The Fourth Fundamental Form IV of Dini-Type Helicoidal Hypersurface in the Four Dimensional Euclidean Space

Erhan Güler

doi:10.3390/axioms10030186

Abstract

We introduce the fourth fundamental form of a Dini-type helicoidal hypersurface in the four dimensional Euclidean space E4. We find the Gauss map of helicoidal hypersurface in E4. We obtain the characteristic polynomial of shape operator matrix. Then, we compute the fourth fundamental form matrix IV of the Dini-type helicoidal hypersurface. Moreover, we obtain the Dini-type rotational hypersurface, and reveal its differential geometric objects.

Highlights

Rotational and helicoidal hyper-surfaces have attracted the attention of scientists such as architects, biologists, physicists, mathematicians, and especially geometers for almost 300 years.Let us review some works about rotational and helicoidal characters in chronological order.Catenoid is a minimal rotational surface described by Euler [1] in 1744
Choi and Kim [21] characterized the helicoid as a ruled surface with pointwise 1-type Gauss map
Dursun and Turgay [31] worked on general rotational surfaces with pointwise 1-type Gauss map in E4

Summary

Introduction

Rotational and helicoidal hyper-surfaces have attracted the attention of scientists such as architects, biologists, physicists, mathematicians, and especially geometers for almost. Choi and Kim [21] characterized the helicoid as a ruled surface with pointwise 1-type Gauss map. Yoon [22] studied rotational surfaces with finite type Gauss map in E4. Ji and Kim [28] introduced mean curvatures and Gauss maps of a pair of isometric helicoidal and rotation surfaces in Minkowski 3-space. Dursun and Turgay [31] worked on general rotational surfaces with pointwise 1-type Gauss map in E4. Bulca, and Kosova [49] studied generalized rotational surfaces in Euclidean spaces. Hacısalihoğlu, and Kim [51] gave the Gauss map and the third Laplace–Beltrami operator of the rotational hypersurface in 4-space. Milousheva, and Turgay worked on [52] general rotational surfaces in pseudo-Euclidean 4-space with neutral metrics. We provide a conclusion in the last section

Preliminaries

Helicoidal Hypersurface

Dini-Type Helicoidal Hypersurface and the Fourth Fundamental Form

Conclusions