On the curvature of nonregular saddle surfaces in the hyperbolic and spherical three-space

Abstract

This paper proves that any nonregular nonparametric saddle surface in a three-dimensional space of nonzero constant curvature $k$, which is bounded by a rectifiable curve, is a space of curvature not greater than $k$ in the sense of Aleksandrov. This generalizes a classical theorem by Shefel′ on saddle surfaces in ${𝔼}^3$.