Remote Access Proceedings of the American Mathematical Society
Green Open Access

Proceedings of the American Mathematical Society

ISSN 1088-6826(online) ISSN 0002-9939(print)

 
 

 

Closed convex hypersurfaces with second fundamental form of constant curvature


Author: Rolf Schneider
Journal: Proc. Amer. Math. Soc. 35 (1972), 230-233
MSC: Primary 53C45
DOI: https://doi.org/10.1090/S0002-9939-1972-0307133-1
MathSciNet review: 0307133
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: It is shown that the Euclidean spheres are the only closed hypersurfaces in Euclidean space on which the second fundamental form defines a (nondegenerate) Riemannian metric of constant curvature.


References [Enhancements On Off] (What's this?)

  • [1] E. Cartan, Les surfaces qui admettent une seconde forme fondamentale donnée, Bull. Sci. Math (2) 67 (1943), 8-32. MR 7, 30. MR 0012495 (7:30c)
  • [2] L. P. Eisenhart, Riemannian geometry, 4th ed., Princeton Univ. Press, Princeton, N.J., 1960. MR 0035081 (11:687g)
  • [3] P. J. Erard, Über die zweite Fundamentalform von Flächen im Raum, Doctoral Thesis, ETH, Zürich, 1968. MR 0288670 (44:5866)
  • [4] R. B. Gardner, Subscalar pairs of metrics with applications to rigidity and uniqueness of hypersurfaces with a non-degenerate second fundamental form (to appear).
  • [5] R. Schneider, Zur affinen Differentialgeometrie im Grossen. I, Math. Z. 101 (1967), 375-406. MR 36 #3255. MR 0220189 (36:3255)
  • [6] U. Simon and A. Weinstein, Anwendungen der de Rhamschen Zerlegung auf Probleme der lokalen Flächentheorie, Manuscripta Math. 1 (1969), 139-146. MR 39 #7538. MR 0246234 (39:7538)
  • [7] U. Simon, II-Verbiegungen von Eiflächen, Arch. Math. 22 (1971), 319-324. MR 0293552 (45:2629)
  • [8] K. Voss, Isometrie von Flächen bezüglich der zweiten Fundamentalform, Nachr. Österr. Math. Ges. 91 (1970), 73.
  • [9] R. Walden, Eindeutigkeitssätze für II-isometrische Eiflächen, Math. Z. 120 (1971), 143-147. MR 0284958 (44:2182)

Similar Articles

Retrieve articles in Proceedings of the American Mathematical Society with MSC: 53C45

Retrieve articles in all journals with MSC: 53C45


Additional Information

DOI: https://doi.org/10.1090/S0002-9939-1972-0307133-1
Keywords: Second fundamental form, closed convex hypersurface, Riemannian metric of constant curvature, scalar curvature, Euclidean sphere
Article copyright: © Copyright 1972 American Mathematical Society

American Mathematical Society