On a linearizability condition for a three-web on a two-dimensional manifold

Abstract

Let W be a three-web formed by three one-parameter families of curves in a two-dimensional differentiable manifold and let K be its curvature. The web W is linearizable (rectifiable) if it is equivalent to a linear three-web, i.e. three-web formed by three one-parameter families of straight lines. A criterion of linearizability is very important in web geometry and especially in its application to nomography. All previous attempts to find the criterion failed. W. Blaschke claimed that it is hopeless to find the criterion. A new approach to the problem allows to find a necessary and sufficient condition for the W to be linearizable in terms of its curvature K, its covariant derivatives and components of the affine deformation tensor of the affine connection induced by W and its covariant derivatives. If a web W is given by an equation u=f(x,y), then this condition is reduced to a condition expressed in terms of the function f(x,y), its partial derivatives and the components of the affine deformation tensor and its covariant derivatives. To find this partial differential equation, a program for symbolic manipulation has been used. To make this condition effective, the components of the affine deformation tensor and its covariant derivatives should are excluded from it.