Improving Pogorelov’s isometric embedding counterexample



We construct a C2,1 metric of non-negative Gauss curvature with no C2 local isometric embedding in \({\mathbb{R}}^{3}.\)


Unable to display preview. Download preview PDF.

Unable to display preview. Download preview PDF.


  1. 1.
    Han Q. (2005). On the isometric embedding of surfaces with Gauss curvature changing sign cleanly. Commun. Pure Appl. Math. 58(2): 285–295 MATHCrossRefGoogle Scholar
  2. 2.
    Han Q. and Hong J.-X. (2006). Isometric Embedding of Riemannian Manifolds in Euclidean spaces. Mathematical Surveys and Monographs, vol. 130. American Mathematical Society, Providence Google Scholar
  3. 3.
    Han Q., Hong J.-X. and Lin C.-S. (2003). Local isometric embedding of surfaces with non-positive Gaussian curvature. J. Differ. Geom. 63(3): 475–520 MATHMathSciNetGoogle Scholar
  4. 4.
    Jacobowitz H. (1971). Local isometric embeddings of surfaces into Euclidean four space. Indiana Univ. Math. J. 21: 249–254 MATHCrossRefMathSciNetGoogle Scholar
  5. 5.
    Lin C.-S. (1985). The local isometric embedding in \({\mathbb{R}}^{3}\) of 2-dimensional Riemannian manifolds with non- negative curvature J. Differ. Geom. 21(2): 213–230 MATHGoogle Scholar
  6. 6.
    Lin C.-S. (1986). The local isometric embedding in \({\mathbb{R}}^{3}\) of two-dimensional Riemannian manifolds with Gaussian curvature changing sign cleanly Commun. Pure Appl. Math. 39(6): 867–887 MATHCrossRefGoogle Scholar
  7. 7.
    Nakamura G. and Maeda Y. (1989). Local smooth isometric embeddings of low-dimensional Riemannian manifolds into Euclidean spaces. Trans. Am. Math. Soc. 313(1): 1–51 MATHCrossRefMathSciNetGoogle Scholar
  8. 8.
    Pogorelov, A.V.: An example of a two-dimensional Riemannian metric admitting no local realization in E3. Dokl. Akad. Nauk SSSR Tom 198(1), 42–43 (1971); English translation in Soviet Math. Dokl. 12, 729–730 (1971)Google Scholar

Copyright information

© Springer-Verlag 2008

Authors and Affiliations

  1. 1.LATPCentre de Mathématiques et InformatiqueMarseille CedexFrance
  2. 2.Department of MathematicsUniversity of WashingtonSeattleUSA

Personalised recommendations