Generalized Monge gauge

Abstract

Monge gauge in differential geometry is generalized. The original Monge gauge is based on a surface defined as a height function $h(x,y)$ 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 Monge gauge toward more complicated surfaces. Here in this study we consider the height function above a curved surface namely a sphere of radius $R$. The proposed height function is a function of $\theta $ and $\varphi $ 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.