Surface Pencils in Euclidean 4-space $\mathbb{E}^{4}$

TL;DR

This paper explores the construction and properties of surface pencils in 4-dimensional Euclidean space, focusing on generalized rotation surfaces, their curvature, and examples of flat surface pencils.

Contribution

It introduces a method to construct surface pencils from curves in $ ext{E}^4$, identifies generalized rotation surfaces as a special case, and examines their curvature properties.

Findings

Generalized rotation surfaces are a special type of surface pencil in $ ext{E}^4$

Curvature properties of these surfaces are characterized

Examples of flat surface pencils are provided

Abstract

In the present paper we study the problem of constructing a family of surfaces (surface pencils) from a given curve in 4-dimensional Euclidean space $\mathbb{E}^{4}$. We have shown that generalized rotation surfaces in $\mathbb{E}^{4}$ are the special type of surface pencils. Further, the curvature properties of these surfaces are investigated. Finally, we give some examples of flat surface pencils in $\mathbb{E}^{4}$.