Visual Geometry and Topology

Anatolij Fomenko as guest Professor at the Mathematical Institute in Heidelberg

Anatolij Fomenko

Visual Geometry and Topology With 50 Full-page Illustrations and 287 Drawings

Springer-Verlag Berlin Heidelberg New York London Paris Tokyo HongKong Barcelona Budapest

Author:

Anatolij T.Fomenko Dept. of Differential Geometry and Applications Faculty of Mathematics and Mechanics Moscow University, Moscow 119899 Russia

Translator:

Marianna V. Tsaplina Moscow, Russia

Title of the Russian edition: Naglyadnaya geometriya i topologia

Moscow University Press, 1993 (abridged version)

Mathematics Subject Classification (1991): 49-XX, 51-XX, 53-xx, 55-XX, 57-XX, 58-XX, 70-XX, 81-XX, 83-XX

Library of Congress Cataloging-in-Publication Data Fomenko, A. T. Visual geometry and topology/ Anatolij Fomenko; [translator, Marianna V. Tsaplinaj. p. em. Includes bibliographical references and index.

ISBN-13: 978-3-642-76237-6 e-ISBN-13: 978-3-642-76235-2 DOl: 10.1007/978-3-642-76235-2

1. Geometry. 2. Topology. I. Title. QA445.F58 1993 516-dc20 92-39676 CIP

This work is subject to copyright. All rights are reserved, whether the whole or part of the material is concerned, specifically the rights of translation, reprint~, reuse of illustrations, recitation, broadcasting, reproduction on microfilms or in any other way, and storage in data banks. Duplication of this publication or parts thereofis permitted only under the provisions of the German Copyright Law of September 9,1965, in its current version, and permission for use must always be obtained from Springer­Verlag. Violations are liable for prosecution under the German Copyright Law.

© Springer-Verlag Berlin Heidelberg 1994 Softcover reprint of the hardcover 1 st edition 1994

Typesetting: Springer T EX in-house system

41/3140 - 5 4 3 2 1 0 - Printed on acid-free paper

Preface

Modem geometry and topology take a special place in mathematics be­cause many of the objects they deal with are treated using visual methods. At the same time, these visual methods now successfully undergo for­malization and far-reaching abstraction which have contributed much to the remarkable progress in modem geometry and its applications. David Hilbert wrote in 1932: "As far as geometry is concerned, the tendency of abstraction in it had led to grand systematic constructions of alge­braic geometry, Riemann geometry and topology where the methods of abstract reasoning, symbolism and analysis are widely used. But non the less, visual perception still plays the leading role in geometry, and not only for being strongly demonstrative in the course of investigation, but also for the understanding and estimation of the results obtained in the course of investigation" [l].

Many geometric concepts arose from concrete problems of mecha­nics, physics, etc., and we shall point out here some of these links. For important modem mechanisms of the appearance of topological ideas in the framework of classical mechanics and mathematical physics see, for example, the papers by Smale [2], Poston and Stewart [3], Hildebrandt and Tromba [4], Novikov [5], Arnold [7], Nitsche [8]. The mathematical life is now being actively intruded by computer geometry which permits, in particular, visualization of intricate mechanical objects that result from long computational experiments and whose geometric character is hardly perdictable. It is relevant to mention here the remarkable works by Ban­choff [9], Mandelbrot [10], Francis [11], Penrose [12], Poston & Stewart [3], and Peitgen & Richter [13]. Some geometrical aspects of the the­ory of probability and mathematical statistics are elucidated in the book by Kolmogorov [14], B.V. Gnedenko [15], Shiryaev [16] and Chentsov [17]. Some of these ideas are reflected in the sections "Visual Material" of the present book.

VI Preface

We have selected only the fragments of geometrical knowledge which are most visual and closely related to applications. See, in par­ticular, Chap.3 devoted to some modem visual aspects of symplectic topology and Hamiltonian mechanics. See also the contributions due to Maslov [18], Faddeev & Zakharov [19], Gelfand [20], Matveev [46], [47], Zieschang [65], Bolsinov [70] and Kozlov [21].

The principal aim of the present book is to narrate, in an accessible and fairly visual language, about some classical and modem achieve­ments of geometry in both intrinsic mathematical problems and applica­tions. We do not restrict our consideration to the classics, and also touch upon current problems which are being rapidly developed today. We lay special emphasis upon visual explanation of the statement of problems, the methods of their solution and the results obtained and try to acquaint the reader with geometric ideas as soon as possible, ignoring for the time being the abstract logical aspect of calculations, considerations, etc. After having read this book, the reader will be able himself to compre­hend, when reading specialized literature, more formal approaches to the problems pointed out in this book. Many modem fields of mathematics admit visual presentations which do not, of course, claim to be logically rigorous but, on the other hand, offer a prompt introduction into the sub­ject matter. In this connection, in Chap. 1 we give a brief presentation of the classical theory of polyhedra and simplicial homologies because these ideas are now widely used in mathematics, physics, etc., but their logical simplification and perfection is often achieved by a higher level of abstraction. In this respect, the geometric approach to the theory of homologies, which goes back to Poincare, is perhaps more cumbersome (in what concerns theoretical grounds) but appreciably simpler and more visual (and therefore more comprehensive at early stages of acquaintance with the subject).

Geometrical intuition plays an essential role in contemporary algebro­topological and geometric studies. Many profound scientific mathemat­ical papers devoted to multi-dimensional geometry use intensively the "visual slang" such as, say, "cut the surface", "glue together the strips", "glue the cylinder", "evert the sphere", etc., typical of the studies of two­and three-dimensional images. Such a terminology is not a caprice of ma­thematicians, but rather a "practical necessity" since its employment and the mathematical thinking in these terms appear to be quite necessary for

Preface VII

the proof of technically very sophisticated results. It happens rather fre­quently that the proof of one or another mathematical fact can at ftrst be "seen", and only after that (and following this visual idea) can we present a logically consistent formulation, which is sometimes a very difficult task requiring serious intellectual efforts. The expediency of these efforts is however psychologically justifted by the visual and beautiful picture already created in the head of the researcher and convincing him that he had taken the right way. Thus, the criterion of beauty of one or another geometric image often serves as a compass for choosing an optimal way of a further formal logical proof. For a more thorough acquaintance with these ideas the reader may be referred to the well-known books by Manin [22], [23], where many interesting details can be found.

It was not our aim to give a systematic presentation of individual ftelds of geometry, but we started on a journey' round its rich world and on our way "took photos" of those fragments which we thought of as particularly interesting. The visual and scientiftcally urgent material being exceedingly abundant, a complete or a systematic presentation is out of the question. What we offer is a short "diary", an attempt to narrate to a wide range of mathematicians, mechanics, physicists about the diversity of methods and applications of modem geometry, to help them recognize really exciting geometric and physical objects against the background of sophisticated abstractions. Each chapter of the book is written as autonomously as possible, so that the reader could plunge into the ideas and concepts of each section and choose for himself the order of reading separete chapters.

The author is greatful to S.V. Matveev, Ya.V. Tatarinov, A.V. Cher­navsky, E.B. Vinberg, V.O. Bugaenko and A.A. Zenkin who kindly pre­sented some materials on visual geometry and topology.

Each mathematician has his own system of concepts of the intrin­sic geometry of his (speciftc) mathematical world and visual images which he associated with some or other abstract concepts of mathematics (including algebra, number theory, analysis, etc.). It is noteworthy that sometimes one and the same abstraction brings about the same visual picture in different mathematicians, but these pictures born by imagina­tion are in most cases very difficult to represent graphically, so to say, to draw. Part of the graphical material contained in the present book is an attempt to ''photograph from within" the sophisticated, peculiar

VIII Preface

mathematical world generously endowed with images and concepts that constitute the subject matter of contemporary. These graphical represen­tations are compiled in sections called "Visual Material". Almost each figure of such a section is endowed with a comment in the text. Some of the figures complement the content of the chapters with concepts not reflected in the main text. In this case, such figures give references to the corresponding literature. Our graphical material is either based on concrete geometric constructions, ideas, theorems and depict real math­ematical objects and processes or reflects various ways of perception of mathematical concepts, for instance, infinity, homotopy, etc. The first at­tempt in this direction was the book "Homotopic Topology" by Fomenko, Fuchs and Gutenmacher [24]. This topic was further developed in a new book "The Course in Homotopic Topology" by Fomenko & Fuchs [25].

The reader can also see the author's art book, Mathematical Impres­sions [26], which contains high-quality reproductions of approximately 80 of the author's works, including some in colour. The album also contains short mathematical and extra-mathematical comments to the pictures. Because they were written with different purposes, these two books complement each other.

Different sections of the present book are intended for different lev­els of mathematical knowledge, but the greater part of the material is in­tended for the first- and second-year students of mathematics or physics.

The book includes some visual aspects of the results in the field of modem computer geometry obtained in the framework of the scientific research "Computer Geometry" headed by AT. Fomenko at the Faculty of Mechanics and Mathematics of Moscow State University. These re­sults have been discussed at the scientific seminar "Computer Geometry" working at the Department of Differential Geometry and Applications, Department of Higher Geometry and Topology and Department of Com­putational Methods headed by AV. Bolsinov, V.L. Golo, I.Kh. Sabitov, E.G. Sklyarenko, V.V. Trofimov and AT. Fomenko in Moscow State University.

The readers who is interested in the details of modem geometri­cal methods can continue his education using the books by Dubrovin, Fomenko, Novikov [27] and the book by Fomenko [28].

Preface IX

The book is intended for natural sciences students (beginning from the first year), post-graduates and specialists interested in applications of modem geometry and topology.

The book included 50 graphical sheets drawn by the author and pre­sented in sections Visual Material. The majority of them are made in pencil and Indian ink on paper and exhibit half tones and sophisticated light and shade technique. The preparation of precise and high-quality photocopies, which the author submitted to Springer-Verlag for reproduc­tion in the book, was therefore a rather difficult task. It was successfully fulfilled by a professional photographer N.S. Moiseenko (Moscow) to whom the author expresses his gratitude.

The author is deeply indebted to Springer Publishers.

Table of Contents

1 Polyhedra. Simplicial Complexes. Homologies

1.1 Polyhedra ........................ . 1.1.1 Introductory Remarks . . . . . . . . . . . . . . . . . . . . 1.1.2 The Concept of an n-Dimensional Simplex

Barycentric Coordinates .................. 5 1.1.3 Polyhedra. Simplicial Subdivisions of Polyhedra.

Simplicial Complexes . . . . . . . . . . . . . . . . . . .. 8 1.1.4 Examples of Polyhedra . . . . . . . . . . . . . . . . . . . 10 1.1.5 Barycentric Subdivision .................. 14 1.1.6 Visual Material ., . . . . . . . . . . . . . . . . . . . . . 16

1.2 Simplicial Homology Groups of Simplicial Complexes (polyhedra) . . . . . . . . . . . . . . . . . . . . . . . .. 18

1.2.1 Simplicial Chains . . . . . . . . . . . . . . . . . . . . .. 18 1.2.2 Chain Boundary ...................... 23 1.2.3 The Simplest Properties of the Boundary

Operator Cycles. Boundaries ..... . . . . . . . . . . . 26 1.2.4 Examples of Calculations of the Boundary Operator . . . . 27 1.2.5 Simplicial Homology Groups . . . . . . . . . . . . . . . . 29 1.2.6 Examples of Calculations of Homology Groups.

Homologies of Two-dimensional Surfaces . . . . . . . . . 32 1.2.7 Visual Material .,. . . . . . . . . . . . . . . . . . . . . 46

1.3 General Properties of Simplicial Homology Groups .. 49

1.3.1 Incidence Matrices ..................... 49 1.3.2 The Method of Calculation of Homology Groups

Using Incidence Matrices . . . . . . . . . . . . . . . . . . 50

XlI Table of Contents

1.3.3 "Traces" of Cell Homologies Inside Simplicial Ones ... 56

1.3.4 Chain Homotopy. Independence of Simplicial Homologies of a Polyhedron of the Choice of Triangulation ...... 59

1.3.5 Visual Material ............... . . . . . . . . 65

2 Low-Dimensional Manifolds

2.1 Basic Concepts of Differential Geometry ........ 75

2.1.1 Coordinates in a Region. Transformations of Curvilinear Coordinates ........ 75

2.1.2 The Concept of a Manifold. Smooth Manifolds. Submanifolds and Ways of Defining Them. Manifolds with Boundary. Tangent Space and Tangent Bundle . . . . 79

2.1.3 Orientability and Non-Orientability. The Differential of a Mapping. Regular Values and Regular Points. Embeddings and Immersions of Manifolds. Critical Points of Smooth Functions on Manifolds. Index of N ondegenerate Critical Points and Morse Functions .................... 85

2.1.4 Vector and Covector Fields. Integral Trajectories. Vector Field Commutators. The Lie Algebra of Vector Fields on a Manifold ............... 91

2.1.5 Visual Material ..................... . . 95

2.2 Visual Properties of One-Dimensional Manifolds . . . . 98

2.2.1 Isotopies, Frames ...................... 98 2.2.2 Visual Material ...... . . . . . . . . . . . . . . . . . 102

2.3 Visual Properties of Two-Dimensional Manifolds . . . . 111

2.3.1 Two-Dimensional Manifolds with Boundary ........ 111 2.3.2 Examples of Two-Dimensional Manifolds ......... 113 2.3.3 Modelling of a Projective Plane

in a Three-Dimensional Space ............... 115 2.3.4 Two Series of Two-Dimensional Closed Manifolds .... 120

Table of Contents XIII

2.3.5 Classification of Closed 2-Manifolds . . . . . . . . . . . . 125 2.3.6 Inversion of a Two-Dimensional Sphere ....... . . . 128 2.3.7 Visual Material ., ..................... 130

2.4 Cohomology Groups and Differential Forms ...... 135

2.4.1 Differential I-Forms on a Smooth Manifold ........ 135 2.4.2 Closed and Exact Forms on a Two-Dimensional Manifold 136 2.4.3 An Important Property of Cohomology Groups ...... 138 2.4.4 Direct Calculation of One-Dimensional Cohomology

Groups of One-Dimensional Manifolds .......... 139 2.4.5 Direct Calculation of One-Dimensional Cohomology

Groups of a Plane, a Two-Dimensional Sphere and a Torus ......................... 141

2.4.6 Direct Calculation of One-Dimensional Cohomology Groups of Oriented Surfaces, i.e. Spheres with Handles . . 148

2.4.7 An Algorithm for Recognition of Two-Dimensional Manifolds. Elements of Two-Dimensional Computer Geometry .................... 152

2.4.8 Calculation of One-Dimensional Cohomologies of a Surface Using Triangulation .............. 154

2.4.9 Visual Material ....................... 154

2.5 Visual Properties of Three-Dimensional Manifolds .. . 159

2.5.1 Heegaard Splittings (or Diagrams) ............. 159 2.5.2 Examples of Three-Dimensional Manifolds ........ 162 2.5.3 Equivalence of Heegaard Splittings ............ 164 2.5.4 Spines ............................. 165 2.5.5 Special Spines ....................... 168 2.5.6 Filtration of 3-Manifolds with Respect to Matveev's

Complexity ......................... 170 2.5.7 Simplification of Special Spines .............. 173 2.5.8 The Use of Computers in Three-Dimensional Topology.

Enumeration of Manifolds in Increasing Order of Complexity ....................... 177

2.5.9 Matveev's Complexity of 3-Manifolds and Simplex Glueings ................... 184

2.5.10 Visual Material ....................... 188

XIV Table of Contents

3 Visual Symplectic Topology and Visual Hamiltonian Mechanics

3.1 Some Concepts of Hamiltonian Geometry . . . . . . . . 193 3.1.1 Hamiltonian Systems on Symplectic Manifolds ...... 193 3.1.2 Invo1utive Integrals and Liouville Tori ........... 196 3.1.3 Momentum Mapping of an Integrable System ..... . . 200 3.1.4 Surgery on Liouville Tori at Critical Energy Values .... 200 3.1.5 Visual Material .................... . . . 204

3.2 Qualitative Questions of Geometric Integration of Some Differential Equations. Classification of Typical Surgeries of Liouville Tori of Integrable Systems with Bott Integrals ........ 208

3.2.1 Nondegenerate (Bott) Integrals ............... 208 3.2.2 Classification of Surgeries of Bott Position

on Liouville Tori ...................... 210 3.2.3 The Topological Structure of Critical Energy Levels

at a Fixed Second Integral ................. 215 3.2.4 Examples from Mechanics. The Equations of Motion

of a Rigid Body. The Poisson Sphere. Geometrical Interpretation of Mechanical Systems . . . . . 216

3.2.5 An Example of an Investigation of a Mechanical System. The Liouville System on the Plane ............. 219

3.2.6 The Liouville System on the Sphere ............ 220 3.2.7 Inertial Motion of a Gyrostat ................ 221 3.2.8 The Case of Chaplygin-Sretensky ............. 223 3.2.9 The Case of Kovalevskaya ...... . . . . . . . . . . . 224 3.2.10 Visual Material ....................... 225

3.3 Three-Dimensional Manifolds and Visual Geometry of Isoenergy Surfaces of Integrable Systems . . . . . . . 231

3.3.1 A One-Dimensional Graph as a Hamiltonian Diagram ... 231 3.3.2 What Familiar Manifolds Are Encountered

Among Isoenergy Surfaces? ................ 234

Table of Contents XV

3.3.3 The Simplest Isoenergy Surfaces (with Boundary) ..... 243 3.3.4 Any Isoenergy Surface of an Integrable Nondegenerate

System Falls into the Sum of Five (or Two) Types of Elementary Bricks . . . . . . . . . . . . . . . . . . . . 244

3.3.5 New Topological Properties of the Isoenergy Surfaces Class ....................... 246

3.3.6 One Example of a Computer Use in Symplectic Topology 249 3.3.7 Visual Material ....................... 252

4 Visual Images in Some Other Fields of Geometry and Its Applications

4.1 Visual Geometry of Soap Films. Minimal Surfaces . . . 255

4.1.1 Boundaries Between Physical Media. Minimal Surfaces .. 255 4.1.2 Some Examples of Minimal Surfaces ........... 258 4.1.3 Visual Material ....................... 260

4.2 Fractal Geometry and Homeomorphisms ........ 264

4.2.1 Various Concepts of Dimension .............. 264 4.2.2 Fractals ........................... 267 4.2.3 Homeomorphisms...................... 268 4.2.4 Visual Material .,. . . . . . . . . . . . . . . . . . . . . 276

4.3 Visual Computer Geometry in the Number Theory .. 284

Appendix 1 Visual Geometry of Some Natural and Nonholonomic Systems

1.1 On Projection of Liouville Tori in Systems with Separation of Variables ................ 293

1.2 What Are Nonholonomic Constraints? ..... . . . . . . 295 1.3 The Variety of Manifolds in the Suslov Problem ..... 297

XVI Table of Contents

Appendix 2 Visual Hyperbolic Geometry

2.1 Discrete Groups and Their Fundamental Region . . . . . . 300 2.2 Discrete Groups Generated by Reflections in the Plane .. 301 2.3 The Gram Matrix and the Coxeter Scheme ........ 303 2.4 Reflection-Generated Discrete Groups in Space ...... 303 2.5 A Model of the Lobachevskian Plane ........... 306 2.6 Convex Polygons on the Lobachevskian Plane ....... 308 2.7 Coxeter Polygons on the Lobachevskian Plane ....... 309 2.8 Coxeter Polyhedra in the Lobachevskian Space ...... 310 2.9 Discrete Groups of Motions of Lobachevskian Space

and Groups of Integer-Valued Automorphisms of Hyperbolic Quadratic Forms . . . . . . . . . . . . . . . 314

2.10 Reflection-Generated Discrete Groups in High-Dimensional Lobachevskian Spaces ........ 315

References ............................ 317