Characterization of quadric surfaces in terms of coordinate finite type Gauss map

In this article, we introduce an important class of surfaces, namely, quadrics in the Euclidean 3-space \(\mathbb{E}^{3}\). We prove that planes, spheres and circular cylinders are the only quadric surfaces whose Gauss map \(\boldsymbol{G}\) satisfies a relation of the form \(\Delta^{I}\boldsymbol{G...

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Bibliographic Details
Published in:arXiv.org 2023-12
Main Authors: Al-Sabbagh, Mutaz, Al-Zoubi, Hassan
Format: Article
Language:English
Subjects:
Euclidean geometry
Euclidean space
Online Access:Get full text
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Summary:In this article, we introduce an important class of surfaces, namely, quadrics in the Euclidean 3-space \(\mathbb{E}^{3}\). We prove that planes, spheres and circular cylinders are the only quadric surfaces whose Gauss map \(\boldsymbol{G}\) satisfies a relation of the form \(\Delta^{I}\boldsymbol{G}= M \boldsymbol{G}\), where \(M\) is a square matrix of order 3 and \(\Delta^{I}\) is the Laplace-Beltrami operator corresponding to the first fundamental form \(I\) of the surface.
ISSN:2331-8422