Constant Scalar Curvature of Three Dimensional Surfaces Obtained by the Equiform Motion of a Sphere

TL;DR

This paper investigates the properties of three-dimensional surfaces generated by the equiform motion of a sphere in seven-dimensional space, focusing on surfaces with constant scalar curvature and establishing bounds on this curvature.

Contribution

It introduces a study of the scalar curvature of surfaces formed by equiform motions in higher-dimensional Euclidean space and proves a bound on the constant scalar curvature.

Findings

Scalar curvature ||<2 for the surfaces considered

Analysis of kinematic surfaces generated by sphere motions in -dimensional space

Conditions under which the scalar curvature remains constant

Abstract

In this paper we consider the equiform motion of a sphere in Euclidean space $\mathbf{E}^7$. We study and analyze the corresponding kinematic three dimensional surface under the hypothesis that its scalar curvature $\mathbf{K}$ is constant. Under this assumption, we prove that $|\mathbf{K}|<2$.