Ruled and Quadric Surfaces Satisfying ΔIIN = ΛN

Al-Zoubi, Hassan; Hamadneh, Tareq; Abu Hammad, Ma’mon; Al-Sabbagh, Mutaz; Ozdemir, Mehmet

doi:10.3390/sym15020300

Open AccessArticle

Ruled and Quadric Surfaces Satisfying Δ^IIN = ΛN

by

Hassan Al-Zoubi

^1,*

,

Tareq Hamadneh

¹,

Ma’mon Abu Hammad

¹,

Mutaz Al-Sabbagh

² and

Mehmet Ozdemir

²

¹

Department of Mathematics, Al-Zaytoonah University of Jordan, P.O. Box 130, Amman 11733, Jordan

²

Department of Basic Engineering Sciences, Imam Abdulrahman Bin Faisal University, Dammam 31441, Saudi Arabia

^*

Author to whom correspondence should be addressed.

Symmetry 2023, 15(2), 300; https://doi.org/10.3390/sym15020300

Submission received: 26 December 2022 / Revised: 14 January 2023 / Accepted: 16 January 2023 / Published: 21 January 2023

(This article belongs to the Section B: Mathematics)

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Abstract

In the 3-dimensional Euclidean space

E^{3}

, a quadric surface is either ruled or of one of the following two kinds

z^{2} = a s^{2} + b t^{2} + c, a b c \neq 0

or

z = \frac{a}{2} s^{2} + \frac{b}{2} t^{2}, a > 0, b > 0

. In the present paper, we investigate these three kinds of surfaces whose Gauss map

N

satisfies the property

Δ^{I I} N = Λ N

, where

Λ

is a square symmetric matrix of order 3, and

Δ^{I I}

denotes the Laplace operator of the second fundamental form

I I

of the surface. We prove that spheres with the nonzero symmetric matrix

Λ

, and helicoids with

Λ

as the corresponding zero matrix, are the only classes of surfaces satisfying the above given property.

Keywords: ruled surfaces; quadric surfaces; surfaces of coordinate finite type in Euclidean 3-space; Laplace operator

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MDPI and ACS Style

Al-Zoubi, H.; Hamadneh, T.; Abu Hammad, M.; Al-Sabbagh, M.; Ozdemir, M. Ruled and Quadric Surfaces Satisfying Δ^IIN = ΛN. Symmetry 2023, 15, 300. https://doi.org/10.3390/sym15020300

AMA Style

Al-Zoubi H, Hamadneh T, Abu Hammad M, Al-Sabbagh M, Ozdemir M. Ruled and Quadric Surfaces Satisfying Δ^IIN = ΛN. Symmetry. 2023; 15(2):300. https://doi.org/10.3390/sym15020300

Chicago/Turabian Style

Al-Zoubi, Hassan, Tareq Hamadneh, Ma’mon Abu Hammad, Mutaz Al-Sabbagh, and Mehmet Ozdemir. 2023. "Ruled and Quadric Surfaces Satisfying Δ^IIN = ΛN" Symmetry 15, no. 2: 300. https://doi.org/10.3390/sym15020300

APA Style

Al-Zoubi, H., Hamadneh, T., Abu Hammad, M., Al-Sabbagh, M., & Ozdemir, M. (2023). Ruled and Quadric Surfaces Satisfying Δ^IIN = ΛN. Symmetry, 15(2), 300. https://doi.org/10.3390/sym15020300

Note that from the first issue of 2016, this journal uses article numbers instead of page numbers. See further details here.

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Ruled and Quadric Surfaces Satisfying Δ^IIN = ΛN

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