Generalized Monge gauge

Abstract

Monge Gauge (MG) in differential geometry is generalized. The original MG is based on a surface defined as a height function [Formula: see text] above a flat reference plane. The total curvature and the Gaussian curvature are found in terms of the height function. Getting benefits from our mathematical knowledge of general relativity, we shall extend the MG toward more complicated surfaces. Here, in this study, we consider the height function above a curved surface, namely a sphere of radius [Formula: see text]. The proposed height function is a function of [Formula: see text] and [Formula: see text] on a closed interval. We find the first, second fundamental forms and the total and Gaussian curvatures in terms of the new height function. Some specific limits are discussed and two illustrative examples are given.