Geometric Characterizations of Canal Hypersurfaces in Euclidean Spaces

TL;DR

This paper provides a comprehensive geometric analysis of canal hypersurfaces in Euclidean spaces, deriving curvature relations and classifying flat and minimal cases in four-dimensional space.

Contribution

It introduces explicit formulas for canal hypersurfaces in Euclidean n-space, especially in E4, and characterizes flat and minimal canal hypersurfaces, including their relation to generalized catenoids.

Findings

Flat canal hypersurfaces in E4 are circular hypercylinders or hypercones.

Minimal canal hypersurfaces in E4 are generalized catenoids.

Derived curvature relations linking mean and Gaussian curvatures.

Abstract

In the present paper, firstly we obtain the general expression of canal hypersurfaces in Euclidean n-space and deal with canal hypersurfaces in Euclidean 4-space E4. We compute Gauss map, Gaussian curvature and mean curvature of canal hypersurfaces in E4 and obtain an important relation between the mean and Gaussian curvatures as 3Hrho = Krho^3-2. We prove that, the flat canal hypersurfaces in Euclidean 4-space are only circular hypercylinders or circular hypercones and minimal canal hypersurfaces are only generalized catenoids. Also, we state the expression of tubular hypersurfaces in Euclidean spaces and give some results about Weingarten tubular hypersurfaces in E4.