Visual Geometry and Topology

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Geometry and topology are strongly motivated by the visualization of ideal objects that have certain special characteristics. A clear formulation of a specific property or a logically consistent proof of a theorem often comes only after the mathematician has correctly "seen" what is going on. These pictures which are meant to serve as signposts leading to mathematical understanding, frequently also contain a beauty of their own. The principal aim of this book is to narrate, in an accessible and fairly visual language, about some classical and modern achievements of geometry and topology in both intrinsic mathematical problems and applications to mathematical physics. The book starts from classical notions of topology and ends with remarkable new results in Hamiltonian geometry. Fomenko lays special emphasis upon visual explanations of the problems and results and downplays the abstract logical aspects of calculations. As an example, readers can very quickly penetrate into the new theory of topological descriptions of integrable Hamiltonian differential equations. The book includes numerous graphical sheets drawn by the author, which are presented in special sections of "Visual material". These pictures illustrate the mathematical ideas and results contained in the book. Using these pictures, the reader can understand many modern mathematical ideas and methods. Although "Visual Geometry and Topology" is about mathematics, Fomenko has written and illustrated this book so that students and researchers from all the natural sciences and also artists and art students will find something of interest within its pages.

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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.- 1.1.3 Polyhedra. Simplicial Subdivisions of Polyhedra. Simplicial Complexes.- 1.1.4 Examples of Polyhedra.- 1.1.5 Barycentric Subdivision.- 1.1.6 Visual Material.- 1.2 Simplicial Homology Groups of Simplicial Complexes (Polyhedra).- 1.2.1 Simplicial Chains.- 1.2.2 Chain Boundary.- 1.2.3 The Simplest Properties of the Boundary Operator Cycles. Boundaries.- 1.2.4 Examples of Calculations of the Boundary Operator.- 1.2.5 Simplicial Homology Groups.- 1.2.6 Examples of Calculations of Homology Groups. Homologies of Two-dimensional Surfaces.- 1.2.7 Visual Material.- 1.3 General Properties of Simplicial Homology Groups.- 1.3.1 Incidence Matrices.- 1.3.2 The Method of Calculation of Homology Groups Using Incidence Matrices.- 1.3.3 "Traces" of Cell Homologies Inside Simplicial Ones.- 1.3.4 Chain Homotopy. Independence of Simplicial Homologies of a Polyhedron of the Choice of Triangulation.- 1.3.5 Visual Material.- 2 Low-Dimensional Manifolds.- 2.1 Basic Concepts of Differential Geometry.- 2.1.1 Coordinates in a Region. Transformations of Curvilinear Coordinates.- 2.1.2 The Concept of a Manifold. Smooth Manifolds. Submanifolds and Ways of Defining Them. Manifolds with Boundary. Tangent Space and Tangent Bundle.- 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 Nondegenerate Critical Points and Morse Functions.- 2.1.4 Vector and Covector Fields. Integral Trajectories. Vector Field Commutators. The Lie Algebra of Vector Fields on a Manifold.- 2.1.5 Visual Material.- 2.2 Visual Properties of One-Dimensional Manifolds.- 2.2.1 Isotopies, Frames.- 2.2.2 Visual Material.- 2.3 Visual Properties of Two-Dimensional Manifolds.- 2.3.1 Two-Dimensional Manifolds with Boundary.- 2.3.2 Examples of Two-Dimensional Manifolds.- 2.3.3 Modelling of a Projective Plane in a Three-Dimensional Space.- 2.3.4 Two Series of Two-Dimensional Closed Manifolds.- 2.3.5 Classification of Closed 2-Manifolds.- 2.3.6 Inversion of a Two-Dimensional Sphere.- 2.3.7 Visual Material.- 2.4 Cohomology Groups and Differential Forms.- 2.4.1 Differential 1-Forms on a Smooth Manifold.- 2.4.2 Closed and Exact Forms on a Two-Dimensional Manifold.- 2.4.3 An Important Property of Cohomology Groups.- 2.4.4 Direct Calculation of One-Dimensional Cohomology Groups of One-Dimensional Manifolds.- 2.4.5 Direct Calculation of One-Dimensional Cohomology Groups of a Plane, a Two-Dimensional Sphere and a Torus.- 2.4.6 Direct Calculation of One-Dimensional Cohomology Groups of Oriented Surfaces, i.e. Spheres with Handles.- 2.4.7 An Algorithm for Recognition of Two-Dimensional Manifolds. Elements of Two-Dimensional Computer Geometry.- 2.4.8 Calculation of One-Dimensional Cohomologies of a Surface Using Triangulation.- 2.4.9 Visual Material.- 2.5 Visual Properties of Three-Dimensional Manifolds.- 2.5.1 Heegaard Splittings (or Diagrams).- 2.5.2 Examples of Three-Dimensional Manifolds.- 2.5.3 Equivalence of Heegaard Splittings.- 2.5.4 Spines.- 2.5.5 Special Spines.- 2.5.6 Filtration of 3-Manifolds with Respect to Matveev's Complexity.- 2.5.7 Simplification of Special Spines.- 2.5.8 The Use of Computers in Three-Dimensional Topology. Enumeration of Manifolds in Increasing Order of Complexity.- 2.5.9 Matveev's Complexity of 3-Manifolds and Simplex Glueings.- 2.5.10 Visual Material.- 3 Visual Symplectic Topology and Visual Hamiltonian Mechanics.- 3.1 Some Concepts of Hamiltonian Geometry.- 3.1.1 Hamiltonian Systems on Symplectic Manifolds.- 3.1.2 Involutive Integrals and Liouville Tori.- 3.1.3 Momentum Mapping of an Integrable System.- 3.1.4 Surgery on Liouville Tori at Critical Energy Values.- 3.1.5 Visual Material.- 3.2 Qualitative Questions of Geometric Integration of Some Differential Equations. Classification of Typical Surgeries of Liouville Tori of Integrable Systems with Bott Integrals.- 3.2.1 Nondegenerate (Bott) Integrals.- 3.2.2 Classification of Surgeries of Bott Position on Liouville Tori.- 3.2.3 The Topological Structure of Critical Energy Levels at a Fixed Second Integral.- 3.2.4 Examples from Mechanics. The Equations of Motion of a Rigid Body. The Poisson Sphere. Geometrical Interpretation of Mechanical Systems.- 3.2.5 An Example of an Investigation of a Mechanical System. The Liouville System on the Plane.- 3.2.6 The Liouville System on the Sphere.- 3.2.7 Inertial Motion of a Gyrostat.- 3.2.8 The Case of Chaplygin-Sretensky.- 3.2.9 The Case of Kovalevskaya.- 3.2.10 Visual Material.- 3.3 Three-Dimensional Manifolds and Visual Geometry of Isoenergy Surfaces of Integrable Systems.- 3.3.1 A One-Dimensional Graph as a Hamiltonian Diagram.- 3.3.2 What Familiar Manifolds Are Encountered Among Isoenergy Surfaces?.- 3.3.3 The Simplest Isoenergy Surfaces (with Boundary).- 3.3.4 Any Isoenergy Surface of an Integrable Nondegenerate System Falls into the Sum of Five (or Two) Types of Elementary Bricks.- 3.3.5 New Topological Properties of the Isoenergy Surfaces Class.- 3.3.6 One Example of a Computer Use in Symplectic Topology.- 3.3.7 Visual Material.- 4 Visual Images in Some Other Fields of Geometry and Its Applications.- 4.1 Visual Geometry of Soap Films. Minimal Surfaces.- 4.1.1 Boundaries Between Physical Media. Minimal Surfaces.- 4.1.2 Some Examples of Minimal Surfaces.- 4.1.3 Visual Material.- 4.2 Fractal Geometry and Homeomorphisms.- 4.2.1 Various Concepts of Dimension.- 4.2.2 Fractals.- 4.2.3 Homeomorphisms.- 4.2.4 Visual Material.- 4.3 Visual Computer Geometry in the Number Theory.- Appendix 1 Visual Geometry of Some Natural and Nonholonomic Systems.- 1.1 On Projection of Liouville Tori in Systems with Separation of Variables.- 1.2 What Are Nonholonomic Constraints?.- 1.3 The Variety of Manifolds in the Suslov Problem.- Appendix 2 Visual Hyperbolic Geometry.- 2.1 Discrete Groups and Their Fundamental Region.- 2.2 Discrete Groups Generated by Reflections in the Plane.- 2.3 The Gram Matrix and the Coxeter Scheme.- 2.4 Reflection-Generated Discrete Groups in Space.- 2.5 A Model of the Lobachevskian Plane.- 2.6 Convex Polygons on the Lobachevskian Plane.- 2.7 Coxeter Polygons on the Lobachevskian Plane.- 2.8 Coxeter Polyhedra in the Lobachevskian Space.- 2.9 Discrete Groups of Motions of Lobachevskian Space and Groups of Integer-Valued Automorphisms of Hyperbolic Quadratic Forms.- 2.10 Reflection-Generated Discrete Groups in High-Dimensional Lobachevskian Spaces.- References.

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