×

Singularities of caustics and wave fronts. Transl. from the Russian. (English) Zbl 0734.53001

Caustics and wave fronts have been classical objects of study. But not until recent years the intrinsic structure of the singularities was discovered to be governed by symplectic geometry and group theory. In this comprehensive expository book the author presents the recent advances in the field - a large portion due to himself and his students. This elegant presentation of the subject uses systematically intrinsic and global techniques and replaces cumbersome and painful long computations by geometric insights. The principal method is to study geometric objects via symplectic and contact geometry on the cotangent bundle. Other techniques used include cobordisms, characteristic classes, groups, Dynkin diagrams, etc. This book will be for some time the definite word on caustics and wave fronts.
Some highlights of the contents: Chapter 1, “symplectic geometry”: Starting with some simple (but also including some unfamiliar) examples the foundation of symplectic manifolds, Hamiltonian systems, symplectic submanifolds, Lagrangian foliations and Lagrangian singularities are laid out. The second chapter, “application of Lagrangian singularities”, heads towards the main subject. It includes applications “to the method of stationary phase, the asymptotics of lattice points in generic large domains, the cosmological theory of large scale structures of the Universe and to the study of perestroicas (metamorphoses) of optical caustics and shock waves”. The description of bifurcations of caustics and optical caustics is followed by a discussion of shock wave singularities.
A chapter on contact geometry and wave fronts follows. Here some of the key results are presented: the equivalence between wave fronts and the non regular orbits of reflection groups. Dynkin diagrams are introduced to describe these orbits. The theory of normal form of perestroicas of wave fronts closes the chapter. Chapter 4 presents some underlying algebraic techniques: The group of diffeomorphisms which reduces the perestroicas to normal forms is fundamental for the following. It’s Lie algebra, which can be described by vector fields tangent to wave fronts is studied in details. Sections on period maps and Poisson structures follow. Chapter 5 treats some global topological problems. These include Lagrangian and Legendre cobordisms and characteristic classes etc. The final chapters apply the previous results to the study of singularities: Chapter 6 treats singularities of projection, Chapter 7 studies the obstacle problem and in Chapter 8 the normal forms of singularities of light hypersurfaces, ray systems and wave fronts are presented.
The book is self-contained but written in such an intensive and original style, that without preparation it may be intimidating. Arnold proceeds from short basic definitions and numerous examples via a few key theorems to the main results. The presentation is very clear and smooth but extremely dense. Many proofs are missing or only sketched but the line of thoughts is always complete. As an inexhaustible source of beautiful results it is an excellent reference book or a textbook for a demanding (to both, teacher and students) graduate course. Anyone who is interested in the mathematics of geometric optics, singularities, bifurcation theory or just in applications of global analysis may find this as one of the few “must have” books.
Reviewer: C.Günther (Libby)

MSC:

53-02 Research exposition (monographs, survey articles) pertaining to differential geometry
58-02 Research exposition (monographs, survey articles) pertaining to global analysis
53C15 General geometric structures on manifolds (almost complex, almost product structures, etc.)
58C25 Differentiable maps on manifolds
58K99 Theory of singularities and catastrophe theory
PDFBibTeX XMLCite