# zbMATH — the first resource for mathematics

##### Examples
 Geometry Search for the term Geometry in any field. Queries are case-independent. Funct* Wildcard queries are specified by * (e.g. functions, functorial, etc.). Otherwise the search is exact. "Topological group" Phrases (multi-words) should be set in "straight quotation marks". au: Bourbaki & ti: Algebra Search for author and title. The and-operator & is default and can be omitted. Chebyshev | Tschebyscheff The or-operator | allows to search for Chebyshev or Tschebyscheff. "Quasi* map*" py: 1989 The resulting documents have publication year 1989. so: Eur* J* Mat* Soc* cc: 14 Search for publications in a particular source with a Mathematics Subject Classification code (cc) in 14. "Partial diff* eq*" ! elliptic The not-operator ! eliminates all results containing the word elliptic. dt: b & au: Hilbert The document type is set to books; alternatively: j for journal articles, a for book articles. py: 2000-2015 cc: (94A | 11T) Number ranges are accepted. Terms can be grouped within (parentheses). la: chinese Find documents in a given language. ISO 639-1 language codes can also be used.

##### Operators
 a & b logic and a | b logic or !ab logic not abc* right wildcard "ab c" phrase (ab c) parentheses
##### Fields
 any anywhere an internal document identifier au author, editor ai internal author identifier ti title la language so source ab review, abstract py publication year rv reviewer cc MSC code ut uncontrolled term dt document type (j: journal article; b: book; a: book article)
Contact geometry and wave propagation. Lectures given at the University of Oxford in November and December 1988 under the sponsorship of the International Mathematical Union. (English) Zbl 0694.53001
Monographie de L’Enseignement Mathématique, 34: Série des Conférences de l’Union Mathématique Internationale, 9. Genève: Univ. de Genève, L’Enseignement Mathématique. 56 p. sFr. 27.00 (1989).

In this monograph the author studies the geometry of contact manifolds and its application to wave propagation. He has been one of the main contributors to this theory, especially regarding this application. See appendices of his own book [Mathematical methods of classical mechanics (1978; Zbl 0386.70001)].

Symplectic geometry is considered as the mathematical structure underlying mechanics and calculus of variations. Contact geometry is its odd dimensional counterpart, and it also has applications to geometrical optics. We can say that the two theories are formally equivalent. However, algebraic calculations are simpler in the symplectic case, but better understood when translated into contact geometry. Also, most of the global, topological results are more natural in contact geometry.

Probably the simplest example of a contact manifold is the 1-jet ${J}^{1}\left(M,R\right)$ of real-valued functions on a manifold M. If ${V}^{2n}$ is a vector space of dimension 2n with a bilinear symplectic structure, its projectivized space ${P}^{2n-1}$ carries a natural contact structure. In the same way, given a cotangent bundle ${T}^{*}M$ with the canonical symplectic structure, the projectivized cotangent bundle $P{T}^{*}M$ has one dimension less and carries a natural contact structure.

Given a ${C}^{\infty }$-function S: $M\to ℝ$, the image of its exterior derivative $dS\left(M\right)\subset {T}^{*}M$ is a particular example of what is called a Lagrangian submanifold, i.e., a maximal dimension manifold where the symplectic form annihilates. In the same way we may define Legendre submanifolds in a contact manifold. For example, in ${J}^{1}\left(M,R\right)$, the function S generates a Legendre submanifold. The Legendre submanifolds are closely related to Legendre transformations, which are particular cases of contact transformations between manifolds of the same dimension. A hypersurface in the space of 1-jets ${J}^{1}\left(M,R\right)$ is a first order nonlinear partial differential equation. The method of characteristics to solve such an equation produces a Legendre submanifold.

The space $P{T}^{*}M$ is interpreted as the space of all contact elements of M. A contact element in M is a hyperplane in a tangent space. Consider for example a hypersurface H in a Riemannian manifold M. The equidistant hypersurfaces are the wavefronts at each time t. They are obtained by moving each point of the hypersurface a distance t along the geodesic orthogonal to H in a given direction. The set of contact elements tangent to H and its image after any time t are always smooth Legendre manifolds in $P{T}^{*}M$. In general they may not be the family of all elements tangent to some smooth manifold, since a wavefront may develop singularities when projected onto M (e.g. caustics in optics, phase transitions in thermodynamics, etc.). The generic singularities of these projections are classified in normal forms by catastrophe theory. For example, simple and stable singularities are classified by simple Lie algebras of types A, D and E, which are related to nonregular orbits of Weyl groups. A more complex situation arises when a wavefront meets an obstacle in the medium (i.e., in a manifold with boundary). Then even the Legendre manifold develops singularities, and we rather talk about Legendre varieties. The best approach to study these singularities has been through Givental triads (describing families of rays and wavefronts at obstacle points) and the particular class of Givental Legendre varieties. This is related to irreducible finite dimensional $s{l}_{2}$- modules and to Coxeter Euclidean reflection groups.

Reviewer: E.A.Lacomba

##### MSC:
 53-02 Research monographs (differential geometry) 53C15 Differential geometric structures on manifolds 78A05 Geometric optics 35L99 Hyperbolic equations and systems 70G45 Differential-geometric methods for dynamical systems 58H05 Pseudogroups and differentiable groupoids on manifolds