On Generalized Spherical Surfaces in Euclidean Spaces

TL;DR

This paper explores generalized spherical surfaces in Euclidean spaces, introducing new types of surfaces, calculating their curvatures, and providing examples to expand understanding of their geometric properties.

Contribution

It introduces generalized spherical surfaces of two kinds in Euclidean spaces and computes their curvature properties, extending classical surface theory.

Findings

Generalized spherical surfaces of the first kind are rotational surfaces in 4.

Generalized spherical surfaces of the second kind are meridian surfaces in 4.

Curvatures of these surfaces are explicitly calculated.

Abstract

In the present study we consider the generalized rotational surfaces in Euclidean spaces. Firstly, we consider generalized spherical curves in Euclidean $(n+1)-$space $\mathbb{E}^{n+1}$. Further, we introduce some kind of generalized spherical surfaces in Euclidean spaces $\mathbb{E}^{3}$ and $% \mathbb{E}^{4}$ respectively. We have shown that the generalized spherical surfaces of first kind in $\mathbb{E}^{4}$ are known as rotational surfaces, and the second kind generalized spherical surfaces are known as meridian surfaces in $\mathbb{E}^{4}$. We have also calculated the Gaussian, normal and mean curvatures of these kind of surfaces. Finally, we give some examples.