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**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

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 \(J^ 1(M,R)\) 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 \(PT^*M\) has one dimension less and carries a natural contact structure.

Given a \(C^{\infty}\)-function S: \(M\to {\mathbb{R}}\), the image of its exterior derivative \(dS(M)\subset T^*M\) 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 \(J^ 1(M,R)\), 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 \(J^ 1(M,R)\) is a first order nonlinear partial differential equation. The method of characteristics to solve such an equation produces a Legendre submanifold.

The space \(PT^*M\) 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 \(PT^*M\). 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 \(sl_ 2\)- modules and to Coxeter Euclidean reflection groups.

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 \(J^ 1(M,R)\) 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 \(PT^*M\) has one dimension less and carries a natural contact structure.

Given a \(C^{\infty}\)-function S: \(M\to {\mathbb{R}}\), the image of its exterior derivative \(dS(M)\subset T^*M\) 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 \(J^ 1(M,R)\), 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 \(J^ 1(M,R)\) is a first order nonlinear partial differential equation. The method of characteristics to solve such an equation produces a Legendre submanifold.

The space \(PT^*M\) 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 \(PT^*M\). 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 \(sl_ 2\)- modules and to Coxeter Euclidean reflection groups.

Reviewer: E.A.Lacomba

### MSC:

53-02 | Research exposition (monographs, survey articles) pertaining to differential geometry |

53C15 | General geometric structures on manifolds (almost complex, almost product structures, etc.) |

78A05 | Geometric optics |

35L99 | Hyperbolic equations and hyperbolic systems |

70G45 | Differential geometric methods (tensors, connections, symplectic, Poisson, contact, Riemannian, nonholonomic, etc.) for problems in mechanics |

58H05 | Pseudogroups and differentiable groupoids |