Facets of contact geometry

Abstract

‘After a while the style settles down a bit and it begins to tell you things you really need to know.’ Douglas Adams, The Hitch Hiker's Guide to the Galaxy This opening chapter is not meant as an introduction in the conventional sense (if only because it is much too long for that). Perhaps it would be appropriate to call it a proem , defined by the Oxford English Dictionary as ‘an introductory discourse at the beginning of a book’. Although some basic concepts are introduced along the way, the later chapters are largely independent of the present one. Primarily this chapter gives a somewhat rambling tour of contact geometry. Specifically, we consider polarities in projective geometry, the Hamiltonian flow of a mechanical system, the geodesic flow of a Riemannian manifold, and Huygens' principle in geometric optics. Contact geometry is the theme that connects these diverse topics. This may serve to indicate that Arnold's claim that ‘contact geometry is all geometry’ ([15], [17]) is not entirely facetious. I also present two remarkable applications of contact geometry to questions in differential and geometric topology: Eliashberg's proof of Cerf's theorem Γ 4 = 0 via the classification of contact structures on the 3—sphere, and the proof of Property P for non-trivial knots by Kronheimer and Mrowka, where symplectic fillings of contact manifolds play a key role.