Surfaces of Revolution with Constant Gaussian Curvature in Four-Space

TL;DR

This paper characterizes surfaces of revolution in four-dimensional space with constant Gaussian curvature, showing it depends solely on the radius of rotation and providing conditions for general rotational surfaces.

Contribution

It establishes necessary and sufficient conditions for constant Gaussian curvature in rotational surfaces in four-space, including parametrizations when curvature and rotation rates are equal.

Findings

Gaussian curvature depends only on the radius of rotation

Conditions for constant Gaussian curvature in general rotational surfaces

Parametrization of meridians when curvature and rotation rates are equal

Abstract

In this paper, we show that the constant property of the Gaussian curvature of surfaces of revolution in both $\mathbb R^4$ and $\mathbb R_1^4$ depend only on the radius of rotation. We then give necessary and sufficient conditions for the Gaussian curvature of the general rotational surfaces whose meridians lie in two dimensional planes in $\mathbb R^4$ to be constant, and define the parametrization of the meridians when both the Gaussian curvature is constant and the rates of rotation are equal.