DOCSLIB.ORG

Finsler Metric, Systolic Inequality, Isometr

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Finsler Metric, Systolic Inequality, Isometr DISCRETE SURFACES WITH LENGTH AND AREA AND MINIMAL FILLINGS OF THE CIRCLE MARCOS COSSARINI Abstract. We propose to imagine that every Riemannian metric on a surface is discrete at the small scale, made of curves called walls. The length of a curve is its number of wall crossings, and the area of the surface is the number of crossings of the walls themselves. We show how to approximate a Riemannian (or self-reverse Finsler) metric by a wallsystem. This work is motivated by Gromov's filling area conjecture (FAC) that the hemisphere minimizes area among orientable Riemannian sur- faces that fill a circle isometrically. We introduce a discrete FAC: every square-celled surface that fills isometrically a 2n-cycle graph has at least n(n−1) 2 squares. We prove that our discrete FAC is equivalent to the FAC for surfaces with self-reverse metric. If the surface is a disk, the discrete FAC follows from Steinitz's algo- rithm for transforming curves into pseudolines. This gives a combinato- rial proof of the FAC for disks with self-reverse metric. We also imitate Ivanov's proof of the same fact, using discrete differential forms. And we prove a new theorem: the FAC holds for M¨obiusbands with self- reverse metric. To prove this we use a combinatorial curve shortening flow developed by de Graaf-Schrijver and Hass-Scott. The same method yields a proof of the systolic inequality for Klein bottles with self-reverse metric, conjectured by Sabourau{Yassine. Self-reverse metrics can be discretized using walls because every normed plane satisfies Crofton's formula: the length of every segment equals the symplectic measure of the set of lines that it crosses. Directed 2- dimensional metrics have no Crofton formula, but can be discretized as well. Their “fine structure" is a triangulation where the length of each edge is 1 in one way and 0 in the other, and the area of the surface is the number of triangles. This structure can be described as a simplicial set, dual to a plabic graph. The role of the walls is played by Postnikov's strands. Keywords: Finsler metric, systolic inequality, isometric filling, integral geometry, lattice polygons, pseudoline arrangements, discrete differential forms, discrete curvature flow, simplicial sets, plabic graphs. Contents 1. Overview 4 1.1. The filling area of the circle 4 1.2. Fillings with Finsler metric 4 1.3. Discretization of self-reverse metrics on surfaces 5 1.4. Discrete FAC for square-celled surfaces 7 1.5. Directed metrics on surfaces and their discretization 7 1.6. Work to do next 8 1 2 MARCOS COSSARINI 1.7. Possibly unconventional terminology 9 2. Finsler manifolds and their Holmes{Thompson volume 10 3. From geodesic scattering to the filling area conjecture 14 3.1. Finsler filling area problem 17 3.2. Rational counterexamples to the Finsler FAC 18 4. Length and area according to integral geometry 20 5. Discrete self-reverse metrics on surfaces and the discrete filling area problem 24 5.1. Wallsystems 25 5.2. Square-celled surfaces 26 5.3. Duality between cellular wallsystems and square-celled decompositions of a surface 28 5.4. Topology of surfaces that fill a circle, and how to make a wallsystem cellular 30 5.5. Even wallsystems and even square-celled surfaces 31 6. Polyhedral approximation of Finsler surfaces 36 6.1. Filling a circle with piecewise-Finsler or polyhedral-Finsler surfaces 38 6.2. Triangulating Finsler surfaces 40 6.3. Polyhedral approximation of piecewise-Finsler manifolds 41 6.4. Smooth approximation of polyhedral-Finsler surfaces 44 6.5. Polygonal approximation of curves 48 6.6. Proof that the discrete FAC implies the continuous FAC 50 7. Discretization of self-reverse Finsler metrics on surfaces 55 7.1. Discretization of planes with self-reverse integral norms 56 7.2. Proof of the discretization theorem for self-reverse Finsler metrics on surfaces 59 8. Continuization of square-celled surfaces 63 9. Proof of the equivalence between discrete FAC and continuous FAC 67 10. Wallsystems on the disk 68 10.1. Steinitz's algorithm for tightening wallsystems on the disk 71 10.2. Tightening wallsystems on the disk by smoothing cusps 75 11. Differential forms on square-celled surfaces 77 11.1. Chains, differential forms and integration 77 11.2. Boundaries, exterior derivatives and Stokes' formula 77 11.3. Exterior products 78 11.4. Proof of the FAC for square-celled disks using cyclic content 80 11.5. Higher cyclic contents 82 12. Isometric fillings homeomorphic to the M¨obiusband and systolic inequality for the Klein bottle 83 12.1. Tightening wallystems on surfaces 84 12.2. Closed curves on the Klein bottle 87 12.3. Crossing numbers of curves on the Klein bottle 89 13. Discretization of directed metrics on surfaces 98 13.1. Fine metrics on graphs and fine surfaces 99 13.2. Directed version of the filling area conjecture 100 13.3. Duality between fine surfaces and plabic graphs 102 DISCRETE SURFACES WITH LENGTH AND AREA 3 14. Work to do next 106 14.1. Minimum filling by integer linear programming 106 14.2. Poset of minlength functions and algorithms 107 14.3. Discretization of d-dimensional metrics 109 14.4. Riemannian rigidity and random surfaces 111 References 112 4 MARCOS COSSARINI 1. Overview In this section we describe the content, main theorems and structure of this thesis. The motivation and context of this work are presented later, in Sections 3 and 4. All definitions and theorems presented here will be repeated later, with more detail. 1.1. The filling area of the circle. Let C be a Riemannian circle, that is, a closed curve of a certain length. A filling of C is a compact surface M with boundary @M = C.A Riemannian filling of C is a filling with a Riemannian metric. It is called an isometric filling if dM (x; y) = dC (x; y) for every x; y 2 C, where the distance dX (x; y) on any space X is the infimum length of paths from x to y along that space X. For example, a Euclidean flat disk fills its boundary non-isometrically, but the Euclidean hemisphere is an isometric filling. The Riemannian filling area of the circle is the infimum area of all Riemannian isometric fillings. Gromov [Gro83] posed the problem of com- puting the Riemannian filling area of the circle, proved the minimality of the hemisphere among Riemannian isometric fillings homeomorphic to a disk, and conjectured its minimality among all orientable Riemannian isometric fillings; this is the (Riemannian) filling area conjecture, or FAC. For ori- entable Riemannian isometric fillings of genus ≤ 1 the Riemannian FAC is known to hold also; the hemisphere is minimal in this class as well [Ban+05]. Computing this single number, the Riemannian filling area of the circle, may seem a very specialized task, but it is related to the problem of de- termining a whole region of space based on how geodesic trajectories are affected when they cross the region, a problem known as \inverse geodesic scattering". In the introductory Section 3 we discuss this connection and what is known about the FAC. 1.2. Fillings with Finsler metric. Ivanov and Burago extended the filling area problem by admitting as fillings surfaces with Finsler metrics, and proved the minimality of the hemisphere among Finsler disks [Iva01; Iva11a; BI02]. Roughly speaking, a Finsler surface is a smooth surface M with a Finsler metric F , that continuously assigns to each tangent vector v a length F (v) ≥ 0, also denoted kvk. The full definition is given in Section 2. A Finsler metric F enables one to define the length of any differentiable R 0 curve γ in M by the usual integration LenF (γ) := kγ (t)k dt. Finsler metrics are more general than Riemannian metrics because they need not satisfy the Pythagorean equation at the infinitesimal scale. More precisely, the Finsler metric restricted to the tangent plane at each point is a norm that need not be related to an inner product. In this thesis, norms, Finsler metrics and distance functions are in principle directed, not self-reverse; this means that we may have k − vk= 6 kvk and d(x; y) 6= d(y; x).1 For the filling area problem, smooth Finsler surfaces are equivalent to piecewise-Finsler surfaces (triangulated surfaces with a Finsler met- ric on each face) and are also equivalent to polyhedral-Finsler surfaces (surfaces made of triangles cut from normed planes), in the sense that when 1Self-reverse Finsler metrics are called \reversible" or \symmetric" by other authors. DISCRETE SURFACES WITH LENGTH AND AREA 5 we fill isometrically the circle, the infimum area that can be attained with isometric fillings of any of the three kinds is the same; this is proved in Theorem 6.3. The definition of area of a Finsler surface that we use here is due to Holmes{Thompson. It involves the standard symplectic form on the cotan- gent bundle T ∗M, which is related to the Hamiltonian approach to the study of geodesic trajectories. However, it is easy to define the Holmes{ Thompson area when the Finsler surface is polyhedral: the area of a triangle cut from some normed plane is its usual (Cartesian) area in any system of linear coordinates xi on the plane, multiplied by the area of the dual unit ball in the dual coordinates pi. Note that we use a non-standard normalization: the Holmes{Thompson area of a Euclidean triangle (or any Riemannian surface) is π times greater than the usual value. Conjecture 1.1 (continuous FAC for surfaces with self-reverse Finsler met- ric).
Load more

Recommended publications
  • "The Newton Polygon of a Planar Singular Curve and Its Subdivision"
    THE NEWTON POLYGON OF A PLANAR SINGULAR CURVE AND ITS SUBDIVISION NIKITA KALININ Abstract. Let a planar algebraic curve C be defined over a valuation field by an equation F (x,y)= 0. Valuations of the coefficients of F define a subdivision of the Newton polygon ∆ of the curve C. If a given point p is of multiplicity m on C, then the coefficients of F are subject to certain linear constraints. These constraints can be visualized in the above subdivision of ∆. Namely, we find a 3 2 distinguished collection of faces of the above subdivision, with total area at least 8 m . The union of these faces can be considered to be the “region of influence” of the singular point p in the subdivision of ∆. We also discuss three different definitions of a tropical point of multiplicity m. 0. Introduction Fix a non-empty finite subset Z2 and any valuation field K. We consider a curve C given by an equation F (x,y) = 0, where A⊂ i j ∗ (1) F (x,y)= aijx y , aij K . ∈ (i,jX)∈A Suppose that we know only the valuations of the coefficients of the polynomial F (x,y). Is it possible to extract any meaningful information from this knowledge? Unexpectedly, many geometric properties of C are visible from such a viewpoint. The Newton polygon ∆=∆( ) of the curve C is the convex hull of in R2. The extended Newton A A2 polyhedron of the curve C is the convex hull of the set ((i, j),s) R R (i, j) ,s val(aij) .
    [Show full text]
  • Riemannian and Finslerian Geometry for Diffusion Weighted Magnetic Resonance Imaging
    Riemannian and Finslerian geometry for diffusion weighted magnetic resonance imaging Luc Florack Computational Brain Connectivity Mapping Winter School Workshop 2017 - November 20-24, Jeans-les-Pins, France © Nicolas Lavarenne, l’exposition ‘A ciel ouvert’ Heuristics Finsler manifold. A Finsler manifold is a space (M,F) of spatial base points x∈M, furnished with a notion of a ‘line’ or ‘length element’ ds ≐ F(x,dx). The ‘infinitesimal displacement vectors’ dx are ‘infinitely scalable’ into finite ‘tangent’ or ‘velocity vectors’ ẋ, viz. dx = ẋ dt. The collection of all (x, ẋ) is called the tangent bundle TM over M. Integrating the line element along a curve C produces the ‘length’ of that curve: L (C)= ds = F (x, dx) ZC ZC Bernhard Riemann Sophus Lie Elie Cartan Albert Einstein Paul Finsler Applications Examples (anisotropic media). Mechanics e.g. F(x,dx) := infinitesimal displacement, or infinitesimal travel time, etc. Optimal control e.g. F(x,dx) := local cost function for infinitesimal movement of Reeds-Shepp car Optics e.g. F(x,dx) := infinitesimal travel time for light propagation Seismology e.g. F(x,dx) := infinitesimal travel time for seismic ray propagation Ecology e.g. F(x,dx) := infinitesimal energy for coral reef state transition Relativity e.g. F(x,dx) := infinitesimal (pseudo-Finslerian) spacetime line element Diffusion MRI e.g. F(x,dx) := infinitesimal hydrogen spin diffusion Axiomatics Literature. © David Bao et al. © Hanno Rund et al. The Riemann-DTI paradigm DWMRI signal attenuation. E(x, q, ⌧)=exp[ ⌧D(x, q, ⌧)] − q = γ¯ g(t)dt q-space variable Z 2⇡iq ⇠ Propagator.
    [Show full text]
  • Heat Flow on Finsler Manifolds
    Heat Flow on Finsler Manifolds Shin-ichi Ohta,∗ Karl-Theodor Sturm August 4, 2008 This paper studies the heat flow on Finsler manifolds. A Finsler manifold is a smooth manifold M equipped with a Minkowski norm F (x, ) : T M R on each tangent space. Mostly, we · x → + will require that this norm is strongly convex and smooth and that it depends smoothly on the base point x. The particular case of a Hilbert norm on each tangent space leads to the important subclasses of Riemannian manifolds where the heat flow is widely studied and well understood. We present two approaches to the heat flow on a Finsler manifold: either as gradient flow on L2(M,m) for the energy • 1 (u)= F 2( u) dm; E 2 ∇ ZM or as gradient flow on the reverse L2-Wasserstein space (M) of probability measures on • P2 M for the relative entropy Ent(u)= u log u dm. ZM Both approaches depend on the choice of a measure m on M and then lead to the same nonlinear evolution semigroup. We prove 1,α-regularity for solutions to the (nonlinear) heat equation on C the Finsler space (M,F,m). Typically, solutions to the heat equation will not be 2. More- C over, we derive pointwise comparison results ´ala Cheeger-Yau and integrated upper Gaussian estimates ´ala Davies. 1 Finsler manifolds 1.1 Finsler structures Throughout this paper, a Finsler manifold will be a pair (M, F ) where M is a smooth, connected n-dimensional manifold and F : T M R is a measurable function (called Finsler structure) → + with the following properties: (i) F (x,cξ)= cF (x,ξ) for all (x,ξ) T M and all c> 0.
    [Show full text]
  • On the Existence of Convex Functions on Finsler Manifolds
    On the existence of convex functions on Finsler manifolds ∗† Sorin V. SABAU and Pattrawut CHANSANGIAM Abstract We show that a non-compact (forward) complete Finsler manifold whose Holmes- Thompson volume is finite admits no non-trivial convex functions. We apply this result to some Finsler manifolds whose Busemann function is convex. 1 Introduction Finsler manifolds are a natural generalization of Riemannian ones in the sense that the metric depends not only on the point, but on the direction as well. This generalization implies the non-reversibility of geodesics, the difficulty of defining angles and many other particular features that distinguish them from Riemannian manifolds. Even though classi- cal Finsler geometry was mainly concerned with the local aspects of the theory, recently a great deal of effort was made to obtain global results in the geometry of Finsler manifolds ([3], [13], [15], [17] and many others). In a previous paper [16], by extending the results in [9], we have studied the ge- ometry and topology of Finsler manifolds that admit convex functions, showing that such manifolds are subject to some topological restrictions. We recall that a func- tion f : (M, F ) R, defined on a (forward) complete Finsler manifold (M, F ), is → called convex if and only if along every geodesic γ :[a, b] M, the composed func- tion ϕ := f γ :[a, b] R is convex, that is → ◦ → f γ[(1 λ)a + λb] (1 λ)f γ(a)+ λf γ(b), 0 λ 1. (1.1) ◦ − ≤ − ◦ ◦ ≤ ≤ arXiv:1811.02121v2 [math.DG] 6 Jan 2019 If the above inequality is strict for all geodesics γ, the function f is called strictly convex, and if the equality holds good for all geodesics γ, then f is called linear.
    [Show full text]
  • Sasakian Finsler Structures on Diffusion of Tangent Bundles Satisfying 푪∗ 푿푯, 풀푯 푺 = ퟎ
    International Journal of Engineering Technology Science and Research IJETSR www.ijetsr.com ISSN 2394 – 3386 Volume 3, Issue 10 October 2016 Sasakian Finsler Structures on Diffusion of Tangent Bundles ∗ 푯 푯 Satisfying 푪 푿 , 풀 푺 = ퟎ Nesrin Caliskan1,, A. Funda Yaliniz2 1Faculty of Education, Department of Elementary Mathematics Education, Usak University, 64200, Usak- TURKEY 2Department of Mathematics, Faculty of Art and Sciences, Dumlupinar University, 43100, Kutahya-TURKEY Abstract. Inthis paper,the quasi-conformal curvature tensor 퐶∗ of Sasakian Finsler structures ontangent bundlesis considered. The type of these manifolds satisfying퐶∗ 푋퐻, 푌퐻 푆 = 0 are discussed where푆 denotes the Ricci tensor field 0 0 of type and 푋퐻, 푌퐻휖 푇 푇푀 퐻 . In addition, relations between these structures and 휂-Einstein structures are 2 2 푢 0 examined. Mathematics Subject Classification (2010). 53D15, 53C05, 53C15, 53C60. Keywords.Quasi-conformal curvature tensor; 휼-Einstein;tangent bundle; Sasakian Finsler structure. 1. Introduction and main results There are many different frameworks to study Finsler geometry (like in [1,2]).In this study, we contextualized by using double tangent bundles of Finsler manifolds (such as [3,4,5]). Quasi-conformal curvature tensor is studied in papers for several structures(like[2,6-9,10]). In this study, the notion of “quasi-conformal curvature tensor” for Sasakian Finsler Structures on diffusions of tangent bundles is defined and we found different invariants. In order to study this, essential characteristics of these structures are given; Let 푀 be an 푚 = 2푛 + 1 -dimensional smooth manifold and푇푀 its tangent bundle, 푢 = 푥, 푦 ∈ 푇푀. Ifa function퐹: 푇푀 → 0, ∞ satisfies 1 ∞ i) 퐹 ∈ 퐶 (푇푀) and 퐹 ∈ 퐶 (푇푀0), 푖 푖 푖 푖 ii) 퐹 푥 , 휆푦 = 휆퐹 푥 , 푦 , 휆 > 0, 1 휕2퐹2 iii) The 푚 × 푚 Hessian matrix 푔 푥, 푦 = is positive definite on 푇푀 , 푖푗 2 휕푦 푖휕푦 푗 0 푚 then퐹 = 푀, 퐹 is called a Finsler manifold and푔 is a Finsler metric tensor with 푔푖푗 coefficients [11].
    [Show full text]
  • Math.DG] 14 Dec 2004 Qain:Tertcladnmrclanalysis’
    FILLING AREA CONJECTURE AND OVALLESS REAL HYPERELLIPTIC SURFACES1 VICTOR BANGERT!, CHRISTOPHER CROKE+, SERGEI V. IVANOV†, AND MIKHAIL G. KATZ∗ Abstract. We prove the filling area conjecture in the hyperellip- tic case. In particular, we establish the conjecture for all genus 1 fillings of the circle, extending P. Pu’s result in genus 0. We trans- late the problem into a question about closed ovalless real surfaces. The conjecture then results from a combination of two ingredi- ents. On the one hand, we exploit integral geometric comparison with orbifold metrics of constant positive curvature on real sur- faces of even positive genus. Here the singular points are Weier- strass points. On the other hand, we exploit an analysis of the combinatorics on unions of closed curves, arising as geodesics of such orbifold metrics. Contents 1. To fill a circle: an introduction 2 2. Relative Pu’s way 5 3. Outline of proof of Theorem 1.7 6 4. Nearoptimalsurfacesandthefootball 7 5. Finding a short figure eight geodesic 9 6. Proof of circle filling: Step 1 10 7. Proof of circle filling: Step 2 12 Appendix A. Ovalless reality and hyperellipticity 15 A.1. Hyperelliptic surfaces 15 arXiv:math/0405583v2 [math.DG] 14 Dec 2004 A.2. Ovalless surfaces 16 1Geometric and Functional Analysis (GAFA), to appear 1991 Mathematics Subject Classification. Primary 53C23; Secondary 52C07 . Key words and phrases. filling area, hyperelliptic surfaces, integral geometry, orbifold metrics, Pu’s inequality, real surfaces, systolic inequality. !Partially Supported by DFG-Forschergruppe ‘Nonlinear Partial Differential Equations: Theoretical and Numerical Analysis’. +Supported by NSF grant DMS 02-02536 and the MSRI.
    [Show full text]
  • Kepler's Area Law in the Principia: Filling in Some Details in Newton's
    Historia Mathematica 30 (2003) 441–456 www.elsevier.com/locate/hm Kepler’s area law in the Principia: filling in some details in Newton’s proof of Proposition 1 Michael Nauenberg Department of Physics, University of California, Santa Cruz, CA 95064, USA Abstract During the past 30 years there has been controversy regarding the adequacy of Newton’s proof of Prop. 1 in Book 1 of the Principia. This proposition is of central importance because its proof of Kepler’s area law allowed Newton to introduce a geometric measure for time to solve problems in orbital dynamics in the Principia.Itis shown here that the critics of Prop. 1 have misunderstood Newton’s continuum limit argument by neglecting to consider the justification for this limit which he gave in Lemma 3. We clarify the proof of Prop. 1 by filling in some details left out by Newton which show that his proof of this proposition was adequate and well-grounded. 2003 Elsevier Inc. All rights reserved. Résumé Au cours des 30 dernières années, il y a eu une controverse au sujet de la preuve de la Proposition 1 telle qu’elle est formulée par Newton dans le premier livre de ses Principia. Cette proposition est d’une importance majeure puisque la preuve qu’elle donne de la loi des aires de Kepler permit à Newton d’introduire une expression géometrique du temps, lui permettant ainsi de résoudre des problèmes dans la domaine de la dynamique orbitale. Nous démontrons ici que les critiques de la Proposition 1 ont mal compris l’argument de Newton relatif aux limites continues et qu’ils ont négligé de considérer la justification pour ces limites donnée par Newton dans son Lemme 3.
    [Show full text]
  • Landsberg Curvature, S-Curvature and Riemann Curvature
    Riemann{Finsler Geometry MSRI Publications Volume 50, 2004 Landsberg Curvature, S-Curvature and Riemann Curvature ZHONGMIN SHEN Contents 1. Introduction 303 2. Finsler Metrics 304 3. Cartan Torsion and Matsumoto Torsion 311 4. Geodesics and Sprays 313 5. Berwald Metrics 315 6. Gradient, Divergence and Laplacian 317 7. S-Curvature 319 8. Landsberg Curvature 321 9. Randers Metrics with Isotropic S-Curvature 323 10. Randers Metrics with Relatively Isotropic L-Curvature 325 11. Riemann Curvature 327 12. Projectively Flat Metrics 330 13. The Chern Connection and Some Identities 333 14. Nonpositively Curved Finsler Manifolds 337 15. Flag Curvature and Isotropic S-Curvature 341 16. Projectively Flat Metrics with Isotropic S-Curvature 342 17. Flag Curvature and Relatively Isotropic L-Curvature 348 References 351 1. Introduction Roughly speaking, Finsler metrics on a manifold are regular, but not neces- sarily reversible, distance functions. In 1854, B. Riemann attempted to study a special class of Finsler metrics | Riemannian metrics | and introduced what is now called the Riemann curvature. This infinitesimal quantity faithfully reveals the local geometry of a Riemannian manifold and becomes the central concept of Riemannian geometry. It is a natural problem to understand general regu- lar distance functions by introducing suitable infinitesimal quantities. For more than half a century, there had been no essential progress until P. Finsler studied the variational problem in a Finsler manifold. However, it was L. Berwald who 303 304 ZHONGMIN SHEN first successfully extended the notion of Riemann curvature to Finsler metrics by introducing what is now called the Berwald connection. He also introduced some non-Riemannian quantities via his connection [Berwald 1926; 1928].
    [Show full text]
  • Of Triangles, Gas, Price, and Men
    OF TRIANGLES, GAS, PRICE, AND MEN Cédric Villani Univ. de Lyon & Institut Henri Poincaré « Mathematics in a complex world » Milano, March 1, 2013 Riemann Hypothesis (deepest scientific mystery of our times?) Bernhard Riemann 1826-1866 Riemann Hypothesis (deepest scientific mystery of our times?) Bernhard Riemann 1826-1866 Riemannian (= non-Euclidean) geometry At each location, the units of length and angles may change Shortest path (= geodesics) are curved!! Geodesics can tend to get closer (positive curvature, fat triangles) or to get further apart (negative curvature, skinny triangles) Hyperbolic surfaces Bernhard Riemann 1826-1866 List of topics named after Bernhard Riemann From Wikipedia, the free encyclopedia Riemann singularity theorem Cauchy–Riemann equations Riemann solver Compact Riemann surface Riemann sphere Free Riemann gas Riemann–Stieltjes integral Generalized Riemann hypothesis Riemann sum Generalized Riemann integral Riemann surface Grand Riemann hypothesis Riemann theta function Riemann bilinear relations Riemann–von Mangoldt formula Riemann–Cartan geometry Riemann Xi function Riemann conditions Riemann zeta function Riemann curvature tensor Zariski–Riemann space Riemann form Riemannian bundle metric Riemann function Riemannian circle Riemann–Hilbert correspondence Riemannian cobordism Riemann–Hilbert problem Riemannian connection Riemann–Hurwitz formula Riemannian cubic polynomials Riemann hypothesis Riemannian foliation Riemann hypothesis for finite fields Riemannian geometry Riemann integral Riemannian graph Bernhard
    [Show full text]
  • Arxiv:1807.08398V1 [Math.DG] 23 Jul 2018 Set by Ifold Usin1.1
    ON FINSLER TRANSNORMAL FUNCTIONS MARCOS M. ALEXANDRINO, BENIGNO O. ALVES, AND HENGAMEH R. DEHKORDI Abstract. In this note we discuss a few properties of transnormal Finsler functions, i.e., the natural generalization of distance functions and isopara- metric Finsler functions. In particular, we prove that critical level sets of an analytic transnormal function are submanifolds, and the partition of M into level sets is a Finsler partition, when the function is defined on a compact analytic manifold M. 1. Introduction Let (M, F ) be a forward complete Finsler manifold. A function f : M R is called F -transnormal function if F ( f)2 = b(f) for some continuous function→ b. Transnormal functions on Riemannian∇ manifolds have been focus of researchers in the last decades. In particular, if b C2(f(M)) then the level sets of f are leaves of the so called singular Riemannian∈ foliation and the regular level sets are equifocal hypersurfaces; see e.g, [14], [15] and [2, Chapter 5]. In Finsler geometry, the study of transnormal functions has just begun, see [6] but there are already some interesting applications in wildfire modeling, see [11]. The most natural example of a transnormal function on a Finsler space is the dis- tance function on a Minkowski space. More precisely consider a Randers Minkowski space (V, Z) and define f(x) := d(0, x). It is well known that in this example b = 1, see [16, Lemma 3.2.3]. And already here one can see a phenomenon that does not −1 exist in the Riemannian case. The regular level set f (c1) is forward parallel to −1 −1 −1 f (c2) if c1 <c2 but f (c2) is not forward parallel to f (c1) and hence the par- −1 tition = f (c) c∈(0,∞) is not a Finsler partition of V 0, recall basic definitions and examplesF { in Sections} 2 and 3 respectively.
    [Show full text]
  • Absolute Irreducibility of Polynomials Via Newton Polytopes
    ABSOLUTE IRREDUCIBILITY OF POLYNOMIALS VIA NEWTON POLYTOPES SHUHONG GAO DEPARTMENT OF MATHEMATICAL SCIENCES CLEMSON UNIVERSITY CLEMSON, SC 29634 USA [email protected] Abstract. A multivariable polynomial is associated with a polytope, called its Newton polytope. A polynomial is absolutely irreducible if its Newton polytope is indecomposable in the sense of Minkowski sum of polytopes. Two general constructions of indecomposable polytopes are given, and they give many simple irreducibility criteria including the well-known Eisenstein’s crite- rion. Polynomials from these criteria are over any field and have the property of remaining absolutely irreducible when their coefficients are modified arbi- trarily in the field, but keeping certain collection of them nonzero. 1. Introduction It is well-known that Eisenstein’s criterion gives a simple condition for a polyno- mial to be irreducible. Over the years this criterion has witnessed many variations and generalizations using Newton polygons, prime ideals and valuations; see for examples [3, 25, 28, 38]. We examine the Newton polygon method and general- ize it through Newton polytopes associated with multivariable polynomials. This leads us to a more general geometric criterion for absolute irreducibility of multi- variable polynomials. Since the Newton polygon of a polynomial is only a small fraction of its Newton polytope, our method is much more powerful. Absolute irre- ducibility of polynomials is crucial in many applications including but not limited to finite geometry [14], combinatorics [47], algebraic geometric codes [45], permuta- tion polynomials [23] and function field sieve [1]. We present many infinite families of absolutely irreducible polynomials over an arbitrary field. These polynomials remain absolutely irreducible even if their coefficients are modified arbitrarily but with certain collection of them nonzero.
    [Show full text]
  • Characterizing and Tuning Exceptional Points Using Newton Polygons
    Characterizing and Tuning Exceptional Points Using Newton Polygons Rimika Jaiswal,1 Ayan Banerjee,2 and Awadhesh Narayan2, ∗ 1Undergraduate Programme, Indian Institute of Science, Bangalore 560012, India 2Solid State and Structural Chemistry Unit, Indian Institute of Science, Bangalore 560012, India (Dated: August 3, 2021) The study of non-Hermitian degeneracies { called exceptional points { has become an exciting frontier at the crossroads of optics, photonics, acoustics and quantum physics. Here, we introduce the Newton polygon method as a general algebraic framework for characterizing and tuning excep- tional points, and develop its connection to Puiseux expansions. We propose and illustrate how the Newton polygon method can enable the prediction of higher-order exceptional points, using a recently experimentally realized optical system. As an application of our framework, we show the presence of tunable exceptional points of various orders in PT -symmetric one-dimensional models. We further extend our method to study exceptional points in higher number of variables and demon- strate that it can reveal rich anisotropic behaviour around such degeneracies. Our work provides an analytic recipe to understand and tune exceptional physics. Introduction{ Energy non-conserving and dissipative Isaac Newton, in 1676, in his letters to Oldenburg and systems are described by non-Hermitian Hamiltoni- Leibniz [54]. They are conventionally used in algebraic ans [1]. Unlike their Hermitian counterparts, they are not geometry to prove the closure of fields [55] and are in- always diagonalizable and can become defective at some timately connected to Puiseux series { a generalization unique points in their parameter space { called excep- of the usual power series to negative and fractional ex- tional points (EPs) { where both the eigenvalues and the ponents [56, 57].
    [Show full text]