DOCSLIB.ORG

Geometry and Topology of Escape II, Homotopic Lobe Dynamics

Total Page：16

File Type：pdf, Size：1020Kb

Download full-text PDF Read full-text
Geometry and Topology of Escape II, Homotopic Lobe Dynamics W&M ScholarWorks Arts & Sciences Articles Arts and Sciences 2003 Geometry and topology of Escape II, Homotopic lobe dynamics K. A. Mitchell J. P. Handley J. B. Delos Stephen Knudson Follow this and additional works at: https://scholarworks.wm.edu/aspubs Part of the Physics Commons Geometry and topology of escape. II. Homotopic lobe dynamics Cite as: Chaos 13, 892 (2003); https://doi.org/10.1063/1.1598312 Submitted: 01 October 2002 . Accepted: 17 June 2003 . Published Online: 15 August 2003 K. A. Mitchell, J. P. Handley, J. B. Delos, and S. K. Knudson ARTICLES YOU MAY BE INTERESTED IN Geometry and topology of escape. I. Epistrophes Chaos: An Interdisciplinary Journal of Nonlinear Science 13, 880 (2003); https:// doi.org/10.1063/1.1598311 Isotope effect on bifurcating reaction path: Valley–ridge inflection point in totally symmetric coordinate The Journal of Chemical Physics 113, 477 (2000); https://doi.org/10.1063/1.481826 The separation of the reaction coordinate in transition state theory: Regularity and dimensionality reduction resulting from local symmetry The Journal of Chemical Physics 150, 164310 (2019); https://doi.org/10.1063/1.5092859 Chaos 13, 892 (2003); https://doi.org/10.1063/1.1598312 13, 892 © 2003 American Institute of Physics. CHAOS VOLUME 13, NUMBER 3 SEPTEMBER 2003 Geometry and topology of escape. II. Homotopic lobe dynamics K. A. Mitchell,a) J. P. Handley, and J. B. Delosb) Department of Physics, College of William and Mary, Williamsburg, Virginia 23187-8795 S. K. Knudson Department of Chemistry, College of William and Mary, Williamsburg, Virginia 23187-8795 ͑Received 1 October 2002; accepted 17 June 2003; published 15 August 2003͒ We continue our study of the fractal structure of escape-time plots for chaotic maps. In the preceding paper, we showed that the escape-time plot contains regular sequences of successive escape segments, called epistrophes, which converge geometrically upon each end point of every escape segment. In the present paper, we use topological techniques to: ͑1͒ show that there exists a minimal required set of escape segments within the escape-time plot; ͑2͒ develop an algorithm which computes this minimal set; ͑3͒ show that the minimal set eventually displays a recursive structure governed by an ‘‘Epistrophe Start Rule:’’ a new epistrophe is spawned ⌬ϭDϩ1 iterates after the segment to which it converges, where D is the minimum delay time of the complex. © 2003 American Institute of Physics. ͓DOI: 10.1063/1.1598312͔ Topological methods and symbolic dynamics have long map required for a point to escape the complex, plotted as a been valuable tools for describing orbits of dynamical function along the line of initial conditions ͑Fig. 2͒. For cha- systems. For example, if a particle in the plane scatters otic systems, such escape-time plots have a complicated set from three fixed disks, labeled A, B, and C, its orbit can of singularities and structure at all levels of resolution.2–16 be characterized by a sequence of symbols, such as These fractal escape-time plots play a central role in a variety ...ABA*BCBCA..., giving the sequence of collisions of classical decay and scattering problems; we have been with the disks. The asterisk gives the location of the par- particularly motivated by the ionization of atoms, especially ticle at the present time; as time goes by the asterisk hydrogen in parallel electric and magnetic fields. takes one step to the right. In this paper, we describe a The escape-time plot of a discrete map is divided into new kind of symbolic dynamics, in which the symbol se- ‘‘escape segments,’’ intervals over which the escape time is quence describes the structure of a curve in the plane. constant. In Paper I, we proved that there exist certain im- The relevant curve is not the trajectory of a particle, but portant sequences of consecutive escape segments, which we rather an ensemble of initial points in phase space—the called epistrophes, at all levels of resolution. The epistrophes line of initial conditions. This line winds around ‘‘holes’’ are characterized by the Epistrophe Theorem, whose core in the plane in a manner described by the symbol se- results are: ͑1͒ each end point of an escape segment spawns quence. The dynamical map applied to the line induces a a new epistrophe which converges upon it; ͑2͒ in the limit !ϱ map on the symbol sequence, which is more complicated ni , every epistrophe converges geometrically, with rate than a simple shift. We use this symbolic dynamics to equal to the Liapunov factor ␣ of the X point; ͑3͒ the derive properties of the epistrophes introduced in the asymptotic tails of any two epistrophes differ by a simple preceding paper. In particular, we use it to obtain a scaling. ‘‘minimal set of escape segments’’ and an ‘‘epistrophe The focus of Paper I was the asymptotic behavior of start rule.’’ epistrophes. In the present paper, we address how epistro- phes begin. We use the topological structure of the ho- moclinic tangle and the line of initial conditions to show that I. INTRODUCTION there is a certain minimal set of escape segments. For this minimal set, we prove the Epistrophe Start Rule ͑Theorem 1 As in the preceding paper ͑Paper I͒, we study maps of 1͒, which says that after a sufficiently large time, each epis- the phase plane, having an unstable fixed point ͑an X point͒ trophe begins some number of iterates ⌬ after the segment and an associated homoclinic tangle of stable and unstable that spawned it ͑i.e., the segment upon which the epistrophe manifolds ͑Fig. 1͒. The stable and unstable manifolds inter- converges͒. The number ⌬ is the same for all epistrophes of sect transversely, bounding a region of phase space ͑called a given map and is dependent on the topological structure of the ‘‘complex’’͒ from which a trajectory may or may not the tangle; explicitly, ⌬ϭDϩ1, where D is the minimum escape. We consider an initial distribution of points along a delay time of the complex, that is, the minimum number of curve passing through the complex ͑the line of initial condi- iterates a scattering trajectory spends inside the complex. tions͒. The escape-time plot is the number of iterates of the The bulk of our effort is devoted to developing an algebraic algorithm for constructing the minimal set of escape seg- a͒Electronic mail: [email protected] ments for a general line of initial conditions. This algorithm b͒Electronic mail: [email protected] allows us to compute the initial structure of the escape-time 1054-1500/2003/13(3)/892/11/$20.00892 © 2003 American Institute of Physics Chaos, Vol. 13, No. 3, 2003 Geometry and topology of escape. II 893 of symbolic dynamics which we call ‘‘homotopic lobe dy- namics.’’ From the symbol sequence, one can readily read off the structure of the minimal set of escape segments. Lines with different symbol sequences may have different minimal sets; however, at long enough times, these minimal sets al- ways obey the Epistrophe Start Rule. Some escape segments, such as that marked with an as- terisk in Fig. 2, are not within the minimal set guaranteed by the topology. These segments are ‘‘surprises’’ which, within the present topological analysis, we cannot predict. Since they break the regular structure and since they often have no obvious connection with any of the epistrophes, we call them ‘‘strophes’’ as in Sec. III B of Paper I. Strophes interfere with the self-similar structure of the fractal and typically do not go away in the asymptotic limit, resulting in what we called FIG. 1. A phase space portrait is shown for our saddle-center map, which ‘‘epistrophic self-similarity’’ in Sec. III C of Paper I. Despite L possesses a homoclinic tangle. The line of initial conditions 0 coincides the presence of these strophes, the minimal set often seems with qϭ1.72. to accurately describe the early and intermediate time struc- ture of the escape-time plot. Patterns similar to our Epistrophe Start Rule have been plot for iterates before the Epistrophe Start Rule sets in. The seen in other work. In the numerical study of Tiyapan and algorithm is then used to prove the Epistrophe Start Rule Jaffe´,14 epistrophes and the Epistrophe Start Rule are evident itself. in the structure of the initial angle-final action plot ͑analo- A critical aspect of this paper is our use of homotopy gous to the escape-time plot͒. Similarly, Jung and theory. We develop the necessary formalism in Sec. III, and co-workers9–11 used symbolic dynamics to construct a tree- prior knowledge of homotopy theory is not required. Homo- diagram that gives a comparable description of a scattering topy theory provides an algebraic framework for describing system. In each case, the authors consider a specific line of the topological structure of curves in the phase plane. As we initial conditions that is far outside of the complex and is shall explain, the phase plane has a set of ‘‘holes’’ into which topologically simple. Easton,17 followed by Rom-Kedar and the line of initial conditions cannot pass. A symbol sequence others,18–20 showed that recursive patterns also apply to ho- can be used to describe how the line circumvents these holes. moclinic intersections between the stable and unstable mani- As the dynamics maps the line forward, there is an induced folds. Thus, it may not be surprising that comparable patterns dynamics on the symbol sequence, representing a new kind should apply to the intersections between the stable manifold and an arbitrary line of initial conditions, at least at suffi- ciently large iterate. But at what iterate does this pattern set in, and what is the minimal set before it sets in? Algorithm 1 answers both these questions, as well as giving a simple proof of the Epistrophe Start Rule.
Load more

Recommended publications
  • Connections Between Differential Geometry And
    VOL. 21, 1935 MA THEMA TICS: S. B. MYERS 225 having no point of C on their interiors or boundaries, the second composed of the remaining cells of 2,,. The cells of the first class will form a sub-com- plex 2* of M,. The cells of the second class will not form a complex, since they may have cells of the first class on their boundaries; nevertheless, their duals will form a complex A. Moreover, the cells of the second class will determine a region Rn containing C. Now, there is no difficulty in extending Pontrjagin's relation of duality to the Betti Groups of 2* and A. Moreover, every cycle of S - C is homologous to a cycle of 2, for sufficiently large values of n and bounds in S - C if and only if the corresponding cycle of 2* bounds for sufficiently large values of n. On the other hand, the regions R. close down on the point set C as n increases indefinitely, in the sense that the intersection of all the RI's is precisely C. Thus, the proof of the relation of duality be- tween the Betti groups of C and S - C may be carried through as if the space S were of finite dimensionality. 1 L. Pontrjagin, "The General Topological Theorem of Duality for Closed Sets," Ann. Math., 35, 904-914 (1934). 2 Cf. the reference in Lefschetz's Topology, Amer. Math. Soc. Publications, vol. XII, end of p. 315 (1930). 3Lefschetz, loc. cit., pp. 341 et seq. CONNECTIONS BETWEEN DIFFERENTIAL GEOMETRY AND TOPOLOG Y BY SumNR BYRON MYERS* PRINCJETON UNIVERSITY AND THE INSTITUTE FOR ADVANCED STUDY Communicated March 6, 1935 In this note are stated the definitions and results of a study of new connections between differential geometry and topology.
    [Show full text]
  • On the Topology of Ending Lamination Space
    1 ON THE TOPOLOGY OF ENDING LAMINATION SPACE DAVID GABAI Abstract. We show that if S is a finite type orientable surface of genus g and p punctures where 3g + p ≥ 5, then EL(S) is (n − 1)-connected and (n − 1)- locally connected, where dim(PML(S)) = 2n + 1 = 6g + 2p − 7. Furthermore, if g = 0, then EL(S) is homeomorphic to the p−4 dimensional Nobeling space. 0. Introduction This paper is about the topology of the space EL(S) of ending laminations on a finite type hyperbolic surface, i.e. a complete hyperbolic surface S of genus-g with p punctures. An ending lamination is a geodesic lamination L in S that is minimal and filling, i.e. every leaf of L is dense in L and any simple closed geodesic in S nontrivally intersects L transversely. Since Thurston's seminal work on surface automorphisms in the mid 1970's, laminations in surfaces have played central roles in low dimensional topology, hy- perbolic geometry, geometric group theory and the theory of mapping class groups. From many points of view, the ending laminations are the most interesting lam- inations. For example, the stable and unstable laminations of a pseudo Anosov mapping class are ending laminations [Th1] and associated to a degenerate end of a complete hyperbolic 3-manifold with finitely generated fundamental group is an ending lamination [Th4], [Bon]. Also, every ending lamination arises in this manner [BCM]. The Hausdorff metric on closed sets induces a metric topology on EL(S). Here, two elements L1, L2 in EL(S) are close if each point in L1 is close to a point of L2 and vice versa.
    [Show full text]
  • William P. Thurston the Geometry and Topology of Three-Manifolds
    William P. Thurston The Geometry and Topology of Three-Manifolds Electronic version 1.1 - March 2002 http://www.msri.org/publications/books/gt3m/ This is an electronic edition of the 1980 notes distributed by Princeton University. The text was typed in TEX by Sheila Newbery, who also scanned the figures. Typos have been corrected (and probably others introduced), but otherwise no attempt has been made to update the contents. Genevieve Walsh compiled the index. Numbers on the right margin correspond to the original edition’s page numbers. Thurston’s Three-Dimensional Geometry and Topology, Vol. 1 (Princeton University Press, 1997) is a considerable expansion of the first few chapters of these notes. Later chapters have not yet appeared in book form. Please send corrections to Silvio Levy at [email protected]. CHAPTER 5 Flexibility and rigidity of geometric structures In this chapter we will consider deformations of hyperbolic structures and of geometric structures in general. By a geometric structure on M, we mean, as usual, a local modelling of M on a space X acted on by a Lie group G. Suppose M is compact, possibly with boundary. In the case where the boundary is non-empty we do not make special restrictions on the boundary behavior. If M is modelled on (X, G) then the developing map M˜ −→D X defines the holonomy representation H : π1M −→ G. In general, H does not determine the structure on M. For example, the two immersions of an annulus shown below define Euclidean structures on the annulus which both have trivial holonomy but are not equivalent in any reasonable sense.
    [Show full text]
  • Floer Homology, Gauge Theory, and Low-Dimensional Topology
    Floer Homology, Gauge Theory, and Low-Dimensional Topology Clay Mathematics Proceedings Volume 5 Floer Homology, Gauge Theory, and Low-Dimensional Topology Proceedings of the Clay Mathematics Institute 2004 Summer School Alfréd Rényi Institute of Mathematics Budapest, Hungary June 5–26, 2004 David A. Ellwood Peter S. Ozsváth András I. Stipsicz Zoltán Szabó Editors American Mathematical Society Clay Mathematics Institute 2000 Mathematics Subject Classification. Primary 57R17, 57R55, 57R57, 57R58, 53D05, 53D40, 57M27, 14J26. The cover illustrates a Kinoshita-Terasaka knot (a knot with trivial Alexander polyno- mial), and two Kauffman states. These states represent the two generators of the Heegaard Floer homology of the knot in its topmost filtration level. The fact that these elements are homologically non-trivial can be used to show that the Seifert genus of this knot is two, a result first proved by David Gabai. Library of Congress Cataloging-in-Publication Data Clay Mathematics Institute. Summer School (2004 : Budapest, Hungary) Floer homology, gauge theory, and low-dimensional topology : proceedings of the Clay Mathe- matics Institute 2004 Summer School, Alfr´ed R´enyi Institute of Mathematics, Budapest, Hungary, June 5–26, 2004 / David A. Ellwood ...[et al.], editors. p. cm. — (Clay mathematics proceedings, ISSN 1534-6455 ; v. 5) ISBN 0-8218-3845-8 (alk. paper) 1. Low-dimensional topology—Congresses. 2. Symplectic geometry—Congresses. 3. Homol- ogy theory—Congresses. 4. Gauge fields (Physics)—Congresses. I. Ellwood, D. (David), 1966– II. Title. III. Series. QA612.14.C55 2004 514.22—dc22 2006042815 Copying and reprinting. Material in this book may be reproduced by any means for educa- tional and scientific purposes without fee or permission with the exception of reproduction by ser- vices that collect fees for delivery of documents and provided that the customary acknowledgment of the source is given.
    [Show full text]
  • Anastasiia Tsvietkova Email A.Tsviet@Rutgers
    1/6 Anastasiia Tsvietkova Email [email protected] Appointments Continuing Assistant Professor, t.-track, Rutgers University, Newark, NJ, USA 09/2016 – present Temporary or Past Von Neumann Fellow, Institute of Advanced Study, Princeton 09/2020 – 08/2021 Assistant Professor and Head of Geometry and Topology of Manifolds 09/2017 – 08/2019 Unit, OIST (research institute), Japan Krener Assistant Professor (non t.-track), University of California, Davis 01/2014 – 08/2016 Postdoctoral Fellow, ICERM, Brown University (Semester program in Fall 2013 Low-dimensional Topology, Geometry, and Dynamics) NSF VIGRE Postdoctoral Researcher, Louisiana State University 08/2012 – 08/2013 Education PhD in Mathematics, University of Tennessee, 05/2012, honored by Graduate Academic Achievement Award Thesis: Hyperbolic structures from link diagrams. Advisor: Prof. Morwen Thistlethwaite. Master’s degree with Honors in Applied Mathematics, Kiev National University, Ukraine, 07/2007 Thesis: Decomposition of cellular balleans into direct products. Advisor: Prof. Ihor Protasov. Bachelor’s degree with Honors in Applied Mathematics, Kiev National University, Ukraine, 05/2005 Thesis: Asymptotic Rays. Advisor: Prof. Ihor Protasov. GPA 5.0/5.0 Research Interests Low-dimensional Topology and Geometry Knot theory Geometric group theory Computational Topology Quantum Topology Hyperbolic geometry Academic Honors, Grants and Other Funding • NSF grant (sole PI), Geometry, Topology and Complexity of 3-manifolds, DMS-2005496, $213,906, 2020-present • NSF grant (sole PI), Hyperbolic
    [Show full text]
  • Topics in Low Dimensional Computational Topology
    THÈSE DE DOCTORAT présentée et soutenue publiquement le 7 juillet 2014 en vue de l’obtention du grade de Docteur de l’École normale supérieure Spécialité : Informatique par ARNAUD DE MESMAY Topics in Low-Dimensional Computational Topology Membres du jury : M. Frédéric CHAZAL (INRIA Saclay – Île de France ) rapporteur M. Éric COLIN DE VERDIÈRE (ENS Paris et CNRS) directeur de thèse M. Jeff ERICKSON (University of Illinois at Urbana-Champaign) rapporteur M. Cyril GAVOILLE (Université de Bordeaux) examinateur M. Pierre PANSU (Université Paris-Sud) examinateur M. Jorge RAMÍREZ-ALFONSÍN (Université Montpellier 2) examinateur Mme Monique TEILLAUD (INRIA Sophia-Antipolis – Méditerranée) examinatrice Autre rapporteur : M. Eric SEDGWICK (DePaul University) Unité mixte de recherche 8548 : Département d’Informatique de l’École normale supérieure École doctorale 386 : Sciences mathématiques de Paris Centre Numéro identifiant de la thèse : 70791 À Monsieur Lagarde, qui m’a donné l’envie d’apprendre. Résumé La topologie, c’est-à-dire l’étude qualitative des formes et des espaces, constitue un domaine classique des mathématiques depuis plus d’un siècle, mais il n’est apparu que récemment que pour de nombreuses applications, il est important de pouvoir calculer in- formatiquement les propriétés topologiques d’un objet. Ce point de vue est la base de la topologie algorithmique, un domaine très actif à l’interface des mathématiques et de l’in- formatique auquel ce travail se rattache. Les trois contributions de cette thèse concernent le développement et l’étude d’algorithmes topologiques pour calculer des décompositions et des déformations d’objets de basse dimension, comme des graphes, des surfaces ou des 3-variétés.
    [Show full text]
  • A Short Course in Differential Geometry and Topology A.T
    A Short Course in Differential Geometry and Topology A.T. Fomenko and A.S. Mishchenko с s P A Short Course in Differential Geometry and Topology A.T. Fomenko and A.S. Mishchenko Faculty of Mechanics and Mathematics, Moscow State University С S P Cambridge Scientific Publishers 2009 Cambridge Scientific Publishers Cover design: Clare Turner All rights reserved. No part of this book may be reprinted or reproduced or utilised in any form or by any electronic, mechanical, or other means, now known or hereafter invented, including photocopying and recording, or in any information storage or retrieval system, without prior permission in writing from the publisher. British Library Cataloguing in Publication Data A catalogue record for this book has been requested Library of Congress Cataloguing in Publication Data A catalogue record has been requested ISBN 978-1-904868-32-3 Cambridge Scientific Publishers Ltd P.O. Box 806 Cottenham, Cambridge CB24 8RT UK www.cambridgescientificpublishers.com Printed and bound in Great Britain by CPI Antony Rowe, Chippenham and Eastbourne Contents Preface ix Introduction to Differential Geometry 1 1.1 Curvilinear Coordinate Systems. Simplest Examples 1 1.1.1 Motivation 1 1.1.2 Cartesian and curvilinear coordinates 3 1.1.3 Simplest examples of curvilinear coordinate systems . 7 1.2 The Length of a Curve in Curvilinear Coordinates 10 1.2.1 The length of a curve in Euclidean coordinates 10 1.2.2 The length of a curve in curvilinear coordinates 12 1.2.3 Concept of a Riemannian metric in a domain of Euclidean space 15 1.2.4 Indefinite metrics 18 1.3 Geometry on the Sphere and Plane 20 1.4 Pseudo-Sphere and Lobachevskii Geometry 25 General Topology 39 2.1 Definitions and Simplest Properties of Metric and Topological Spaces 39 2.1.1 Metric spaces 39 2.1.2 Topological spaces 40 2.1.3 Continuous mappings 42 2.1.4 Quotient topology 44 2.2 Connectedness.
    [Show full text]
  • Calculus of the Embedding Functor and Spaces of Knots
    CALCULUS OF THE EMBEDDING FUNCTOR AND SPACES OF KNOTS ISMAR VOLIC´ Abstract. We give an overview of how calculus of the embedding functor can be used for the study of long knots and summarize various results connecting the calculus approach to the rational homotopy type of spaces of long knots, collapse of the Vassiliev spectral sequence, Hochschild homology of the Poisson operad, finite type knot invariants, etc. Some open questions and conjectures of interest are given throughout. Contents 1. Introduction 1 1.1. Taylor towers for spaces of long knots arising from embedding calculus 2 1.2. Mapping space model for the Taylor tower 3 1.3. Cosimplicial model for the Taylor tower 4 1.4. McClure-Smith framework and the Kontsevich operad 5 2. Vassiliev spectral sequence and the Poisson operad 5 3. Rational homotopy type of Kn,n> 3 6 3.1. Collapse of the cohomology spectral sequence 6 3.2. Collapse of the homotopy spectral sequence 9 4. Case K3 and finite type invariants 9 5. Orthogonal calculus 11 5.1. Acknowledgements 12 References 12 1. Introduction Fix a linear inclusion f of R into Rn, n ≥ 3, and let Emb(R, Rn) and Imm(R, Rn) be the spaces of smooth embeddings and immersions, respectively, of R in Rn which agree with f outside a compact set. It is not hard to see that Emb(R, Rn) is equivalent to the space of based knots in the sphere Sn, and is known as the space of long knots. Let Kn be the homotopy fiber of the (inclusion Emb(R, Rn) ֒→ Imm(R, Rn) over f.
    [Show full text]
  • Geometry and Topology SHP Spring ’17
    Geometry and Topology SHP Spring '17 Henry Liu Last updated: April 29, 2017 Contents 0 About Mathematics 3 1 Surfaces 4 1.1 Examples of surfaces . 4 1.2 Equivalence of surfaces . 5 1.3 Distinguishing between surfaces . 7 1.4 Planar Models . 9 1.5 Cutting and Pasting . 11 1.6 Orientability . 14 1.7 Word representations . 16 1.8 Classification of surfaces . 19 2 Manifolds 20 2.1 Examples of manifolds . 21 2.2 Constructing new manifolds . 23 2.3 Distinguishing between manifolds . 25 2.4 Homotopy and homotopy groups . 27 2.5 Classification of manifolds . 30 3 Differential Topology 31 3.1 Topological degree . 33 3.2 Brouwer's fixed point theorem . 34 3.3 Application: Nash's equilibrium theorem . 36 3.4 Borsuk{Ulam theorem . 39 3.5 Poincar´e{Hopftheorem . 41 4 Geometry 44 4.1 Geodesics . 45 4.2 Gaussian curvature . 47 4.3 Gauss{Bonnet theorem . 51 1 5 Knot theory 54 5.1 Invariants and Reidemeister moves . 55 5.2 The Jones polynomial . 57 5.3 Chiral knots and crossing number . 60 A Appendix: Group theory 61 2 0 About Mathematics The majority of this course aims to introduce all of you to the many wonderfully interesting ideas underlying the study of geometry and topology in modern mathematics. But along the way, I want to also give you a sense of what \being a mathematician" and \doing mathematics" is really about. There are three main qualities that distinguish professional mathematicians: 1. the ability to correctly understand, make, and prove precise, unambiguous logical statements (often using a lot of jargon); 2.
    [Show full text]
  • Arxiv:2004.00097V2 [Math.MG] 3 May 2021 the Aigcnetdcmoet,Alioere of Isometries All Components, Connected Taking G Htpeevstedsac Ucin H E Isom( Set the Function
    LIFTING ISOMETRIES OF ORBIT SPACES RICARDO A. E. MENDES Abstract. Given an orthogonal representation of a compact group, we show that any element of the connected component of the isom- etry group of the orbit space lifts to an equivariant isometry of the original Euclidean space. Corollaries include a simple formula for this connected component, which applies to “most” representa- tions. 1. Introduction Given a metric space X, an isometry of X is defined as a bijection that preserves the distance function. The set Isom(X) of all isometries of X forms a group under composition, which is called the isometry group of X. An early result [DW28] (see also [KN96, pages 46–50]) states that if X is connected and locally compact, then Isom(X), endowed with the compact-open topology, is also locally compact, and the isotropy group Isom(X)x is compact for each x ∈ X. Moreover, if X is itself compact, then so is Isom(X). Shortly thereafter, Myers and Steenrod proved that Isom(X) is a Lie transformation group when X is a Riemannian manifold, see [MS39], [Kob95, page 41]. Generalizing this, Fukaya and Yamaguchi have shown [FY94] that Isom(X) is a Lie group whenever X is an Alexandrov space. Since then isometries of Alexandrov spaces have been actively studied, see for example [GGG13, HS17]. arXiv:2004.00097v2 [math.MG] 3 May 2021 In the present article, we consider a special class of Alexandrov spaces. Namely, given a compact group G acting linearly by isome- tries on a Euclidean vector space V , the orbit space X = V/G is an Alexandrov space with curvature bounded from below by 0.
    [Show full text]
  • Arxiv:2009.10470V2 [Math.GR] 17 Feb 2021 Yaisgoer-Tutr.Teatoswr Loprilys Partially UMO-2018/31/G/ST1/02681
    ABSTRACT GROUP ACTIONS OF LOCALLY COMPACT GROUPS ON CAT(0) SPACES PHILIP MOLLER¨ AND OLGA VARGHESE Abstract. We study abstract group actions of locally compact Hausdorff groups on CAT(0) spaces. Under mild assumptions on the action we show that it is continuous or has a global fixed point. This mirrors results by Dudley and Morris- Nickolas for actions on trees. As a consequence we obtain a geometric proof for the fact that any abstract group homomorphism from a locally compact Hausdorff group into a torsion free CAT(0) group is continuous. 1. Introduction This article is located in the area of geometric group theory, a field at the inter- section of algebra, geometry and topology. The basic principle of geometric group theory is to investigate algebraic properties of infinite groups using geometric and topological methods. In this project we want to understand the ways in which topo- logical groups can act on spaces of non-positive curvature with the focus on automatic continuity. The main idea of automatic continuity is to establish conditions on topo- logical groups G and H under which an abstract group homomorphism ϕ: G → H is necessarily continuous. We will always assume that topological groups have the Hausdorff property. There are several results in this direction in the literature, see [5], [15], [17], [19], [23], [28]. Here, the group G will be a locally compact group while H will be the isometry group of a CAT(0) space equipped with the discrete topology. One of the powerful theorems in this direction is due to Dudley [19].
    [Show full text]
  • Course on Differential Geometry and Topology
    Course on Differential Geometry and Topology Bai-Ling Wang Institute of Mathematics Zurich University Winterthurerstrasse 190 CH-8057 Zurich, Switzerland 1 Contents 1 Manifolds and de Rham Cohomology 6 1.1 Differentiablemanifolds . 6 1.2 Directional derivatives and the chain rule . ...... 10 1.3 Tangent space and cotangent space . 11 1.4 Submanifolds ................................. 14 1.5 VectorfieldsandLiebracket . 16 1.6 Tangent bundle and cotangent bundle . 18 1.7 The exterior algebra, wedge and contraction operations ......... 20 1.8 Differential form and the exterior derivative . ....... 21 1.9 DeRhamcohomology ............................ 25 2 Lie Groups and Lie Algebras 28 2.1 Liegroupsandexamples........................... 28 2.2 Left invariant vector fields and Lie algebras . ...... 29 2.3 Exponential map and the Adjoint representation . ...... 34 2.3.1 TheLiederivative.......................... 38 2.4 Maurer-Cartan forms and Maurer-Cartan equations . ...... 41 2.5 Groupactionsonmanifolds . 42 3 Principal Bundles 50 3.1 Definition, examples and the transition functions . ........ 50 3.2 Associatedbundles.............................. 56 3.3 Universalbundles............................... 60 4 Connections and Curvatures 63 4.1 Connections.................................. 63 4.1.1 Connections on vector bundles . 63 4.1.2 Connections on principal bundles . 68 4.2 Curvatures .................................. 74 4.2.1 The holonomy of a connection . 78 2 5 Characteristic Classes 80 5.1 TheChern-Weilhomomorphism. 81 5.2 TheChernclasses .............................. 84 5.3 ThePontrjaginClasses. 89 5.4 TheEulerClass................................ 92 5.5 Appendix: Characteristic classes for classifying spaces .......... 97 3 Introduction Differential geometry is the language of modern physics as well as mathematics. Typically, one considers sets which are manifolds (that is, locally resemble Euclidean space) and which come equipped with a measure of distances.
    [Show full text]