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Journal for Geometry and Graphics 20 (2016), No. 2, 147--157
Copyright Heldermann Verlag 2016

Ruled and Quadric Surfaces Satisfying ΔIIIx = Λ x

Hassan Al-Zoubi
Dept. of Basic Sciences, Al-Zaytoonah University, Amman, Jordan

Stylianos Stamatakis
Dept. of Mathematics, Aristotle University, 54124 Thessaloniki, Greece

We consider ruled and quadric surfaces in the 3-dimensional Euclidean space which are of coordinate finite type with respect to the third fundamental form III, i.e., their position vector x satisfies the relation ΔIIIx = Λ x where Λ is a square matrix of order 3. We show that helicoids and spheres are the only classes of surfaces mentioned above satisfying this equation.

Keywords: Surfaces in Euclidean space, surfaces of coordinate finite type, Beltrami operator.

MSC: 53A05; 47A75

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