Blair, D. E.; Koufogiorgos, Th. Ruled surfaces with vanishing second Gaussian curvature. (English) Zbl 0765.53003 Monatsh. Math. 113, No. 3, 177-181 (1992). For a surface free of points of vanishing Gaussian curvature in Euclidean space the second Gaussian curvature is defined formally. It is first pointed out that a minimal surface has vanishing second Gaussian curvature but that a surface with vanishing second Gaussian curvature need not be minimal. Ruled surfaces for which a linear combination of the second Gaussian curvature and the mean curvature is constant along the rulings are then studied. In particular the only ruled surface in Euclidean space with vanishing second Gaussian curvature is a piece of a helicoid. Reviewer: D.E.Blair (East Lansing) Cited in 2 ReviewsCited in 12 Documents MSC: 53A05 Surfaces in Euclidean and related spaces Keywords:ruled surfaces; second Gaussian curvature; minimal surface; mean curvature PDFBibTeX XMLCite \textit{D. E. Blair} and \textit{Th. Koufogiorgos}, Monatsh. Math. 113, No. 3, 177--181 (1992; Zbl 0765.53003) Full Text: DOI EuDML References: [1] Berger, M., Gostiaux, B.: Differential Geometry: Manifolds, Curves and Surfaces. Berlin-Heidelberg-New York: Springer. 1988. · Zbl 0629.53001 [2] Glässner, E.: Über die Minimalflächen der zweiten Fundamentalform. M. Math.78, 193–214 (1974). · Zbl 0284.53006 · doi:10.1007/BF01297274 [3] Koufogiorgos, Th., Hasanis, T.: A characteristic property of the sphere. Proc. Am. Math. Soc.67, 303–305 (1977). · Zbl 0379.53030 · doi:10.1090/S0002-9939-1977-0487927-7 [4] Struik, D. J.: Differential Geometry. Reading, MA: Addison-Wesley. 1961. · Zbl 0105.14707 This reference list is based on information provided by the publisher or from digital mathematics libraries. Its items are heuristically matched to zbMATH identifiers and may contain data conversion errors. In some cases that data have been complemented/enhanced by data from zbMATH Open. This attempts to reflect the references listed in the original paper as accurately as possible without claiming completeness or a perfect matching.