On the properties of the normal mapping generated by the equations $rt-s^2=-f^2(x,y)$

S. P. Geisberg

Sbornik. Mathematics, Tome 11 (1970) no. 2, pp. 201-208

Résumé

In this paper the following theorem is proved: Let $z=z(x,y)\in C^2$ be a solution of the equation $rt-s^2=-f^2(x,y)$ defined in the entire $(x,y)$ plane, and let $p=z_x(x,y)$, $q=z_y(x,y)$ be the normal image of this plane in the $(p,q)$ plane. Let one of the following conditions be satisfied: 1) $f(x,y)$ is a convex function, $f(x,y)>\varepsilon>0$; 2) $f^2(x, y)$ is a polynomial, $f(x,y)>\varepsilon>0$. \noindent Then the image of the $(x,y)$ plane cannot be a strip between parallel lines. This theorem gives an answer, in an important particular case, to a question posed by N. V. Efimov at the 2nd All-Union Symposium on Geometry in the Large in 1967. Bibliography: 2 titles.