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1985 | Buch

The simplest examples Kapitel lesen Erstes Kapitel lesen

Singularities of Differentiable Maps

Volume I: The Classification of Critical Points Caustics and Wave Fronts

verfasst von: V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko

Verlag: Birkhäuser Boston

Buchreihe : Monographs in Mathematics

Enthalten in: Professional Book Archive

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Über dieses Buch

... there is nothing so enthralling, so grandiose, nothing that stuns or captivates the human soul quite so much as a first course in a science. After the first five or six lectures one already holds the brightest hopes, already sees oneself as a seeker after truth. I too have wholeheartedly pursued science passionately, as one would a beloved woman. I was a slave, and sought no other sun in my life. Day and night I crammed myself, bending my back, ruining myself over my books; I wept when I beheld others exploiting science fot personal gain. But I was not long enthralled. The truth is every science has a beginning, but never an end - they go on for ever like periodic fractions. Zoology, for example, has discovered thirty-five thousand forms of life ... A. P. Chekhov. "On the road" In this book a start is made to the "zoology" of the singularities of differentiable maps. This theory is a young branch of analysis which currently occupies a central place in mathematics; it is the crossroads of paths leading from very abstract corners of mathematics (such as algebraic and differential geometry and topology, Lie groups and algebras, complex manifolds, commutative algebra and the like) to the most applied areas (such as differential equations and dynamical systems, optimal control, the theory of bifurcations and catastrophes, short-wave and saddle-point asymptotics and geometrical and wave optics).

Inhaltsverzeichnis

Frontmatter

Basic Concepts

Frontmatter
1. The simplest examples
Abstract
Here we describe the classification due to H. Whitney of the singularities of smooth maps of spaces of small dimensions.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
2. The classes Σ I
Abstract
Here singularities are classified according to the rank of the first differential of the map and the ranks of its restrictions to submanifolds of singularities.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
3. The quadratic differential of a map
Abstract
The rank of the first differential f x determines the singularity classes Σ i . Consideration of the quadratic part of the map gives a more precise classification: we associate to each singularity a family of quadratic forms invariantly associated with it.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
4. The local algebra of a map and the Weierstrass preparation theorem
Abstract
Every geometrical object may be described in two ways — in terms of points of manifolds and in terms of functions on them.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
5. The local multiplicity of a holomorphic map
Abstract
It is proved in this Chapter that the algebraic multiplicity of a holomorphic map coincides with its geometric multiplicity, that is with the index of the singular point of the corresponding holomorphic field. Although this result was known classically it seems that a detailed proof was published only in the paper [139] of V. P. Palamodov. The idea of the elementary proof presented below is due to A. G. Kushnirenko [106].
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
6. Stability and infinitesimal stability
Abstract
In this Chapter a linearisation method is described for determining whether a given differentiable map-germ is stable. The gist of the method consists in reducing the question to the linear problem of infinitesimal stability and to the practically more easily solved problem of infinitesimal V-stability. We develop the technique necessary for the foundation of the method and apply it to the simplest situation, proving a theorem about the equivalence of a function to its Taylor polynomial in a neighbourhood of a critical point of finite multiplicity.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
7. The proof of the stability theorem
Abstract
In this Chapter the stability of an infinitesimally stable map-germ is proved. The proof consists of two parts. Firstly it is proved that the k-jet of an infinitesimally stable germ is stable for sufficiently large k, that is that every sufficiently near map has at a suitably near point a left-right equivalent k-jet. Secondly it is proved that the k-jet of an infinitesimally stable germ is sufficient for sufficiently large k. From these two facts stability clearly follows.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
8. Versal deformations
Abstract
Ordinarily in the consideration of all possible singularities the principal interest focusses on the generic case, since all the more complicated singularities may be got rid of by a small perturbation. For example, a generic function has only nondegenerate critical points; degenerate critical points, like the critical point of the function x3, decompose into nondegenerate ones under a suitable small deformation, for example x3 − εx.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
9. The classification of stable germs by genotype
Abstract
Here we prove the theorem of Mather, reducing the classification of RL-stable germs to the V-classification of germs of maps of spaces of smaller dimension (see [126]).
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
10. Review of further results
Abstract
In this Chapter an account is given of a number of generalisations of the stability and versality theorems: we consider diagrams of maps, maps with a compact symmetry group and the local topological theory of singularities.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko

Critical Points of Smooth Functions

Frontmatter
11. A start to the classification of critical points
Abstract
In this Chapter we describe the basic stages in the classification of critical points of holomorphic functions; the results of the classification and the computations necessary for carrying out the various stages are given in the following sections. All the functions under consideration are holomorphic and have a critical point at 0 with critical value 0. By equivalence we mean R-equivalence (two function-germs are equivalent, if they can be turned into one another under the action of biholomorphic changes of the independent variables).
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
12. Quasihomogeneous and semiquasihomogeneous singularities
Abstract
Here we construct the apparatus of quasihomogeneous and semiquasihomogeneous diffeomorphisms for reducing to normal form quasihomogeneous and semiquasihomogeneous singularities.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
13. The classification of quasihomogeneous functions
Abstract
Here are described methods for reducing quasihomogeneous functions to normal form by means of quasihomogeneous diffeomorphisms.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
14. Spectral sequences for the reduction to normal forms
Abstract
Here is described a method for the reduction to normal forms, based on a spectral sequence, constructed from the filtration of the Koszul complex determined by the partial derivatives of the function under discussion. We do not use explicitly any properties of spectral sequences or of Koszul complexes, but prove directly everything that is necessary for the practical calculations. The correspondence between our constructions and the ordinary algebraic constructions is described in [18].
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
15. Lists of singularities
Abstract
In this Section the beginning of the hierarchy of classes of singularities of holomorphic functions is described.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
16. The determinator of singularities
Abstract
The 105 theorems given below allow one to find the place of any singularity in the lists of Chapter 15.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
17. Real, symmetric and boundary singularities
Abstract
Three generalisations of the theory of critical points of functions are considered here. Tables are given of the simplest degeneracies in the real case, in the symmetric case and in the case of functions on manifolds with boundary.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko

The Singularities of Caustics and Wave Fronts

Frontmatter
18. Lagrangian singularities
Abstract
A caustic may be seen on a wall, illuminated by rays of light reflected from a concave surface, for example the surface of a cup. By moving the cup one can see that generic caustics have only standard singularities, while more complicated singularities decompose into standard ones under small perturbations. In this Chapter we construct a tool for studying the singularities of caustics — the theory of Lagrangian singularities.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
19. Generating families
Abstract
A germ of an n-dimensional Lagrangian manifold can be given by a generating function in n variables. In this sense the manifold of Lagrangian germs in ℝ2n has an atlas, each of whose 2 n charts is the space of germs of functions of n variables (modulo additive constants).
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
20. Legendrian singularities
Abstract
Lay out along each inner normal to an ellipse a segment of length t. The free ends of these segments form a curve, called a front (of a disturbance flowing out into the ellipse in time t).
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
21. The classification of Lagrangian and Legendrian singularities
Abstract
The theory of generating families reduces the study of Lagrangian and Legendrian singularities to the study of singularities of families of functions and hypersurfaces. The tools developed in the preceding parts for studying the singularities of functions therefore give valuable information about caustics and fronts. In the pages which follow we give the results that have been obtained in this direction.
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
22. Bifurcations of caustics and wave fronts
Abstract
An evolving wave front will not at all moments of time be a generic front: at particular moments it will bifurcate. The study of such bifurcations reduces to a problem on generic singularities in one-parameter families of Legendrian maps. In this Chapter generic one-parameter families of Lagrangian and Legendrian maps are studied. Normal forms are given in those cases where the dimension of the front does not exceed four (or where the dimension of the caustic does not exceed two).
V. I. Arnold, S. M. Gusein-Zade, A. N. Varchenko
Backmatter
Metadaten
Titel
Singularities of Differentiable Maps
verfasst von
V. I. Arnold
S. M. Gusein-Zade
A. N. Varchenko
Copyright-Jahr
1985
Verlag
Birkhäuser Boston
Electronic ISBN
978-1-4612-5154-5
Print ISBN
978-1-4612-9589-1
DOI
https://doi.org/10.1007/978-1-4612-5154-5