Classification of separable surfaces with constant Gaussian curvature

TL;DR

This paper classifies all separable surfaces with constant Gaussian curvature in Euclidean 3-space, identifying their geometric types and providing explicit parametrizations for zero curvature cases.

Contribution

It provides a complete classification of separable constant Gaussian curvature surfaces, including explicit parametrizations for zero curvature surfaces and a proof that non-zero curvature surfaces are surfaces of revolution.

Findings

Surfaces with zero Gaussian curvature are surfaces of revolution, cylinders, or cones.

Explicit parametrizations are obtained for zero curvature surfaces.

Surfaces with non-zero Gaussian curvature are necessarily surfaces of revolution.

Abstract

We classify all surfaces with constant Gaussian curvature $K$ in Euclidean $3$-space that can be expressed as an implicit equation of type $f(x)+g(y)+h(z)=0$, where $f$, $g$ and $h$ are real functions of one variable. If $K=0$, we prove that the surface is a surface of revolution, a cylindrical surface or a conical surface, obtaining explicit parametrizations of such surfaces. If $K\not=0$, we prove that the surface is a surface of revolution.