$\epsilon$-isothermic surfaces in pseudo-Euclidean 3-space

TL;DR

This paper studies $$-isothermic surfaces in pseudo-Euclidean 3-space, classifies Dupin surfaces with distinct principal curvatures, and provides explicit solutions to the pseudo-Calapso equation.

Contribution

It introduces the concept of $$-isothermic surfaces in pseudo-Euclidean space and classifies Dupin surfaces with explicit coordinate representations.

Findings

Derived the pseudo-Calapso equation.

Classified Dupin surfaces with distinct principal curvatures.

Provided explicit solutions to the pseudo-Calapso equation.

Abstract

In this paper we describe the $\epsilon$-isothermic surfaces in the pseudo-Euclidean 3-space and we obtain the pseudo-Calapso equation. In sequence, we classify the Dupin surfaces in pseudo-Euclidean 3-space having distinct principal curvatures and provide explicit coordinates for such surfaces. As application of the theory, we give explicit solutions to the pseudo-Calapso equation.