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Über dieses Buch

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.

Inhaltsverzeichnis

Frontmatter

1. Polyhedra. Simplicial Complexes. Homologies

Abstract
Geometry and topology most often deal with geometrical figures, objects realized as a set of points in a Euclidean space (maybe of many dimensions). It is useful to view these objects not as rigid (solid) bodies, but as figures that admit continuous deformation preserving some qualitative properties of the object. Recall that the mapping of one object onto another is called continuous if it can be determined by means of continuous functions in a Cartesian coordinate system in space. The mapping of one figure onto another is called homeomorphism if it is continuous and one-to-one, i.e. establishes a one-to-one correspondence between points of both figures. In this case there exists the inverse mapping with the same properties. For example, a straight line segment and a continuous arc (without self-intersections) on a plane are homeomorphic (Fig. 1.1.1). Homeomorphic are also a square and a circle (Fig. 1.1.1), a cube and a tetrahedron (sometimes called a simplex) (Fig. 1.1.1), a plane and a sphere with one punctured (discarded) point (Fig. 1.1.1). In the latter case the homeomorphism can be realized using the so-called stereographic projection (Fig. 1.1.2). To this end one should put a standard sphere onto a plane, take the point of tangency for the south pole and the very top point for the north pole N.
Anatolij T. Fomenko

2. Low-Dimensional Manifolds

Abstract
Modern differential geometry is an independent scientific discipline, exceedingly branched and connected with numerous applications. Today’s edifice of differential geometry exhibits two basic layers. The one which was historically the first to appear may be conditionally called local differential geometry which usually develops in a region of a Euclidean space. This is the foundation and the first storeys of the whole edifice. Then there appeared next storeys which took their shape later than the first ones. They are referred to as “global differential geometry”. Here local concepts are closely interwoven with global, topological ones. The backbone of the whole building is the theory of smooth manifolds. On lower storeys, it is at first local, i.e. events occur in a small enough domain of a manifold. Ascending, the theory develops global aspects and finally, on the top, the modern geometry operates with essentially nonlocal effects.
Anatolij T. Fomenko

3. Visual Symplectic Topology and Visual Hamiltonian Mechanics

Abstract
We shall consider a vector field υ on a smooth manifold. Relative to local coordinates x1,..., x n this field can be written as dx i /dt = υ i (x1,..., x n ) where 1 ≤ in and υ i (x1,..., x n ) are smooth functions which are components of the field υ. Thus, each vector field is interpreted as a system of ordinary differential equations on a manifold. Inversely, each system or ordinary differential equations can be represented as the vector field on a corresponding manifold. Many physical laws are written in the form of differential equations, and therefore the study of topological properties of vector fields provides a good deal of qualitative information on the behaviour of certain physical systems. Among the variety of mechanical and physical systems there exists an important class described using so-called Hamiltonian equations. These systems can be realized on even-dimensional manifolds only.
Anatolij T. Fomenko

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

Abstract
Suppose in ℝ3 there are several adjoining but not mixing physical media, for instance, in a large vessel there are several immiscible fluids. Suppose, the whole system is in equilibrium. Since the media are immiscible, the boundaries (interfaces) between them are determined. These interfaces J can be tought of (in the first approximation) as two-dimensional piece-wise smooth surfaces separating the adjoining media. We consider for simplicity the case of two media which we denote by A1 and A2. Let the pressures in the media be respectively equal to p1 and p2. The equilibrium condition for the media proves to impose a strong restriction upon the geometry of their interface. To formulate this restriction, we require an important concept of local differential geometry, namely, the concept of mean curvature of the surface.
Anatolij T. Fomenko

Backmatter

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