Web Geometry of Solutions of First Order ODEs

A first order ordinary differential equation of one valuable (ODE) is f(x, y, y′) = 0 (∗) where y ′ stands for the derivative dy/dx and f is a germ of function at (0, 0, p0). f = 0 is equivalent to an equation g(x, y, y′) = 0 if there exists a germ of diffeomorphism φ of xy-plane which sends the solu-tions of f to those of g. In order to solve (*) one may solve the equation in y ′ locally as y ′ = fi(x, y) , i = 1,..., d using implicit functions fi. The solutions of each explicit differential equation form a germ of foliation by curves, hence the solutions of the equation (∗) form a configuration of d foliations. Such a structure is called a d-WEB, and has been long studied. The above figure shows a tipical singular 3-web structure, where p0 =∞. One of basic ideas to extract geometric invariants is to extend Bott connection of these foliations (if possible) to an equal affine connection ∇ of the xy-space. In the case d = 3, such a connection exists and is called Chern connection. In order to intro-duce it let ωi = Ui (dy − fi dx), i = 1, 2, 3 with functions Ui 6 = 0. and assume the normaliza-tion condition ω1 + ω2 + ω3 = 0. Then it is seen that there exists a unique θ such tha