×

Hypersurfaces with constant inner curvature of the second fundamental form, and the non-rigidity of the sphere. (English) Zbl 0869.53003

Summary: We classify the hypersurfaces of revolution in Euclidean space whose second fundamental form defines an abstract pseudo-Riemannian metric of constant sectional curvature. In particular we find such piecewise analytic hypersurfaces of class \(C^2\) where the second fundamental form defines a complete space of constant positive, zero, or negative curvature. Among them are closed convex hypersurfaces distinct from spheres, in contrast to a theorem of R. Schneider [Proc. Am. Math. Soc. 35, 230-233 (1972; Zbl 0251.53043)] saying that such a hypersurface of class \(C^4\) has to be a round sphere. In particular, the sphere is not \(II\)-rigid in the class of all convex \(C^2\)-hypersurfaces.

MSC:

53A07 Higher-dimensional and -codimensional surfaces in Euclidean and related \(n\)-spaces
53C40 Global submanifolds
53A15 Affine differential geometry

Citations:

Zbl 0251.53043
PDFBibTeX XMLCite
Full Text: DOI EuDML

References:

[1] Blair, D.E., Koufogiorgos, Th.: Ruled surfaces with vanishing second Gaussian curvature, Monatsh. Math.113, 177–181 (1992) · Zbl 0765.53003 · doi:10.1007/BF01641765
[2] Blaschke, W., Leichtweiß, K.: Elementare Differentialgeometrie, 5. Aufl., Springer 1973 · Zbl 0264.53001
[3] Cartan, E.: Les surfaces qui admettent une seconde forme fondamentale donnée, Bull. Sci. Math.67, 8–32 (1943) · Zbl 0027.42503
[4] Erard, P.J.: Über die zweite Fundamentalform von Flächen im Raum, Dissertation, ETH Zürich, 1968
[5] Hopf, H.: Über Flächen mit einer Relation zwischen den Haupt-Krümmungen, Math. Nachr.4, 232–249 (1951) · Zbl 0042.15703
[6] Huck, H., Roitzsch, R., Simon, U., Vortisch, W., Walden, R., Wegner, B., Wendland, W.: Beweismethoden der Differentialgeometrie im Großen, Lecture Notes in Mathematics335, Springer 1973
[7] Koufogiorgos, Th., Hasanis, Th.: A characteristic property of the sphere, Proc. AMS67, 303–305 (1977) · Zbl 0379.53030 · doi:10.1090/S0002-9939-1977-0487927-7
[8] Koutroufiotis, D.: Two characteristic properties of the sphere, Proc. AMS44, 176–178 (1974) · Zbl 0283.53002 · doi:10.1090/S0002-9939-1974-0339025-8
[9] Kühnel, W.: Zur inneren Krümmung der zweiten Grundform, Monatsh. f. Math.91, 241–251 (1981) · Zbl 0449.53043 · doi:10.1007/BF01301791
[10] Kühnel, W.: On the inner cuvature of the second fundamental form, Proc. 3rd Congress of Geometry (N.K. Stephanidis, ed.), Thessaloniki, 248–253 (1991)
[11] Kühnel, W.: RuledW-surfaces, Arch. Math.62, 475–480 (1994) · Zbl 0794.53008 · doi:10.1007/BF01196440
[12] Leite, M.L.: Rotational hypersurfaces of space forms with constant scalar curvature, manuscr. math.67, 285–304 (1990) · Zbl 0695.53040 · doi:10.1007/BF02568434
[13] O’Neill, B.: Semi-Riemannian Geometry, Academic Press 1983
[14] Schneider, R.: Zur affinen Differentialgeometrie im Großen. I, Math. Z.101, 375–406 (1967) · Zbl 0156.20101 · doi:10.1007/BF01109803
[15] Schneider, R.: Closed convex hypersurfaces with second fundamental form of constant curvature, Proc. AMS35, 230–233 (1972) · Zbl 0222.53047 · doi:10.1090/S0002-9939-1972-0307133-1
[16] Simon, U.: Characterizations of the sphere by the curvature of the second fundamental form, Proc. AMS55, 282–284 (1976)
[17] Spivak, M.: A Comprehensive Introduction to Differential Geometry, Publish or Perish, 1970–75
[18] Stamou, G.: Global characterizations of the sphere, Proc. AMS68, 328–330 (1978) · Zbl 0379.53029 · doi:10.1090/S0002-9939-1978-0467620-8
[19] Struik, D.: Lectures on Classical Differential Geometry, Dover 1988 · Zbl 0697.53002
[20] Voss, K.: Isometrie von Flächen bezüglich der zweiten Fundamentalform, Nachr. Österr. Math. Ges., Sonderheft Nr.91, 78 (1970)
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.