Local isometric immersions of two-dimensional metrics inE 4 with prescribed Gaussian torsion

Suppose that in a domain R(, B) of variables (r, ): (0 r , ₁ +B(r–r₀) ₂–B(r–r₀), where > 0, B > 0, ₁ < 0 < ₂ are numbers) a metric ds² = dr² +G(r, )d² and a function k(r, ) are given. The problem of isometrically immersing ds²in E⁴with prescribed Gaussian torsion is considered. The following is proved: The class C⁵metric ds²is locally realized in the form of a class C³surface F²whose Gaussian torsion is the prescribed class C³function (r, ).Translated from Ukrainskii Geometricheskii Sbornik, No. 35, pp. 38–47, 1992.