Remote Access Transactions of the American Mathematical Society
Green Open Access

Transactions of the American Mathematical Society

ISSN 1088-6850(online) ISSN 0002-9947(print)

Request Permissions   Purchase Content 


Relative isometric embeddings of Riemannian manifolds

Authors: Mohammad Ghomi and Robert E. Greene
Journal: Trans. Amer. Math. Soc. 363 (2011), 63-73
MSC (2010): Primary 53C42, 53A07
Published electronically: August 16, 2010
MathSciNet review: 2719671
Full-text PDF

Abstract | References | Similar Articles | Additional Information

Abstract: We prove the existence of $ C^1$ isometric embeddings, and $ C^\infty$ approximate isometric embeddings, of Riemannian manifolds into Euclidean space with prescribed values in a neighborhood of a point.

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

  • 1. S. Alexander, M. Ghomi, and J. Wong.
    Topology of Riemannian submanifolds with prescribed boundary, Duke Math. J., 152 (2010), 533-565.
  • 2. Ben Andrews, Notes on the isometric embedding problem and the Nash-Moser implicit function theorem, Surveys in analysis and operator theory (Canberra, 2001) Proc. Centre Math. Appl. Austral. Nat. Univ., vol. 40, Austral. Nat. Univ., Canberra, 2002, pp. 157–208. MR 1953483
  • 3. Y. Eliashberg and N. Mishachev, Introduction to the ℎ-principle, Graduate Studies in Mathematics, vol. 48, American Mathematical Society, Providence, RI, 2002. MR 1909245
  • 4. Mohammad Ghomi, The problem of optimal smoothing for convex functions, Proc. Amer. Math. Soc. 130 (2002), no. 8, 2255–2259 (electronic). MR 1896406, 10.1090/S0002-9939-02-06743-6
  • 5. Robert E. Greene, Isometric embeddings of Riemannian and pseudo-Riemannian manifolds., Memoirs of the American Mathematical Society, No. 97, American Mathematical Society, Providence, R.I., 1970. MR 0262980
  • 6. Mikhael Gromov, Partial differential relations, Ergebnisse der Mathematik und ihrer Grenzgebiete (3) [Results in Mathematics and Related Areas (3)], vol. 9, Springer-Verlag, Berlin, 1986. MR 864505
  • 7. Matthias Günther, Isometric embeddings of Riemannian manifolds, Proceedings of the International Congress of Mathematicians, Vol. I, II (Kyoto, 1990) Math. Soc. Japan, Tokyo, 1991, pp. 1137–1143. MR 1159298
  • 8. Matthias Günther, On the perturbation problem associated to isometric embeddings of Riemannian manifolds, Ann. Global Anal. Geom. 7 (1989), no. 1, 69–77. MR 1029846, 10.1007/BF00137403
  • 9. Qing Han and Jia-Xing Hong, Isometric embedding of Riemannian manifolds in Euclidean spaces, Mathematical Surveys and Monographs, vol. 130, American Mathematical Society, Providence, RI, 2006. MR 2261749
  • 10. Noel J. Hicks, Notes on differential geometry, Van Nostrand Mathematical Studies, No. 3, D. Van Nostrand Co., Inc., Princeton, N.J.-Toronto-London, 1965. MR 0179691
  • 11. H. Jacobowitz, Extending isometric embeddings, J. Differential Geometry 9 (1974), 291–307. MR 0377773
  • 12. Erwin Kreyszig, Differential geometry, Dover Publications, Inc., New York, 1991. Reprint of the 1963 edition. MR 1118149
  • 13. Nicolaas H. Kuiper, On 𝐶¹-isometric imbeddings. I, II, Nederl. Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math. 17 (1955), 545–556, 683–689. MR 0075640
  • 14. John Nash, 𝐶¹ isometric imbeddings, Ann. of Math. (2) 60 (1954), 383–396. MR 0065993
  • 15. John Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. (2) 63 (1956), 20–63. MR 0075639
  • 16. Barrett O’Neill, Elementary differential geometry, 2nd ed., Elsevier/Academic Press, Amsterdam, 2006. MR 2351345
  • 17. A. V. Pogorelov, Extrinsic geometry of convex surfaces, American Mathematical Society, Providence, R.I., 1973. Translated from the Russian by Israel Program for Scientific Translations; Translations of Mathematical Monographs, Vol. 35. MR 0346714
  • 18. I. Kh. Sabitov, Local theory of bendings of surfaces [ MR1039820 (91c:53004)], Geometry, III, Encyclopaedia Math. Sci., vol. 48, Springer, Berlin, 1992, pp. 179–256. MR 1306736, 10.1007/978-3-662-02751-6_3
  • 19. Michael Spivak, A comprehensive introduction to differential geometry. Vol. V, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. MR 532834

Similar Articles

Retrieve articles in Transactions of the American Mathematical Society with MSC (2010): 53C42, 53A07

Retrieve articles in all journals with MSC (2010): 53C42, 53A07

Additional Information

Mohammad Ghomi
Affiliation: School of Mathematics, Georgia Institute of Technology, Atlanta, Georgia 30332

Robert E. Greene
Affiliation: Department of Mathematics, University of California, Los Angeles, Los Angeles, California 90095

Keywords: Relative isometric embedding, Riemannian manifold, short map
Received by editor(s): April 28, 2008
Published electronically: August 16, 2010
Additional Notes: The first-named author was supported by NSF Grant DMS-0336455 and CAREER award DMS-0332333.
Article copyright: © Copyright 2010 American Mathematical Society
The copyright for this article reverts to public domain 28 years after publication.