
Journal for Geometry and Graphics 18 (2014), No. 2, 217223 Copyright Heldermann Verlag 2014 Characterizations of Ruled Surfaces in R^{3} and of Hyperquadrics in R^{n+1} via Relative Geometric Invariants Stylianos Stamatakis Dept. of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece stamata@math.auth.gr Ioannis Kaffas Dept. of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece IoannaIris Papadopoulou Dept. of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece We consider hypersurfaces in the real Euclidean space R^{n+1} (n ≥ 2) which are relatively normalized. We give necessary and sufficient conditions (a) for a surface of negative Gaussian curvature in R^{3} to be ruled, (b) for a hypersurface of positive Gaussian curvature in R^{n+1} to be a hyperquadric and (c) for a relative normalization to be constantly proportional to the equiaffine normalization. Keywords: Ruled surfaces, ovaloids, hyperquadrics, equiaffine normalization, Pickinvariant. MSC: 53A05; 53A07, 53A15, 53A25, 53A40 [ Fulltextpdf (107 KB)] for subscribers only. 