Relative isometric embeddings of Riemannian manifolds

Ghomi, Mohammad; Greene, Robert

doi:10.1090/S0002-9947-2010-05095-0

Relative isometric embeddings of Riemannian manifolds
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by Mohammad Ghomi and Robert E. Greene

Trans. Amer. Math. Soc. 363 (2011), 63-73

DOI: https://doi.org/10.1090/S0002-9947-2010-05095-0

Published electronically: August 16, 2010

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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

S. Alexander, M. Ghomi, and J. Wong. Topology of Riemannian submanifolds with prescribed boundary, Duke Math. J., 152 (2010), 533–565.
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
Y. Eliashberg and N. Mishachev, Introduction to the $h$-principle, Graduate Studies in Mathematics, vol. 48, American Mathematical Society, Providence, RI, 2002. MR 1909245, DOI 10.1090/gsm/048
Mohammad Ghomi, The problem of optimal smoothing for convex functions, Proc. Amer. Math. Soc. 130 (2002), no. 8, 2255–2259. MR 1896406, DOI 10.1090/S0002-9939-02-06743-6
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
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, DOI 10.1007/978-3-662-02267-2
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
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, DOI 10.1007/BF00137403
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, DOI 10.1090/surv/130
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
H. Jacobowitz, Extending isometric embeddings, J. Differential Geometry 9 (1974), 291–307. MR 377773
Erwin Kreyszig, Differential geometry, Dover Publications, Inc., New York, 1991. Reprint of the 1963 edition. MR 1118149
Nicolaas H. Kuiper, On $C^1$-isometric imbeddings. I, II, Nederl. Akad. Wetensch. Proc. Ser. A. 58 = Indag. Math. 17 (1955), 545–556, 683–689. MR 0075640
John Nash, $C^1$ isometric imbeddings, Ann. of Math. (2) 60 (1954), 383–396. MR 65993, DOI 10.2307/1969840
John Nash, The imbedding problem for Riemannian manifolds, Ann. of Math. (2) 63 (1956), 20–63. MR 75639, DOI 10.2307/1969989
Barrett O’Neill, Elementary differential geometry, 2nd ed., Elsevier/Academic Press, Amsterdam, 2006. MR 2351345
A. V. Pogorelov, Extrinsic geometry of convex surfaces, Translations of Mathematical Monographs, Vol. 35, American Mathematical Society, Providence, R.I., 1973. Translated from the Russian by Israel Program for Scientific Translations. MR 0346714
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, DOI 10.1007/978-3-662-02751-6_{3}
Michael Spivak, A comprehensive introduction to differential geometry. Vol. V, 2nd ed., Publish or Perish, Inc., Wilmington, Del., 1979. MR 532834