On geometry of the first and second fundamental forms of canal surfaces

TL;DR

This paper investigates the geometric properties of canal surfaces in Euclidean 3-space, focusing on conditions for flatness, minimality, and related curvature properties, and classifies degenerate cases and non-existence results.

Contribution

It provides a comprehensive analysis of curvature conditions on canal surfaces, including non-existence theorems and classification of degenerate cases based on radius functions.

Findings

Non-existence of flat, minimal, II-flat, and II-minimal canal surfaces with non-zero curvature center curves.

Classification of degenerate canal surfaces based on their radii.

Identification that only cylinders and cones can have certain curvature properties.

Abstract

In this study, we analyze the general canal surfaces in terms of the features flat, II-flat minimality and II-minimality, namely we study under which conditions the first and second Gauss and mean curvature vanishes, i.e. K=0, H=0, K_{II}=0 and H_{II} =0. We give a non-existence result for general canal surfaces in E^3 with vanishing the curvatures K, H, K_{II} and H_{II} except the cylinder and cone.We classify the general canal surfaces for which are degenerate according to their radiuses. Finally we obtain that there are no flat, minimal, II-flat and II-minimal general canal surfaces in the Euclidean 3-space such that the center curve has non-zero curvatures.