Isometric Surfaces and the Third Laplace-Beltrami

TL;DR

This paper investigates classical isometric helicoidal and rotational surfaces, generalizes them via Bour's theorem, and derives the third Laplace-Beltrami operators for these surfaces in three-dimensional Euclidean space.

Contribution

It introduces a generalization of classical surfaces using Bour's theorem and computes the third Laplace-Beltrami operators for these surfaces.

Findings

Derived the third Laplace-Beltrami operators for classical surfaces.

Generalized classical isometric surfaces via Bour's theorem.

Enhanced understanding of geometric properties of these surfaces.

Abstract

In this paper, classical isometric helicoidal and rotational surfaces are studied, and generalized by Bour's theorem in three dimensional Euclidean space. Moreover, the third Laplace-Beltrami operators of two classical surfaces are obtained.