A sufficient condition for global regularity of nets of curves in the Euclidean plane

Sufficient conditions are obtained for a net of lines in a plane given by
$$ \alpha\,dx^2+2\beta\,dx\,dy+\gamma\,dy^2=0,\qquad\alpha\gamma-\beta^2<0, $$
to be homeomorphic to the cartesian net (globally or in some region). These conditions are expressed in terms of the integral of the modulus of the second Chebyshev vector of the net. We consider separately the special case when the net is formed by the characteristics of a hyperbolic transformation.
Figures: 7.
Bibliography: 6 titles.