introduce two different families of metrics, containing as special cases the arithmetic and the max symmetrisations respectively of the distance functions associated with these Finsler metrics. We are interested in various natural questions concerning metrics in such a family, regarding its geodesics, its completeness, conditions under which such a metric is Finsler, the shape of its unit ball in the case where it is Finsler, etc. We address such questions in particular in the setting of Funk and Hilbert geometries, and in that of the Teichm{ü}ller spaces of several kinds of surfaces, equipped with Thurstonlike asymmetric metrics.

math.DG↗

arXiv2025

On the Marinus--Ptolemy and Delisle--Euler conical maps

We examine connections between the mathematics behind methods of drawing geographical maps due, on the one hand to Marinos and Ptolemy (1st-2nd c. CE) and on the other hand to Delisle and Euler (18th century). A recent work by the first two authors of this article shows that methods of Delisle and Euler for drawing geographical maps, which are improvements of methods of Marinus and Ptolemy, are best among a collection of geographical maps we term ``conical''. This is an instance where after practitioners and craftsmen (here, geographers) have used a certain tool during several centuries, mathematicians prove that this tool is indeed optimal. Many connections among geography, astronomy and geometry are highlighted. The fact that the Marinos--Ptolemy and the Delisle--Euler methods of drawing geographical maps share many non-trivial properties is an important instance of historical continuity in mathematics.

math.HO↗

arXiv2024

The infinitesimal and global Thurston geometry of Teichm{ü}ller space

We undertake a systematic study of the infinitesimal geometry of the Thurston metric, showing that the topology, convex geometry and metric geometry of the tangent and cotangent spheres based at any marked hyperbolic surface representing a point in Teichm{ü}ller space can recover the marking and geometry of this marked surface. We then translate the results concerning the infinitesimal structures to global geometric statements for the Thurston metric, most notably deriving rigidity statements for the Thurston metric analogous to the celebrated Royden theorem.

math.GT↗

arXiv2023

On the Teichm{ü}ller space of acute triangles

We continue the study of the analogue of Thurston's metric on the Teichm{ü}ller space of Euclidean triangle which was started by Saglam and Papadopoulos in [1].By direct calculation, we give explicit expressions of the distance function and the Finsler structure of the metric restricted to the subspace of acute triangles.We deduce from the form of the Finsler unit sphere a result on the infinitesimal rigidity of the metric.We give a description of the maximal stretching loci for a family of extreme Lipschitz maps.

math.GT↗

arXiv2022

The Teichm{ü}ller-Randers metric

In this paper, we introduce a new asymmetric weak metric on the Teichm{ü}ller space of a closed orientable surface with (possibly empty) punctures.This new metric, which we call the Teichm{ü}ller-Randers metric, is an asymmetric deformation of the Teichm{ü}ller metric, and is obtained by adding to the infinitesimal form of the Teichm{ü}ller metric a differential 1-form. We study basic properties of the Teichm{ü}ller-Randers metric. In the case when the 1-form is exact, any Teichm{ü}ller geodesic between two points is a unique Teichm{ü}ller--Randers geodesic between them. A particularly interesting case is when the differential 1-form is (up to a factor) the differential of the logarithm of the extremal length function associated with a measured foliation. We show that in this case the Teichm{ü}ller-Randers metric is incomplete in any Teichm{ü}ller disc, and we give a characterisation of geodesic rays with bounded length in this disc in terms of their directing measured foliations.

math.CV↗

arXiv2021

Tangent spaces of the Teichm{ü}ller space of the torus with Thurston's weak metric

In this paper, we show that the analogue of Thurston's asymmetric metric on the Teichm{ü}ller space of flat structures on the torus is weak Finsler and we give a geometric description of its unit sphere at each point in the tangent space to Teichm{ü}ller space. We then introduce a family of weak Finsler metrics which interpolate between Thurston's asymmetric metric and the Teichm{ü}ller metric of the torus (which coincides with the the hyperbolic metric). We describe the infinitesimal unit spheres of the metrics in this family.The final version of this paper will appear in Annales Academiæ \ Scientiarum Fennicæ\ Mathematica.

math.CV↗

arXiv2020

A glimpse into Thurston's work

We present an overview of some significant results of Thurston and their impact on mathematics. The final version of this paper will appear as Chapter 1 of the book "In the tradition of Thurston: Geometry and topology", edited by K. Ohshika and A. Papadopoulos (Springer, 2020).

math.GT↗

arXiv2018

Bijections of geodesic lamination space preserving left Hausdorff convergence

We introduce an asymmetric distance function, which we call the `left Hausdorff distance function', on the space of geodesic laminations on a closed hyperbolic surface of genus at least 2. This distance is an asymmetric version of the Hausdorff distance between compact subsets of a metric space. We prove a rigidity result for the action of the extended mapping class group of the surface on the space of geodesic laminations equipped with the topology induced from this distance. More specifically, we prove that there is a natural homomorphism from the extended mapping class group into the group of bijections of the space of geodesic laminations that preserve left Hausdorff convergence and that this homomorphism is an isomorphism.

math.GT↗

arXiv2018

Homeomorphisms and intersection number

We prove two rigidity results for automorphism groups of the spaces ML(S) of measured laminations on a closed hyperbolic surface S and PML(S) of projective measured laminations on this surface. The results concern the homeomorphisms of ML(S) that preserve the geometric intersection between laminations and the homeomorphisms of PML(S) that preserve the zero sets of these intersection functions.

math.GT↗