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

## TL;DR

This paper investigates the scalar curvature of three-dimensional surfaces generated by the equiform motion of a helix in seven-dimensional Euclidean space, proving that such surfaces can only have zero scalar curvature.

## Contribution

It establishes that the scalar curvature of these surfaces must be zero when the surface is generated by a helix undergoing equiform motion in seven-dimensional space.

## Key findings

- Scalar curvature of the surface is necessarily zero.
- The study focuses on surfaces generated by a helix in $ extbf{E}^7$.
- Provides conditions under which the scalar curvature remains constant.

## Abstract

In this paper we consider the equiform motion of a helix 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 if the scalar curvature $\mathbf{K}$ is constant, then $\mathbf{K}$ must equal zero.