Ruled and Quadric Surfaces Satisfying Δ[sup.II]IN/I = IΛN/I

• Published in: Symmetry (Basel) 2023-01, Vol.15 (2)
• Main Authors: Al-Zoubi, Hassan, Hamadneh, Tareq, Abu Hammad, Ma’mon, Al-Sabbagh, Mutaz, Ozdemir, Mehmet
• Format: Article
• Language: English
• ISSN: 2073-8994; 2073-8994
• DOI: 10.3390/sym15020300

In the 3-dimensional Euclidean space E[sup.3], a quadric surface is either ruled or of one of the following two kinds z[sup.2]=as[sup.2]+bt[sup.2]+c,abc≠0 or z=a/2s[sup.2]+b/2t[sup.2],a>0, b>0. In the present paper, we investigate these three kinds of surfaces whose Gauss map N satisfies the property Δ[sup.II]N=ΛN, where Λ is a square symmetric matrix of order 3, and Δ[sup.II] denotes the Laplace operator of the second fundamental form II 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.