Journal for Geometry and Graphics 18 (2014), No. 2, 217--223
Copyright Heldermann Verlag 2014
Characterizations of Ruled Surfaces in R3 and of Hyperquadrics in Rn+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
Ioanna-Iris Papadopoulou
Dept. of Mathematics, Aristotle University of Thessaloniki, 54124 Thessaloniki, Greece
We consider hypersurfaces in the real Euclidean space Rn+1 (n ≥ 2) which are relatively normalized. We give necessary and sufficient conditions (a) for a surface of negative Gaussian curvature in R3 to be ruled, (b) for a hypersurface of positive Gaussian curvature in Rn+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, Pick-invariant.
MSC: 53A05; 53A07, 53A15, 53A25, 53A40
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