
Journal for Geometry and Graphics 20 (2016), No. 2, 147157 Copyright Heldermann Verlag 2016 Ruled and Quadric Surfaces Satisfying Δ^{III}x = Λ x Hassan AlZoubi Dept. of Basic Sciences, AlZaytoonah University, Amman, Jordan dr.hassanz@zuj.edu.jo Stylianos Stamatakis Dept. of Mathematics, Aristotle University, 54124 Thessaloniki, Greece stamata@math.auth.gr We consider ruled and quadric surfaces in the 3dimensional 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 Δ^{III}x = 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 [ Fulltextpdf (158 KB)] for subscribers only. 