# Generalized Monge gauge

## TL;DR

This paper extends the Monge gauge from flat to curved surfaces like spheres, deriving curvature formulas for more complex geometries using differential geometry and general relativity insights.

## Contribution

It introduces a generalized Monge gauge for curved surfaces, providing formulas for fundamental forms and curvatures on a spherical reference surface.

## Key findings

- Derived expressions for fundamental forms and curvatures on a sphere
- Analyzed limits and special cases of the generalized gauge
- Provided illustrative examples demonstrating the approach

## 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.