Contact geometry

Abstract

Contact geometry forms the mathematical basis of geometrical optics, in the same sense in which symplectic geometry forms the basis of classical mechanics. The ‘optical-mechanical analogy’ of Hamilton allows one to translate problems and results from symplectic geometry to the language of contact geometry and vice versa. However, a direct approach in terms of contact geometry is in many cases preferable, at least from the point of view of geometrical intuition: it shows the geometrical content of formulas from the symplectic theory. The relation between symplectic and contact geometry is similar to the relation between the geometries of linear spaces and projective geometry: in order to obtain a contact counterpart of a symplectic piece of theory one has to replace functions by hyp er surf aces, aißne spaces by projective spaces, etc.