Covariant description of isothermic surfaces

TL;DR

This paper develops a covariant framework for describing isothermic surfaces in curved 3D spaces, providing invariant conditions and methods for constructing special coordinate systems.

Contribution

It introduces a coordinate-invariant condition for isothermic surfaces and details how to construct isothermic coordinates in curved spaces.

Findings

Derived covariant Gauss-Weingarten and Gauss-Mainardi-Codazzi equations.

Established a necessary and sufficient invariant condition for isothermic surfaces.

Provided a method to construct isothermic coordinates in curved geometries.

Abstract

We present a covariant formulation of the Gauss-Weingarten equations and the Gauss-Mainardi-Codazzi equations for surfaces in 3-dimensional curved spaces. We derive a coordinate invariant condition on the first and second fundamental form which is locally necessary and sufficient for the surface to be isothermic. We show how to construct isothermic coordinates.